Various Multinational Corporations minimize their effective global tax rate, and hence their contribution to public services, through Corporate Tax Avoidance. Taxpaying citizens, however, cannot reap these benefits of country-specific legislation under the international tax system, and frequently carry the majority of the tax burden. Hence, corporations are subject to accusations of not paying a "fair share". Based on equity theory, our paper analyses citizens' perception of fairness in regard to corporate taxation. By executing a mediation analysis, we determine which corporate tax rate is perceived as fair, mediating the relationship between equity theory determinants (individuals' tax system satisfaction, a social comparison with other entities, and cultural value-based cognition) and possible system-supportive or detrimental consequences. We confirm that a perception of inequity is prevalent among the 218 participants in our survey, and "fair burden-sharing" is perceived to be non-existent. We contribute to theory by classifying the social comparison determinant as most relevant for the fairness perceptions among individuals towards questionable business practices. Moreover, we emphasize that CTA needs to be considered a possible legitimacy threat for societal and institutional functioning since it may increase citizens' tax avoidant behavior, and jeopardizes social cohesion. However, the cultural values of power distance and masculinity were found to mitigate these generally detrimental consequences of CTA. Our practical and institutional implications put great emphasis on further promoting fairness within the international tax system since the recently suggested global corporate tax rate of 15% is still not considered as fair by our survey participants.
Citation: Marcus Conrad. Do multinational corporations pay their 'Fair Share'?[J]. Green Finance, 2022, 4(1): 88-114. doi: 10.3934/GF.2022005
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Various Multinational Corporations minimize their effective global tax rate, and hence their contribution to public services, through Corporate Tax Avoidance. Taxpaying citizens, however, cannot reap these benefits of country-specific legislation under the international tax system, and frequently carry the majority of the tax burden. Hence, corporations are subject to accusations of not paying a "fair share". Based on equity theory, our paper analyses citizens' perception of fairness in regard to corporate taxation. By executing a mediation analysis, we determine which corporate tax rate is perceived as fair, mediating the relationship between equity theory determinants (individuals' tax system satisfaction, a social comparison with other entities, and cultural value-based cognition) and possible system-supportive or detrimental consequences. We confirm that a perception of inequity is prevalent among the 218 participants in our survey, and "fair burden-sharing" is perceived to be non-existent. We contribute to theory by classifying the social comparison determinant as most relevant for the fairness perceptions among individuals towards questionable business practices. Moreover, we emphasize that CTA needs to be considered a possible legitimacy threat for societal and institutional functioning since it may increase citizens' tax avoidant behavior, and jeopardizes social cohesion. However, the cultural values of power distance and masculinity were found to mitigate these generally detrimental consequences of CTA. Our practical and institutional implications put great emphasis on further promoting fairness within the international tax system since the recently suggested global corporate tax rate of 15% is still not considered as fair by our survey participants.
The bipolar Navier-Stokes-Poisson system has been used to simulate the transport of charged particles under the influence of electrostatic force governed by the self-consistent Poisson. In this paper, we are concerned with the Cauchy problem of the bipolar Navier-Stokes-Poisson system in 3 dimensions:
{∂tρ1+div(ρ1u1)=0,∂tu1+div(ρ1u1⊗u1)+∇P1(ρ1)=μ1Δu1+μ2∇divu1+ρ1∇Φ,∂tρ2+div(ρ2u2)=0,∂tu2+div(ρ2u2⊗u2)+∇P2(ρ2)=μ1Δu2+μ2∇divu2−ρ2∇Φ,ΔΦ=ρ1−ρ2,lim|x|→∞Φ(x,t)=0, | (1.1) |
with initial data
(ρ1,u1,ρ2,u2,∇Φ)(x,0)=(ρ1,0,u1,0,ρ2,0,u2,0,∇Φ0)(x),x∈R3. | (1.2) |
Here ρi(x,t),ui(x,t),Φ(x,t), and Pi(ρ)((x,t) represent the fluid density, velocity, self-consistent electric potential and pressure. The viscosity coefficients satisfy the usual physical conditions μ1>0, 3μ1+2μ2>0. We assume that Pi(ρ) satisfies P′i(ρ)>0 for all ρ>0 and P′i(ˉρ)=1, where ¯ρ>0 denotes the prescribed density of positive charged background ions, and in this paper is taken as a positive constant. Without loss of generality, we take ¯ρ to be 1. For the initial data (ρ1,0,u1,0,ρ2,0,u2,0), we consider small perturbations of (¯ρ,0,¯ρ,0), in which ¯ρ is defined as before and taken to be 1, and we assume that ρ1,0,ρ2,0 has positive lower bound and upper bound.
Now, we review some previous works on the Cauchy problem for some related models. There has been a lot of studies for the compressible Navier-Stokes system (CNS) for either isentropic or non-isentropic cases on the existence, stability, and Lp-decay rates with p≥2. For the results of small solutions, see [1,2] and [3,4,5,6,7], where the authors use the (weighted) energy method together with spectrum analysis, and for the results of partial small solutions (under the setting that the initial data is of small energy but possibly large oscillations), see [8,9] and the references therein. On the other hand, many scholars also use the method of Green's function to analyze the asymptotic behavior of a specific system, for example, by using the method of Green's function, Liu and Zeng [10] first studied the point-wise estimates of solutions to the general hyperbolic-parabolic equations in one dimension. Later, Liu and Wang [11] give the point-wise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimension and explain the generalized Huygens' principle for the Navier-Stokes systems.
For the unipolar Navier-Stokes-Poisson system (NSP), there is also a mass of results for the Cauchy problem when the initial data (ρ0,u0) is a small perturbation around the constant state (ˉρ,0). For instance, the global existence of weak solutions was obtained by [12,13]; the framework of Matsumura and Nishda [14,15] shows the global existence of small strong solutions in HN Sobolev spaces. In [16,17], the authors obtain the global existence of small solutions in some Besov spaces. For the solutions which are of small energy but possibly large oscillations, see [18,19] and the references therein. In fact, the NSP system is a hyperbolic-parabolic system with a nonlocal term arising from the electric field ∇Φ. From the analysis of Green's function, the symbol of this nonlocal term is singular in the long wave of the Green's function and it destroys the time-decay rate for the velocity. As we know, for the CNS system, when the initial perturbation ρ0−1,u0∈Lp∩HN, with p near 1, and N≥3 is a large enough integer for the nonlinear system, then the solutions have L2 optimal decay rate
‖(ρ−1,u)(t)‖L2≤C(1+t)−32(1p−12). |
In [15,20,21], the authors survey the decay rate of solutions for the NSP system and they observed that the electric field destroys the decay rate of the solutions, i.e., when the initial perturbation ρ0−1,u0∈Lp∩HN, with p∈[1,2], then the solutions have L2 optimal decay rate
‖(ρ−1)(t)‖L2≤C(1+t)−32(1p−12), ‖u(t)‖L2≤C(1+t)−32(1p−12)+12. |
In [22], the author gives another comprehension toward the effect of the electric field on the decay rate of the solutions for the NSP system. The author believes that it is natural to assume that ∇Φ0∈L2, and with this condition in hand, we can obtain the L2 optimal decay rate for the linear NSP system as follows:
‖(ρ−1)(t)‖L2≤C(1+t)−32(1p−12)−12, ‖u(t)‖L2≤C(1+t)−32(1p−12). |
In this sense, we see that the electric force enhances the decay rate of the density with the factor 12 compared with the CNS system.
For the bipolar Navier-Stokes-Poisson system (BNSP), Wang and Xu [23] obtain the global existence of small solutions and the decay rate of the solutions. [24,25,26] observed the large time behavior of the BNSP system. The global existence of the solutions for the BNSP system under the partial smallness of the initial value is still an open problem, and the aim of this paper is to obtain the global existence of the solution to the system (1.1) and (1.2) provided that the initial data is partially small, which means that we require the initial value itself to be small and its derivative only to be bounded. In other words, its initial value is large in some classes, that is, its derivative can be large except for the initial value itself. In this paper, we first establish a regularity criterion to obtain the uniform boundedness of the solutions, and then combine abstract bootstrap argument to extend the local solutions. In order to prove the global existence of solution under the condition of initial data is partially small, the general regularity hypothesis requires that the solution is bounded with respect to time in the sense of some Sobolev norm, and another important condition is that the solution is integrable with respect to the time variable, which is necessary to obtain a consistent estimate of the solution through Gronwall's inequality.
However, in this paper, it is difficult to obtain the regularity condition of integrability due to the more complex nonlinear structure in BNSP equations. The classical regularity criterion is established based on the energy method and the decay property in time variable is usually captured by integrability. It is always difficult and not applicable for the bootstrap argument. In fact, in [27] for the global existence of solutions for shallow water equations with partially large initial data, we applied Green's function method and replaced the integrability condition by detailed decaying structure in time variable, which makes the bootstrap argument more clear and concise.
The linear and nonlinear structures of BNSP equations discussed in this paper are far more complicated, and we can imagine that it could be quite difficult and complex for the energy method. Even in the construction of Green's function, compared with the previous case for shallow water equations, BNSP equations are hyperbolic-parabolic-elliptic coupled ones and the structure is very complicated. The elliptic structure provides a nonlocal operator and also causes the lack of symmetry. These are all troubles we need to overcome in this paper.
Before we list the main result, we introduce some notations. Throughout this paper, ∂t stands for the derivative with respect to time variable and ft=∂tf. The symbol ∂if(i=1,2,3) means partial derivative with respect to xi,
∂if=∂f∂xi. |
We use the notation ∇kf to mean the partial derivative of order k. That is, if k is a nonnegative integer, then
∇kf:={∇αf|α=(α1,α2,α3),|α|=k} |
is a set of all partial derivatives of order k, endowed with the norm
‖∇kf‖2L2=∑|α|=k‖∇αf‖2L2, |
where
∇αf:=∂α1x1∂α2x2∂α3x3f, αi≥0. |
Let Λ be a quasi-differential operator defined as follows:
Λα=(−Δ)α2. |
The main result of this paper is the following:
Theorem 1.1. Let (ρ1,0−1,u1,0,ρ2,0−1,u2,0,∇Φ0)∈Hs+1(s≥4), and
‖(ρ1,0−1,u1,0,ρ2,0−1,u2,0,∇Φ0)‖L2(R3)⋂L1(R3)≤E0, |
where E0 is sufficiently small, then the Cauchy problem (1.1) and (1.2) has a global solution in time that satisfies
(ρ1−1,u1,ρ2−1,u2)∈L∞([0,∞);Hs+1), ∇(u1,u2)∈L2([0,∞);Hs+1), |
For 2≤p≤+∞ and α=(α1,α2,α3), |α|≤s−1, it holds
‖Dα(ρ1−1,u1,ρ2−1,u2)(⋅,t)‖Lp≤C(1+t)−32(1−1p)−|α|2. |
Remark 1.1. In Theorem 1.1, we only assume the norm of the initial datum ρ1,0,ρ2,0,u1,0,u2,0 and ∇Φ0 are small enough, but for the derivatives of ρ1,0,ρ2,0,u1,0,u2,0, and ∇Φ0, we assume that they are bounded.
The rest of this paper is organized as follows. In Section 2, we establish the uniform time estimate of solutions. In Section 3, we analyze the Green's function of the linear BNSP system, and the different properties of the Green function at high and low frequencies are obtained. In Section 4, we complete the partial proof of Theorem 1.1, i.e., we mainly obtain the existence of the solutions. In this paper, we also obtain the decay rate of the solutions, and the reader can see Sections 4 and 5 for details.
Throughout this paper, we denote by C a positive constant that varies from line to line.
We reformulate the Cauchy problem (1.1) and (1.2) about constant state (1,0,1,0) as follows:
{∂tρ1+divu1=−div(ρ1u1),∂tu1−μ1Δu1−μ2∇divu1+∇ρ1−∇Φ=−u1⋅∇u1−(P′1(1+ρ1)1+ρ1−1)∇ρ1−ρ11+ρ1(μ1Δu1+μ2∇divu1),∂tρ2+divu2=−div(ρ2u2),∂tu2−μ1Δu2−μ2∇divu2+∇ρ2+∇Φ=−u2⋅∇u2−(P′2(1+ρ2)1+ρ2−1)∇ρ2−ρ21+ρ2(μ1Δu2+μ2∇divu2),ΔΦ=ρ1−ρ2,lim|x|→∞Φ(x,t)=0, | (2.1) |
where we also note (ρ1−1,u1,ρ2−1,u2) as (ρ1,u1,ρ2,u2) without causing confusion. In this section, we will get the estimates of the solutions for the system (2.1) under the assumption that for any fixed 0<T<+∞, t∈[0,T],
(‖∇ρ1(t)‖L∞(R3),‖∇u1(t)‖L∞(R3),‖∇ρ2(t)‖L∞(R3),‖∇u2(t)‖L∞(R3))≤C(1+t)−2, | (2.2) |
and
‖(ρ1,0,ρ2,0,u1,0,u2,0,∇Φ0)‖L2⋂L1≤E0, | (2.3) |
where E0 is a positive constant which is sufficiently small. Similar to reference [28,29], we call Eq (2.2) the regularity criterion. In this section, based on this regularity criterion, we get a consistent estimate of the solutions of the equations and its derivatives.
We first define
F1(ρ)=P′1(1+ρ)1+ρ−1, F2(ρ)=P′2(1+ρ)1+ρ−1, H(ρ)=ρ1+ρ. |
To do this end, from (2.2), (2.3), and Gagliardo-Nirenberg's inequality, we know that there exists a sufficiently small ε1>0 that satisfies
‖ρ1‖L∞(R3)≤ε1 and ‖ρ2‖L∞(R3)≤ε1. | (2.4) |
First of all, from (2.4), we obtain
23≤1+ρ1≤43, 23≤1+ρ2≤43. |
Hence, by the definition of F1(ρ),F2(ρ), and H(ρ), we immediately have
|F1(ρ1)|,|H(ρ1)|≤C|ρ1|, |F2(ρ2)|,|H(ρ2)|≤C|ρ2|,|F(k)1(ρ1)|,|F(k)2(ρ2)|,|H(k)(ρ1)|,|H(k)(ρ2)|≤C for any k≥1. | (2.5) |
Let us start with a lemma that will be frequently used later.
Lemma 2.1. [22] Assume that ‖ρ‖L∞(R3)≤1, and f(ρ) is a smooth function of ρ with bounded derivatives of any order, then for any integer k≥1, we have
‖∇k(f(ρ))‖L∞(R3)≤C‖∇kρ‖L∞(R3). |
We obtain the estimates of the low order derivatives of the system (2.1) first.
Lemma 2. Under the assumption (2.2) and (2.4), we have
‖(ρ1,u1,ρ2,u2,∇Φ)‖L∞(0,T;L2(R3)),‖(∇u1,∇u2)‖L2(0,T;L2(R3))≤C, |
where C is a constant depending on ‖(ρ1,0,ρ2,0,u1,0,u2,0,∇Φ0)‖L2.
Proof. Multiplying the Eqs (2.1)1, (2.1)2, (2.1)3, and (2.1)4 by ρ1,u1,ρ2,u2, respectively, and integrating the equations over R3, we obtain
12∂t∫R3(ρ21+|u1|2+ρ22+|u2|2)dx+μ1∫R3(|∇u1|2+|∇u2|2)dx+μ2∫R3(|divu1|2+|divu2|2)dx=−∫R3∇Φ(u2−u1)dx−∫R3div(ρ1u1)ρ1dx−∫R3div(ρ2u2)ρ2dx−∫R3u1⋅∇u1⋅u1dx−∫R3u2⋅∇u2⋅u2dx−∫R3F1(ρ1)u1⋅∇ρ1dx−∫R3F2(ρ2)u2⋅∇ρ2dx−∫R3H(ρ1)(μ1Δu1+μ2∇divu1)u1dx−∫R3H(ρ2)(μ1Δu2+μ2∇divu2)u2dx=:9∑i=1Ai. | (2.6) |
Now we estimate Ai one by one. Because the self-consistent potential Φ(x,t) is coupled with the density through the Poisson equation, using Hölder's inequality and Cauchy's inequality, for A1, it holds
A1=−∫R3∇Φ(u2−u1)dx=−∫R3Φdiv(u1−u2)dx=−∫R3Φ(−∂tρ1−div(ρ1u1)+∂tρ2+div(ρ2u2))dx=∫R3Φ∂t(ρ1−ρ2)dx+∫R3Φ(div(ρ1u1)−div(ρ2u2))dx=∫R3Φ∂tΔΦdx+∫R3Φ(div(ρ1u1)−div(ρ2u2))dx=−12∂t∫R3|∇Φ|2dx−∫R3∇Φ(ρ1u1−ρ2u2))dx≤−12∂t∫R3|∇Φ|2dx+(‖∇Φ‖2L2+‖u1‖2L2+‖u2‖2L2)(‖ρ1‖L∞+‖ρ2‖L∞). | (2.7) |
For A4 and A5, by Hölder's inequality, we easily check that
|A4|+|A5|≤‖∇u1‖L∞‖u1‖2L2+‖∇u2‖L∞‖u2‖2L2. | (2.8) |
Integrating by parts, and using Hölder's inequality and Cauchy's inequality, it holds
|A2|+|A3|≤‖∇ρ1‖L∞(‖u1‖2L2+‖ρ1‖2L2)+‖∇ρ2‖L∞(‖u2‖2L2+‖ρ2‖2L2). | (2.9) |
By the definition of F1(ρ) and F2(ρ), using Hölder's inequality and Cauchy's inequality, and from (2.5), we have
|A6|+|A7|≤‖∇ρ1‖L∞(‖u1‖2L2+‖ρ1‖2L2)+‖∇ρ2‖L∞(‖u2‖2L2+‖ρ2‖2L2). | (2.10) |
Also integrating by parts, using Hölder's inequality and Cauchy's inequality, and by (2.5) and Lemma (2.1), we have
|A8|+|A9|=|−∫R3∇H(ρ1)(μ1∇u1+μ2divu1)u1dx−∫R3H(ρ1)(μ1∇u1+μ2divu1)∇u1dx|+|−∫R3∇H(ρ2)(μ1∇u2+μ2divu2)u2dx−∫R3H(ρ2)(μ1∇u2+μ2divu2)∇u2dx|≤‖∇ρ1‖2L∞‖u1‖2L2+ϵ‖∇u1‖2L2+‖∇ρ2‖2L∞‖u2‖2L2+ϵ‖∇u2‖2L2, | (2.11) |
where we take ϵ small enough such that ϵ≪1. Plugging the estimates for A1–A9, i.e., (2.7)–(2.11) into (2.6), we get
12∂t∫R3(ρ21+|u1|2+ρ22+|u2|2+|∇Φ|2)dx+μ1∫R3(|∇u1|2+|∇u2|2)dx+μ2∫R3(|divu1|2+|divu2|2)dx≤(‖ρ1‖L∞+‖ρ2‖L∞)(‖∇Φ‖2L2+‖u1‖2L2+‖u2‖2L2)+(‖∇ρ1‖L∞+‖∇ρ1‖2L∞+‖∇u1‖L∞)(‖u1‖2L2+‖ρ1‖2L2)+(‖∇ρ2‖L∞+‖∇ρ2‖2L∞+‖∇u2‖L∞)(‖u2‖2L2+‖ρ2‖2L2). | (2.12) |
Using Gronwall's inequality, we complete the proof of the lemma.
In the following, we would like to give the high regularity estimates of the solutions.
Lemma 2.3. Under the assumption (2.2) and (2.4), we have
‖(∇ρ1,∇u1,∇ρ2,∇u2,∇2Φ)‖L∞(0,T;L2(R3)),‖(∇2u1,∇2u2)‖L2(0,T;L2(R3))≤C, |
where C is a constant depending on ‖(∇ρ1,0,∇ρ2,0,∇u1,0,∇u2,0,∇2Φ0)‖L2.
Proof. We operate each equation of (2.1) with operator ∇ to derive
{∂t∇ρ1+∇divu1=−∇div(ρ1u1),∂t∇u1−μ1∇Δu1−μ2∇∇divu1+∇∇ρ1−∇∇Φ=−∇(u1⋅∇u1)−∇((P′1(1+ρ1)1+ρ1−1)∇ρ1)−∇(ρ11+ρ1(μ1Δu1+μ2∇divu1)),∂t∇ρ2+∇divu2=−∇div(ρ2u2),∂t∇u2−μ1∇Δu2−μ2∇∇divu2+∇∇ρ2+∇∇Φ=−∇(u2⋅∇u2)−∇((P′2(1+ρ2)1+ρ2−1)∇ρ2)−∇(ρ21+ρ2(μ1Δu2+μ2∇divu2)), | (2.13) |
and multiplying the above equations by ∇ρ1,∇u1,∇ρ2,and∇u2, respectively, and integrating over R3 yields
12∂t∫R3(|∇ρ1|2+|∇u1|2+|∇ρ2|2+|∇u2|2)dx+μ1∫R3(|∇2u1|2+|∇2u2|2)dx+μ2∫R3(|∇divu1|2+|∇divu2|2)dx+∫R3∇2Φ∇(u2−u1)dx=−∫R3∇div(ρ1u1)∇ρ1dx−∫R3∇div(ρ2u2)∇ρ2dx−∫R3∇(u1⋅∇u1)∇u1dx−∫R3∇(u2⋅∇u2)∇u2dx−∫R3∇(F1(ρ1)∇ρ1)∇u1dx−∫R3∇(F2(ρ2)∇ρ2)∇u2dx−∫R3∇(H(ρ1)(μ1Δu1+μ2∇divu1))∇u1dx−∫R3∇(H(ρ2)(μ1Δu2+μ2∇divu2))∇u2dx=:8∑i=1Bi. | (2.14) |
For the last term on the left-hand side of (2.14), since Φ(x,t) satisfies the Poisson equation, we have
∫R3∇2Φ∇(u2−u1)dx=∫R3∇Φ∇div(u1−u2)dx=−∫R3∇2Φ(−∂t(ρ1−ρ2)+div(ρ2u2−ρ1u1)dx=12∂t∫R3|∇2Φ|2dx+∫R3∇2Φdiv(ρ1u1−ρ2u2)dx. | (2.15) |
For the term ∫R3∇2Φdiv(ρ1u1−ρ2u2)dx, using Hölder's inequality and Young's inequality, we have
|∫R3∇2Φdiv(ρ1u1−ρ2u2)dx|≤‖ρ1‖L∞(‖∇u1‖2L2+‖∇2Φ‖2L2)+‖u1‖L∞(‖∇ρ1‖2L2+‖∇2Φ‖2L2)+‖ρ2‖L∞(‖∇u2‖2L2+‖∇2Φ‖2L2)+‖u2‖L∞(‖∇ρ2‖2L2+‖∇2Φ‖2L2). | (2.16) |
Now, we estimate each term on the righthand side of (2.14). Hölder's inequality and Young's inequality gives
|B1|≤∫R3|ρ1||∇2u1||∇ρ1|dx+∫R3|∇u1||∇ρ1|2dx≤C(ϵ)‖ρ1‖2L∞‖∇ρ1‖2L2+ϵ‖∇2u1‖2L2+‖∇u1‖L∞‖∇ρ1‖2L2, | (2.17) |
where ϵ is a positive number that is small enough to be determined, as ϵ appears in the following inequalities. Similar to the estimate of B1, we obtain
|B2|≤∫R3|ρ2||∇2u2||∇ρ2|dx+∫R3|∇u2||∇ρ2|2dx≤C(ϵ)‖ρ2‖2L∞‖∇ρ2‖2L2+ϵ‖∇2u2‖2L2+‖∇u2‖L∞‖∇ρ2‖2L2. | (2.18) |
Simple computation gives
|B3|+|B4|≤‖∇u1‖L∞‖∇u1‖2L2+‖∇u2‖L∞‖∇u2‖2L2. |
By the definition of F1 and F2, integrating by parts, and using Hölder's inequality and Young's inequality, from (2.5) we have
|B5|+|B6|=|−∫R3f1(ρ1)∇ρ1div∇u1dx|+|−∫R3f2(ρ2)∇ρ2div∇u2dx|≤C(ϵ)‖ρ1‖2L∞‖∇ρ1‖2L2+ϵ‖∇2u1‖2L2+C(ϵ)‖ρ2‖2L∞‖∇ρ2‖2L2+ϵ‖∇2u2‖2L2. | (2.19) |
Integrating by parts, by (2.5) and Lemma 2.1, we obtain
|B7|+|B8|=|−μ1∫R3H(ρ1)(|∇2u1|2+∇2u1∇divu1)dx|+|−μ2∫R3H(ρ2)(|∇2u2|2+∇2u2∇divu2)dx|≤ϵ(‖∇2u1‖2L2+‖∇divu2‖2L2). | (2.20) |
Consequently, by (2.14)–(2.20) and taking ϵ≪1, we deduce
12∂t∫R3(|∇ρ1|2+|∇u1|2+|∇ρ2|2+|∇u2|2+|∇2Φ|2)dx+μ1∫R3(|∇2u1|2+|∇2u2|2)dx+μ2∫R3(|∇divu1|2+|∇divu2|2)dx≤(‖ρ1‖2L∞+‖∇u1‖L∞+‖u1‖L∞+‖u2‖L∞)‖∇ρ1‖2L2+(‖ρ2‖2L∞+‖∇u2‖L∞+‖u1‖L∞+‖u2‖L∞)‖∇ρ2‖2L2+(‖∇u1‖L∞+‖ρ1‖L∞+‖ρ2‖L∞)‖∇u1‖2L2+(‖∇u2‖L∞+‖ρ1‖L∞+‖ρ2‖L∞)‖∇u2‖2L2+(‖ρ1‖L∞+‖ρ2‖L∞+‖u1‖L∞+‖u2‖L∞)‖∇2Φ‖2L2. | (2.21) |
With the help of Gronwall's inequality, we complete the proof of the lemma.
Lemma 2.4. Under the assumption (2.2) and (2.4), we have
‖(∇2ρ1,∇2u1,∇2ρ2,∇2u2,∇3Φ)‖L∞(0,T;L2(R3)),‖(∇3u1,∇3u2)‖L2(0,T;L2(R3))≤C, |
where C is a constant depending on ‖(∇2ρ1,0,∇2ρ2,0,∇2u1,0,∇2u2,0,∇3Φ0)‖L2.
Proof. Operating ∇2 on each equation of (2.1) gives
{∂t∇2ρ1+∇2divu1=−∇2div(ρ1u1),∂t∇2u1−μ1∇2Δu1−μ2∇2∇divu1+∇2∇ρ1−∇2∇Φ=−∇2(u1⋅∇u1)−∇2((P′1(1+ρ1)1+ρ1−1)∇ρ1)−∇2(ρ11+ρ1(μ1Δu1+μ2∇divu1)),∂t∇2ρ2+∇2divu2=−∇2div(ρ2u2),∂t∇2u2−μ1∇2Δu2−μ2∇2∇divu2+∇2∇ρ2+∇2∇Φ=−∇2(u2⋅∇u2)−∇2((P′2(1+ρ2)1+ρ2−1)∇ρ2)−∇2(ρ21+ρ2(μ1Δu2+μ2∇divu2)), | (2.22) |
and multiplying the above equations by ∇2ρ1,∇2u1,∇2ρ2,and ∇2u2, respectively, and integrating over R3 gives
12∂t∫R3(|∇2ρ1|2+|∇2u1|2+|∇2ρ2|2+|∇2u2|2)dx+μ1∫R3(|∇3u1|2+|∇3u2|2)dx+μ2∫R3(|∇2divu1|2+|∇2divu2|2)dx+∫R3∇2∇Φ∇2(u2−u1)dx=−∫R3∇2div(ρ1u1)∇2ρ1dx−∫R3∇2div(ρ2u2)∇2ρ2dx−∫R3∇2(u1⋅∇u1)∇2u1dx−∫R3∇2(u2⋅∇u2)∇2u2dx−∫R3∇2(F1(ρ1)∇ρ1)∇2u1dx−∫R3∇2(F2(ρ2)∇ρ2)∇2u2dx−∫R3∇2(H(ρ1)(μ1Δu1+μ2∇divu1))∇2u1dx−∫R3∇2(H(ρ2)(μ1Δu2+μ2∇divu2))∇2u2dx=:8∑i=1Ci. | (2.23) |
Similar to the proof of Lemma 2.3, for the last term on the left-hand side of (2.23), we get
∫R3∇2∇Φ∇2(u2−u1)dx=∫R3∇2Φ∇2div(u1−u2)dx=∫R3∇2Φ∇2(−∂t(ρ1−ρ2)+div(ρ2u2−ρ1u1)dx=−∫R3∇2Φ∇2∂t∇2Φdx+∫R3∇3Φ∇2(ρ1u1−ρ2u2)dx=12∂t∫R3|∇3Φ|2dx+∫R3∇3Φ∇2(ρ1u1−ρ2u2)dx. | (2.24) |
For the term ∫R3∇3Φ∇2(ρ1u1−ρ2u2)dx, we can easily check that
∫R3∇3Φ∇2(ρ1u1−ρ2u2)dx≤(‖ρ1‖L∞+‖u1‖L∞)(‖∇3Φ‖2L2+‖∇2ρ1‖2L2+‖∇2u1‖2L2)+‖∇ρ1‖L∞(‖∇3Φ‖2L2+‖∇u1‖2L2)+(‖ρ2‖L∞+‖u2‖L∞)(‖∇3Φ‖2L2+‖∇2ρ2‖2L2+‖∇2u2‖2L2)+‖∇ρ2‖L∞(‖∇3Φ‖2L2+‖∇u2‖2L2). | (2.25) |
Now we estimate Ci. We hereby declare that ϵ occurring in the following inequalities is a sufficiently small positive number to be determined. To begin, it is easy to check that
|C3|+|C4|=|−∫R3∇(u1⋅∇u1)∇3u1dx|+|−∫R3∇(u2⋅∇u2)∇3u2dx|≤ϵ‖∇3u1‖2L2+C(ϵ)‖∇u1‖2L∞‖∇u1‖2L2+‖∇u1‖L∞‖∇2u1‖2L2+ϵ‖∇3u2‖2L2+C(ϵ)‖∇u2‖2L∞‖∇u2‖2L2+‖∇u2‖L∞‖∇2u2‖2L2. | (2.26) |
Using Hölder's inequality and Young's inequality, we have
|C1|=|∫R3∇2(ρ1divu1+u1⋅∇ρ1)∇2ρ1dx|≤∫R3|∇2ρ1||∇u1||∇2ρ1|dx+∫R3ρ1|∇3u1||∇2ρ1|dx+∫R3|∇ρ1||∇2u1||∇2ρ1|dx≤‖∇u1‖L∞‖∇2ρ1‖2L2+ϵ‖∇3u1‖2L2+C(ϵ)‖ρ1‖2L∞‖∇2ρ1‖2L2+‖∇ρ1‖L∞(‖∇2ρ1‖2L2+‖∇2u1‖2L2). | (2.27) |
Similarly, we also have
|C2|≤‖∇u2‖L∞‖∇2ρ2‖2L2+ϵ‖∇3u2‖2L2+C(ϵ)‖ρ2‖2L∞‖∇2ρ2‖2L2+‖∇ρ2‖L∞(‖∇2ρ2‖2L2+‖∇2u2‖2L2). | (2.28) |
Also, integrating by parts, using Hölder's inequality and Cauchy's inequality, and by (2.5) and Lemma 2.1, we have
|C5|=|∫R3∇(F1(ρ1)∇ρ1)∇3u1dx|=|∫R3∇F1∇ρ1∇3u1dx+∫R3F1∇2ρ1∇3u1dx|≤ϵ‖∇3u1‖2L2+C(ϵ)‖∇ρ1‖2L∞‖∇ρ1‖2L2+C(ϵ)‖ρ1‖2L∞‖∇2ρ1‖2L2. | (2.29) |
Similarly to the estimate of C5, we obtain that
|C6|≤ϵ‖∇3u2‖2L2+C(ϵ)‖∇ρ2‖2L∞‖∇ρ2‖2L2+C(ϵ)‖ρ2‖2L∞‖∇2ρ2‖2L2. | (2.30) |
For the rest estimates of Ci, it is easy to check that
|C7|=|∫R3∇(H(ρ1)(μ1Δu1+μ2∇divu1))∇3u1dx|=|∫R3∇H(ρ1)(μ1Δu1+μ2∇divu1)∇3u1dx+∫R3H(ρ1)∇(μ1Δu1+μ2∇divu1)∇3u1dx|≤ϵ‖∇3u1‖2L2+C(ϵ)‖∇ρ1‖2L∞‖∇2u1‖2L2, | (2.31) |
and
|C8|≤ϵ‖∇3u2‖2L2+C(ϵ)‖∇ρ2‖2L∞‖∇2u2‖2L2. | (2.32) |
Combining (2.23)–(2.32), we have
12∂t∫R3(|∇2ρ1|2+|∇2u1|2+|∇2ρ2|2+|∇2u2|2+|∇3Φ|2)dx+μ1∫R3(|∇3u1|2+|∇3u2|2)dx+μ2∫R3(|∇2divu1|2+|∇2divu2|2)dx≤C(‖ρ1‖L∞+‖u1‖L∞+‖ρ2‖L∞+‖u2‖L∞+‖∇ρ1‖L∞+‖∇ρ2‖L∞)‖∇3Φ‖2L2+C(‖ρ1‖L∞+‖u1‖L∞+‖ρ1‖2L∞+‖∇ρ1‖L∞+‖∇u1‖L∞)‖∇2ρ1‖2L2+C(‖ρ1‖L∞+‖u1‖L∞+‖∇ρ1‖L∞+‖∇u1‖L∞+‖∇ρ1‖2L∞)‖∇2u1‖2L2+C(‖ρ2‖L∞+‖u2‖L∞+‖ρ2‖2L∞+‖∇ρ2‖L∞+‖∇u2‖L∞)‖∇2ρ2‖2L2+C(‖ρ2‖L∞+‖u2‖L∞+‖∇ρ2‖L∞+‖∇u2‖L∞+‖∇ρ2‖2L∞)‖∇2u2‖2L2+C(‖∇ρ1‖2L∞‖∇ρ1‖2L2+‖∇ρ2‖2L∞‖∇ρ2‖2L2)+C(‖∇ρ1‖L∞+‖∇u1‖2L∞)‖∇u1‖2L2+C(‖∇ρ2‖L∞+‖∇u2‖2L∞)‖∇u2‖2L2. | (2.33) |
By Gronwall's inequality, we complete the proof of the lemma.
Lemma 2.5. Under the assumption (2.2) and (2.4), for 3≤l≤s+1, we have
‖(∇lρ1,∇lu1,∇lρ2,∇lu2,∇l+1Φ)‖L∞(0,T;L2(R3)),‖(∇l+1u1,∇l+1u2)‖L2(0,T;L2(R3))≤C, |
where C is a constant that depends only on ‖(∇lρ1,0,∇lρ2,0,∇lu1,0,∇lu2,0,∇l+1Φ0)‖L2.
Proof. Similar to the proof of Lemma 2.4, we can obtain the conclusion of the lemma. So we omit it.
In order to see the Green's function of the linear part of the system better, we reformulate the system (2.1) slightly. Let
n=ρ1+ρ2,m=ρ2−ρ1,v=u1+u2,w=u2−u1, | (3.1) |
which equivalently gives
ρ1=n−m2,ρ2=n+m2,u1=v−w2,u2=v+w2. | (3.2) |
Then, the Cauchy problem (2.1) can be reformulated into the following form:
{∂tn+divv=Q1(n,v,m,w),∂tv−μ1Δv−μ2∇divv+∇n=Q2(n,v,m,w),∂tm+divw=Q3(n,v,m,w),∂tw−μ1Δw−μ2∇divw+∇m+2∇Φ=Q4(n,v,m,w),ΔΦ=−m,(n,v,m,w,∇Φ)(x,0)=(n0,v0,m0,w0,∇Φ0)(x), | (3.3) |
where (n0,v0,m0,w0,∇Φ0)(x)=(ρ1,0+ρ2,0,u1,0+u2,0,ρ2,0−ρ1,0,u2,0−u1,0,∇Φ0)(x) and
Q1(n,v,m,w)=−div(n−m2v−w2)−div(n+m2v+w2) | (3.4) |
Q2=−v−w2⋅∇(v−w2)−v+w2⋅∇(v+w2)−(P′1(1+n−m2)1+n−m2−1)∇(n−m2)−(P′2(1+n+m2)1+n+m2−1)∇(n+m2)−n−m21+n−m2(μ1Δ(v−w2)+μ2∇div(v−w2))−n+m21+n+m2(μ1Δ(v+w2)+μ2∇div(v+w2)), | (3.5) |
Q3(n,v,m,w)=div(n−m2v−w2)−div(n+m2v+w2) | (3.6) |
Q4=v−w2⋅∇(v−w2)−v+w2⋅∇(v+w2)+(P′1(1+n−m2)1+n−m2−1)∇(n−m2)−(P′2(1+n+m2)1+n+m2−1)∇(n+m2)+n−m21+n−m2(μ1Δ(v−w2)+μ2∇div(v−w2))−n+m21+n+m2(μ1Δ(v+w2)+μ2∇div(v+w2)). | (3.7) |
The linearized system of (3.3) is
{∂tn+divv=0,∂tv−μ1Δv−μ2∇divv+∇n=0,∂tm+divw=0,∂tw−μ1Δw−μ2∇divw+∇m+2∇Φ=0,ΔΦ=−m. | (3.8) |
We can also rewrite (3.8) as
(∂t+A(D))V=0, | (3.9) |
where
A(D)=(0div00∇−μ1Δ−μ2∇div00000div00∇+2∇(−Δ)−1−μ1Δ−μ2∇div), V=(nvmw). | (3.10) |
Consider the Green's function G of (3.9), i.e., the solution to the following Cauchy problem
{(∂t+A(D))G(x,t)=0,G(x,0)=δ(x)I8×8, | (3.11) |
where δ(x) denotes the Dirac function and I8×8 denotes the unit matrix. By direct calculation, we obtain the Fourier transform of the Green's function G as
ˆG=(ˆG100ˆG2), | (3.12) |
where
ˆG1(ξ,t)=(λ+eλ−t−λ−eλ+tλ+−λ−√−1eλ−t−eλ+tλ+−λ−ξT√−1eλ−t−eλ+tλ+−λ−ξ−e−μ1|ξ|2tI+(λ+eλ+t−λ−eλ−tλ+−λ−+e−μ1|ξ|2t)ξξT|ξ|2), |
ˆG2(ξ,t)=(˜λ+e˜λ−t−˜λ−e˜λ+t˜λ+−˜λ−√−1e˜λ−t−e˜λ+t˜λ+−˜λ−ξT√−1(1+|ξ|−2)e˜λ−t−e˜λ+t˜λ+−˜λ−ξ−e−μ1|ξ|2tI+(˜λ+e˜λ+t−˜λ−e˜λ−t˜λ+−˜λ−+e−μ1|ξ|2t)ξξT|ξ|2), |
and
λ±=−μ|ξ|2±√μ2|ξ|4−4|ξ|22, |
˜λ±=−μ|ξ|2±√μ2|ξ|4−4(|ξ|2+2)2, μ=μ1+μ2. |
For the convenience of writing, we also give the following definition,
ˆG=(ˆG100ˆG1), | (3.13) |
where ˆG is the Fourier transform of the Green's function G. In this paper, we divide Green's function into a high frequency part and a low frequency part since Green's function has different properties in high and low frequency. Let χ(ξ) be a smooth cutoff function
χ(ξ)={1,|ξ|≤12,0,|ξ|>1. | (3.14) |
We denote G=GL+GRH+S, where GL stands for the lower frequency part, GRH stands for the regular part of the high frequency, and the S stands for the sigular part. GL,GRH, and S have the following forms:
GL=χ(D)G, GRH=(1−χ(D))G−S,S=e−1μtδ(x)(1−χ(D)000000000000000000000000000000000001−χ(D)000000000000000000000000000), | (3.15) |
Here, μ=μ1+μ2. For the convenience of the description in the fifth part of this paper, we redefine smooth cutoff functions χ1(ξ) and χ2(ξ)
χ1(ξ)={1,|ξ|≤1,0,|ξ|>76.χ2(ξ)={1,|ξ|≤13,0,|ξ|>12. | (3.16) |
Let us redefine the low frequency of G and the high frequency of G,
G˜L=χ1(D)G, G˜H=(1−χ2(D))G=:G~RH+G˜S, |
where
G˜S=e−1μtδ(x)(1−χ2(D)000000000000000000000000000000000001−χ2(D)000000000000000000000000000). | (3.17) |
Here, μ=μ1+μ2. From the definition of χ(ξ), χ1(ξ), and χ2(ξ) in (3.14) and (3.16), we can obtain
χ(D)G˜L=χ(D)G, (1−χ(D))G˜H=(1−χ(D))G. |
In this paper, we use Gi,j to represent the element in row i and column j of G.
Below, we list some properties of Green's function, and the readers can refer to [11,21,30,31] for details.
Lemma 3.1. If ϵ1>0 is small enough, then for |ξ|<ϵ1, we have
λ±=−μ2|ξ|2±√−1|ξ|(1+∞∑j=1dj|ξ|2j), |
and
˜λ±=−μ2|ξ|2+∞∑j=2aj|ξ|2j±√−1(√2+∞∑j=1bj|ξ|2j). |
Proof. The readers can refer to [11,21,30,31] for details, so we omit the proof.
Lemma 3.2. If 1≤p≤+∞ and α=(α1,α2,α3), αi≥0, x∈R3, we have
‖DαGL(⋅,t)‖Lp≤C(1+t)−32(1−1p)−|α|2. |
Proof. By the representation of G and Lemma 3.1, we can obtain the proof of the lemma, so we omit it.
Lemma 3.3. If K>0 is large enough, then for |ξ|>K, we have
λ+=−1μ+μ2∞∑j=1ej|ξ|−2j, λ−=−μ|ξ|2+1μ−∞∑j=1ej|ξ|−2j, |
˜λ+=−1μ+∞∑j=1lj|ξ|−2j, ˜λ−=−μ|ξ|2+1μ−∞∑j=1lj|ξ|−2j. |
Here, μ=μ1+μ2, and all ej,lj are real constants.
Proof. See [11,21,30,31] for details, and we omit the proof.
Remark: This lemma states that in G, only the terms related to
λ−(ξ)λ+(ξ)−λ−(ξ)eλ+(ξ)t or ˜λ−(ξ)˜λ+(ξ)−˜λ−(ξ)e˜λ+(ξ)t |
will occur in a singular part, and the other terms will bear at least a first derivative.
Lemma 3.4. If 1≤p≤+∞ and α=(α1,α2,α3), αi≥0, |α|≤1, we have
‖DαGRH(⋅,t)‖Lp≤Ce−C0t, |
‖DαG~RH(⋅,t)‖Lp≤Ce−C0t, |
‖S(⋅,t)∗V(⋅,t)‖Lp≤Ce−C0t‖V(⋅,t)‖Lp, |
‖G˜S(⋅,t)∗V(⋅,t)‖Lp≤Ce−C0t‖V(⋅,t)‖Lp, |
where C0>0 is the fixed normal number associated with μ.
Proof. The readers can refer to [11,21,30,31] for details, so we omit the proof.
Below we derive an estimation method that combines the advantages of the Green's function and energy estimate. We consider the system:
{(∂t+B(D))U(x,t)=R(U(x,t)),U(x,0)=U0(x), | (3.18) |
where B(D) is an operator and R(U) is nonlinear terms. The Green's function G(x,t) corresponding to the system (3.18) is the fundamental solution of the Cauchy problem of the linear equations of its corresponding system, i.e., G(x,t) is the solution of the following Cauchy problem:
{(∂t+B(D))G(x,t)=0,G(x,0)=δ(x)In×n. | (3.19) |
where δ(x) denotes the Dirac function and In×n denotes the unit matrix.
The solution of (3.18) is usually discussed in terms of energy estimate or the Green's function. The following lemma combines the advantages of Green's function and energy estimate, which we may call the G-E estimate. The advantage of this estimate is that on the one hand, the fine decaying estimate of the solution can be obtained with the Green's function; on the other hand, the derivative in the singular part of the high frequency can be shared through integration by parts similar to the energy estimate.
Lemma 3.5. [27] If B(ξ) is a complex normal matrix (i.e., B∗B=BB∗, B∗=¯BT), then it holds
‖U(⋅,t)‖2L2=∫Rn(GT(⋅,t)∗U0)TG(⋅,t)∗U0dx+2∫t0∫Rn(GT(x−⋅,t−τ)∗U(⋅,τ))TG(x−⋅,t−τ)∗R(U(⋅,τ))dxdτ, | (3.20) |
where G is Green's function about (3.19).
Remark 3.1. If GT=G, then we have
‖U(⋅,t)‖2L2=‖G(⋅,t)∗U0‖2L2+2∫t0∫RnG(x−⋅,t−τ)∗U(⋅,τ)⋅G(x−⋅,t−τ)∗R(U(⋅,τ))dxdτ. | (3.21) |
The Eq (3.20) is in the form of row vector times column vector, while the Eq (3.21) is in the form of the inner product of vectors.
Remark 3.2. As can be seen from (3.10), G, which we defined in (3.13), satisfies GT=G.
In this section, we first give the local existence theory.
According to (2.1), we construct the approximate solution sequence {˜Vn(t)} by the following linearized iteration scheme:
{∂tρn+11+divun+11+div(ρn+11un1)=0,∂tun+11−μ1Δun+11−μ2∇divun+11+∇ρn+11−∇Φn+1=−un1⋅∇un1−(P′1(1+ρn1)1+ρn1−1)∇ρn1−ρn11+ρn1(μ1Δun1+μ2∇divun1),∂tρn+12+divun+12+div(ρn+12un2)=0,∂tun+12−μ1Δun+12−μ2∇divun+12+∇ρn+12+∇Φn+1=−un2⋅∇un2−(P′2(1+ρn2)1+ρn2−1)∇ρn2−ρn21+ρn2(μ1Δun2+μ2∇divun2),ΔΦn+1=ρn+11−ρn+12,lim|x|→∞Φn+1(x,t)=0, | (4.1) |
where {˜Vn(t)} is defined as ˜Vn(t)=(ρn1(t),un1(t),ρn2(t),un2(t),∇Φn(t)),n≥0, and ˜V0(t)=(0,0,0,0,0). For any given integer s≥[32]+3, we define
XsT,E={˜V(t)|‖˜V‖Xs<E} |
as the suitable space for the solutions, where
‖˜V‖Xs=sup0≤t≤T‖˜V(t)‖Hs. |
It is easy to show that XsT,E, equipped with the norm ‖⋅‖Xs, is a nonempty Banach space. To obtain the local soluiton, we need the following two lemmas. To do this end, we first give a prior assumption. For sufficiently small ε1>0, we have
‖ρ1‖L∞(R3)≤ε1 and ‖ρ2‖L∞(R3)≤ε1. | (4.2) |
Lemma 4.1. Under the assumption (4.2), when T is small enough, there exists a constant E>0 such that {˜Vn(x,t)}⊆Xs+1T,E.
Proof. We imply the inductive method to accomplish the proof. To start, when n=0, we have
{∂tρ11+divu11=0,∂tu11−μ1Δu11−μ2∇divu11+∇ρ11−∇Φ1=0,∂tρ12+divu12=0,∂tu12−μ1Δu12−μ2∇divu12+∇ρ12+∇Φ1=0,ΔΦ1=ρ11−ρ12,lim|x|→∞Φ1(x,t)=0, | (4.3) |
By the energy estimate, we have
12∂t(‖ρ11‖2Hs+‖u11‖2Hs+‖ρ12‖2Hs+‖u12‖2Hs+‖∇Φ1‖2Hs≤0. |
We take E=2‖(‖ρ1(x,0),u1(x,0),ρ2(x,0),u2(x,0))‖Hs, then we get ˜V1(x,t)∈XsT,E. Now, assuming that {˜Vj(x,t)}∈XsT,E for all j≤n, we need to prove it holds for j=n+1.
Applying the energy method to (4.1), we have
12∂t∫R3(|ρn+11|2+|un+11|2+|ρn+12|2+|un+12|2)dx+μ1∫R3(|∇un+11|2+|∇un+12|2)dx+μ2∫R3(|divun+11|2+|divun+12|2)dx=−∫R3∇Φn+1(un+12−un+11)dx−∫R3div(ρn+11un1)ρn+11dx−∫R3div(ρn+12un2)ρn+12dx−∫R3un1⋅∇un1⋅un+11dx−∫R3un2⋅∇un2⋅un+12dx−∫R3F1(ρn1)un+11⋅∇ρn1dx−∫R3F2(ρn2)un+12⋅∇ρn2dx−∫R3H(ρn1)(μ1Δun1+μ2∇divun1)un+11dx−∫R3H(ρn2)(μ1Δun2+μ2∇divun2)un+12dx=:9∑i=1Gi. | (4.4) |
Similar to the proof of Lemma 2.2, we have the following estimates for each Gi, 1≤i≤9.
|G1|≤−12∂t∫R3|∇Φ|2dx+C‖∇Φn+1‖2L2(‖un1‖2L∞+‖un2‖2L∞)+C‖ρn+11‖2L2+‖ρn+11‖2L2, |
|G2+G3|≤C‖divun1‖L∞‖ρn+11‖2L2+C‖divun2‖L∞‖ρn+12‖2L2, |
|G4+G5|≤C‖∇un1‖2L∞‖un1‖2L2+C‖∇un2‖2L∞‖un2‖2L2+C‖un+11‖2L2+C‖un+12‖2L2, |
|G6+G7|≤C‖∇ρn1‖2L∞‖ρn1‖2L2+C‖un+11‖2L2+C‖∇ρn2‖2L∞‖ρn2‖2L2+C‖un+12‖2L2, |
|G8+G9|≤C‖∇ρn1‖2L∞‖un+11‖2L2+C(1+‖ρn1‖2L∞)‖∇un1‖2L2+ϵ‖∇un+11‖2L2+C‖∇ρn2‖2L∞‖un+12‖2L2+C(1+‖ρn2‖2L∞)‖∇un2‖2L2+ϵ‖∇un+12‖2L2. |
where we take ϵ small enough such that ϵ≪1. Plugging the estimates for G1−G9 into (4.4), and integrating with respect to t, we obtain
sup0≤t≤T(‖ρn+11(t)‖2L2+‖un+11(t)‖2L2+‖ρn+12(t)‖2L2+‖un+12(t)‖2L2+‖∇Φn+1(t)‖2L2)+μ1∫t0∫R3(|∇un+11|2+|∇un+12|2)dx+μ2∫t0∫R3(|divun+11|2+|divun+12|2)dx≤(12E)2+CTE2sup0≤t≤T‖∇Φn+1‖2L2+CT(1+E)sup0≤t≤T‖ρn+11‖2L2+CT(1+E)sup0≤t≤T‖ρn+12‖2L2+CT(1+E2)sup0≤t≤T‖un+11‖2L2+CT(1+E2)sup0≤t≤T‖un+12‖2L2+CTE2(1+E2). | (4.5) |
Taking T small enough, we can get the following estimate from (4.5),
sup0≤t≤T‖˜Vn+1(t)‖L2≤E. | (4.6) |
To derive higher-order estimates, similar to the proof of Lemmas 2.3–2.5, we can obtain
sup0≤t≤T‖Dα˜Vn+1(t)‖L2≤E, 1≤α≤s+1. | (4.7) |
Combining (4.6) and (4.7) yields
sup0≤t≤T‖˜Vn+1(t)‖Hs+1≤E, |
which means ˜Vn+1(x,t)∈Xs+1T,E. We complete the proof of the lemma.
Lemma 4.2. Under the assumption (4.2), when T is small enough, {˜Vn(x,t)} is a Cauchy sequence in XsT,E.
Proof. We set
fn+11=ρn+11−ρn1, fn+12=ρn+12−ρn2, gn+11=un+11−un1, |
gn+12=un+12−un2, Ψn+1=Φn+1−Φn, |
and define
Yn+1=(fn+11,fn+12,gn+11,gn+12,Ψn+1), |
then we only need to verify that
‖Yn+1‖Hs≤κ‖Yn‖Hs, |
where 0<κ<1. From (4.1), we get Yn+1 satisfies
{∂tfn+11+divgn+11+div(ρn+11un1−ρn1un−11)=0,∂tgn+11−μ1Δgn+11−μ2∇divgn+11+∇fn+11−∇Ψn+1=−(un1⋅∇un1−un−11⋅∇un−11)−((P′1(1+ρn1)1+ρn1−1)∇ρn1−(P′1(1+ρn−11)1+ρn−11−1)∇ρn−11)−(ρn11+ρn1(μ1Δun1+μ2∇divun1)−ρn−111+ρn−11(μ1Δun−11+μ2∇divun−11)),∂tfn+12+divgn+12+div(ρn+12un2−ρn2un−12)=0,∂tgn+12−μ1Δgn+12−μ2∇divgn+12+∇fn+12+∇Ψn+1=−(un2⋅∇un2−un−12⋅∇un−12)−((P′2(1+ρn2)1+ρn2−1)∇ρn2−(P′2(1+ρn−12)1+ρn−12−1)∇ρn−12)−(ρn21+ρn2(μ1Δun2+μ2∇divun2)−ρn−121+ρn−12(μ1Δun−12+μ2∇divun−12)),ΔΨn+1=fn+11−fn+12,lim|x|→∞Ψn+1(x,t)=0. | (4.8) |
Applying the energy method to (4.8), we have
12∂t∫R3(|fn+11|2+|gn+11|2+|fn+12|2+|gn+12|2)dx+μ1∫R3(|∇gn+11|2+|∇gn+12|2)dx+μ2∫R3(|divgn+11|2+|divgn+12|2)dx=−∫R3∇Ψn+1(gn+12−gn+11)dx−∫R3div(ρn+11un1−ρn1un−11)fn+11dx−∫R3div(ρn+12un2−ρn2un−12)fn+12dx−∫R3(un1⋅∇un1−un−11⋅∇un−11)⋅gn+11dx−∫R3(un2⋅∇un2−un−12⋅∇un−12)⋅gn+12dx−∫R3(F1(ρn1)⋅∇ρn1−F1(ρn−11)⋅∇ρn−11)gn+11dx−∫R3(H(ρn1)(μ1Δun1+μ2∇divun1)−H(ρn−11)(μ1Δun−11+μ2∇divun−11))gn+11dx−∫R3(H(ρn2)(μ1Δun2+μ2∇divun2)−H(ρn−12)(μ1Δun−12+μ2∇divun−12))gn+12dx−∫R3(F2(ρn2)⋅∇ρn2−F2(ρn−12)⋅∇ρn−12)gn+12dx=:9∑i=1Ji. | (4.9) |
Similar to the proof of Lemma 2.2, we have the following estimates for each Ji, 1≤i≤9.
For J1, we have
J1=−12∂t‖∇Ψn+1‖2L2+∫R3∇Ψn+1(fn+12un2+ρn2gn2−fn+11un1−ρn1gn1)dx≤−12∂t‖∇Ψn+1‖2L2+4‖∇Ψn+1‖2L2+‖un2‖2L∞‖fn+12‖2L2+‖un1‖2L∞‖fn+11‖2L2+‖ρn2‖2L∞‖gn2‖2L2+‖ρn1‖2L∞‖gn+11‖2L2. |
For J2, we have
J2=∫R3div(fn+11un1+ρn1gn1)fn+11dx≤‖∇un1‖L∞‖fn+11‖2L2+‖∇ρn1‖L∞‖fn+11‖2L2+‖gn1‖2L2+‖∇gn1‖2L2+‖ρn1‖L∞‖fn+11‖2L2. |
Similar to the proof of J2, for J3, we can obtain
J3=∫R3div(fn+12un2+ρn2gn2)fn+12dx≤‖∇un2‖L∞‖fn+12‖2L2+‖∇ρn2‖L∞‖fn+12‖2L2+‖gn2‖2L2+‖∇gn2‖2L2+‖ρn2‖L∞‖fn+12‖2L2. |
For J4, we have
J4=−∫R3(gn1⋅∇un1+un−11⋅∇gn1)gn+11dx≤‖gn1‖2L2+(‖∇un1‖2L∞+‖∇un−11‖2L∞)‖gn+11‖2L2+ϵ‖∇gn+11‖2L2+C‖gn1‖2L2‖un−11‖2L∞. |
Similar to the estimate of J4, for J5, we can obtain
J5=−∫R3(gn2⋅∇un2+un−12⋅∇gn2)gn+12dx≤‖gn2‖2L2+(‖∇un2‖2L∞+‖∇un−12‖2L∞)‖gn+12‖2L2+ϵ‖∇gn+12‖2L2+C‖gn2‖2L2‖un−12‖2L∞. |
For J6, we have
J6=−∫R3((F1(ρn1)−F1(ρn−11)∇ρn1+F1(ρn−11)∇fn1)gn+11dx≤‖∇ρn1‖2L∞‖gn+11‖2L2+2‖ρn1‖2L2+2‖ρn−11‖2L2+‖fn1‖2L2+‖∇ρn−11‖2L∞‖gn+11‖2L2+ϵ‖∇gn+11‖2L2+C‖ρn−11‖2L∞‖fn1‖2L2. |
Similar to the estimate of J6, for J9, we can obtain
J9=−∫R3((F2(ρn2)−F2(ρn−12)∇ρn2+F2(ρn−12)∇fn2)gn+12dx≤‖∇ρn2‖2L∞‖gn+12‖2L2+2‖ρn2‖2L2+2‖ρn−12‖2L2+‖fn2‖2L2+‖∇ρn−12‖2L∞‖gn+12‖2L2+ϵ‖∇gn+12‖2L2+C‖ρn−12‖2L∞‖fn2‖2L2. |
For J7, we have
J7=−∫R3((H(ρn1)−H(ρn−11))(μ1Δun1+μ2∇divun1)+H(ρn−11)(μ1Δgn1+μ2∇divgn1))gn+11dx≤C(‖∇ρn1‖2L∞+‖∇ρn−11‖2L∞)‖gn+11‖2L2+C‖∇un1‖2L2+ϵ‖∇gn+11‖2L2+C(ρn1‖2L∞+‖ρn−11‖2L∞)‖∇un1‖2L2+C‖∇gn1‖2L2+C‖∇ρn−11‖2L∞‖gn+11‖2L2. |
Similar to the estimate of J7, for J8, we have
J8=−∫R3((H(ρn2)−H(ρn−12))(μ1Δun2+μ2∇divun2)+H(ρn−12)(μ1Δgn2+μ2∇divgn2))gn+12dx≤C(‖∇ρn2‖2L∞+‖∇ρn−12‖2L∞)‖gn+12‖2L2+C‖∇un2‖2L2+ϵ‖∇gn+12‖2L2+C(ρn2‖2L∞+‖ρn−12‖2L∞)‖∇un2‖2L2+C‖∇gn2‖2L2+C‖∇ρn−12‖2L∞‖gn+12‖2L2. |
Plugging the estimates for J1–J9 into (4.9), and integrating with respect to t over [0, T], we obtain
sup0≤t≤T‖Yn+1(t)‖2L2≤12sup0≤t≤T‖Yn(t)‖2L2, | (4.10) |
where T is small enough. To derive higher-order estimates, similar to the proof of Lemma (2.3)–(2.5), we can obtain
sup0≤t≤T‖DαYn+1(t)‖L2≤12sup0≤t≤T‖DαYn(t)‖2L2, 1≤α≤s. | (4.11) |
Combining (4.10) and (4.11) yields
sup0≤t≤T‖Yn+1(t)‖Hs≤12sup0≤t≤T‖Yn(t)‖Hs, |
so we complete the proof of the lemma.
So far, we complete the proof of local existence.
Lemma 4.3. Let (ρ1,0−1,u1,0,ρ2,0−1,u2,0,∇Φ0)∈Hs+1(s≥4), and
‖(ρ1,0−1,u1,0,ρ2,0−1,u2,0,∇Φ0)‖L2(R3)⋂L1(R3)≤E0, |
where E0 is sufficiently small, then there exists a time T>0, such that the Cauchy problem (1.1) and (1.2) admits a unique classical solution in [0, T) that satisfies
(ρ1−1,u1,ρ2−1,u2)∈L∞([0,T);Hs+1), ∇(u1,u2)∈L2([0,T);Hs+1). |
In this subsection, we will establish the global solution to the systems (1.1) and (1.2) by using the bootstrap argument if the initial data satisfies
‖(ρ1,0,ρ2,0,u1,0,u2,0,∇Φ0)‖L2⋂L1≤E0, | (4.12) |
where E0 is sufficiently small. However, for the derivatives of ρ1,0,ρ2,0,u1,0,u2,0, and ∇Φ0, we only assume that they are bounded. Now, we first give the following abstract bootstrap argument.
Lemma 4.4. [32] Let T>0. Assume that two statements C(t) and H(t) with t∈[0,T] satisfy the following conditions:
1) If H(t) holds for some t∈[0,T], then C(t) holds for the same t;
2) If C(t) holds for some t0∈[0,T], then H(t) holds for t in a neighborhood of t0;
3) If C(t) holds for tm∈[0,T] and tm→t, then C(t) holds;
4) C(t) holds for at least one t1∈[0,T].
Then, C(t) holds on [0,T].
For any fixed 0<T<∞, t∈[0,T], through the regularity criterion in the Section 2, we know that if (ρ1,ρ2,u1,u2,∇Φ)(x,t) satisfies
(‖∇ρ1(t)‖L∞(R3),‖∇u1(t)‖L∞(R3),‖∇ρ2(t)‖L∞(R3),‖∇u2(t)‖L∞(R3))≤C(1+t)−2, |
and
‖ρ1(t)‖L∞(R3)≤ε1 and ‖ρ2(t)‖L∞(R3)≤ε1, |
where ε1>0 is small enough, then
‖(ρ1,u1,ρ2,u2,∇Φ)‖L∞([0,T];Hs+1)+‖(∇u1,∇u2)‖L2([0,T];Hs+1)≤C, |
where s≥4. From (3.1) and (3.2), we get
‖(ρ1,0,u1,0,ρ2,0,u2,0,∇Φ0)‖L2⋂L1≤E0, |
is equal to
‖(n0,v0,m0,w0,∇Φ0)‖L2⋂L1≤E0, | (4.13) |
and
(‖∇ρ1(t)‖L∞(R3),‖∇u1(t)‖L∞(R3),‖∇ρ2(t)‖L∞(R3),‖∇u2(t)‖L∞(R3))≤C(1+t)−2, |
is equal to
(‖∇n(t)‖L∞(R3),‖∇v(t)‖L∞(R3),‖∇m(t)‖L∞(R3),‖∇w(t)‖L∞(R3))≤C(1+t)−2, | (4.14) |
as well as
‖ρ1‖L∞(R3)≤ε1 and ‖ρ2‖L∞(R3)≤ε1, |
is equal to
‖n‖L∞(R3)≤ε1 and ‖m‖L∞(R3)≤ε1, |
For convenience, in this subsection, we use the condition (4.13) and the assumption (4.14). Let δ be a fixed positive number, say,
2‖V0‖L1∩L2+2‖∇V0‖L∞≤δ2, |
where V0 is defined as V0=(n0,v0,m0,w0,∇Φ0)T. For any fixed 0<T<∞,t∈[0,T], let us denote
C(T):(‖∇n(t)‖L∞(R3),‖∇v(t)‖L∞(R3),‖∇m(t)‖L∞(R3),‖∇w(t)‖L∞(R3))≤δ2(1+t)−2 |
and
H(T):(‖∇n(t)‖L∞(R3),‖∇v(t)‖L∞(R3),‖∇m(t)‖L∞(R3),‖∇w(t)‖L∞(R3))≤δ(1+t)−2, | (4.15) |
Based on the local existence of solutions, we only need to verify the condition 1 in Lemma (4.4) under the condition (4.12), i.e., given H(T) as the condition, to derive C(T) is valid. Before we check the condition 1, we need some lemmas.
Lemma 4.5. Under the assumption (4.15) and the condition (4.13), the following estimate holds:
‖(n,v,m,w,∇Φ)‖L∞(0,T;L2(R3))≤CE0, T∈(0,+∞). | (4.16) |
Proof. By (4.15) and Lemma 2.2, we complete the proof of the lemma.
Lemma 4.6. Under the assumption (4.15) and the condition (4.13), we have
(‖(n,v,m,w)(t)‖L∞(R3)≤CE250(1+t)−65, |
‖∇(n,v,m,w)(t)‖L2(R3)≤CE17280(1+t)−34, |
‖∇(v,m)(t)‖L4(R3)≤C(1+t)−118, t∈[0,T]. |
Proof. By the Gagliardo-Nirenberg inequality, we know
‖(n,v,m,w)(t)‖L∞(R3)≤C‖(n,v,m,w)(t)‖25L2‖∇(n,v,m,w)(t)‖35L∞(R3)≤CE250(1+t)−65,‖∇v(t)‖L2≤C‖Λ1+ϵv(t)‖θL∞‖v(t)‖1−θL2, | (4.17) |
and
‖Λ1+ϵv(t)‖L∞≤C‖∇v(t)|1−βL∞‖Λ1+kϵv(t)‖βL∞, | (4.18) |
where θ=22ϵ+5,β=1k. Then, from (4.17) and (4.18), we have
‖∇v(t)‖L2≤C‖v(t)‖1−22ϵ+5L2‖∇v(t)‖(1−1k)22ϵ+5L∞≤CE1−22ϵ+50(1+t)−(1−1k)42ϵ+5. |
Taking k=1ϵ, ϵ=122 in the above inequality, we obtain
‖∇v(t)‖L2(R2)≤CE17280(1+t)−34. |
The same computation also gives
‖∇(n,m,w)(t)‖L2(R2)≤CE17280(1+t)−34. |
By the interpolation inequality, it holds
‖∇(v,w)(t)‖L4≤C‖∇(v,w)‖12L2‖∇(v,w)‖12L∞.≤C(1+t)−(12⋅34+12⋅2)=C(1+t)−118. |
Thus we complete the proof of the lemma.
Lemma 4.7. Under the condition (4.13), we have
‖(n(t),m(t))‖L1(R3)≤CE0. |
Proof. By (3.3)1 and (3.3)3, and using the condition (4.13), we can obtain the result.
Now let us use Green's function to prove some lemmas. To begin, by Duhamel's principle, we know
V(t)=G(t)∗V0+∫t0G(t−s)∗N(V)(s)ds, | (4.19) |
where
V=(nvmw), V0=(n0v0m0w0)=:(V10V20V30V40), N(V)=:(Q1Q2Q3Q4). | (4.20) |
where the Qi(i=1,2,3,4) are defined by (3.4)–(3.7). From (4.19), for the component v and w of V, we have
v=(4∑k=1G2,k(t))∗Vk0+∫t04∑k=1G2,k(t−s)∗Qk(s))ds4∑k=1G3,k(t))∗Vk0+∫t04∑k=1G3,k(t−s)∗Qk(s))ds4∑k=1G4,k(t))∗Vk0+∫t04∑k=1G3,k(t−s)∗Qk(s))ds)=:(M1M2M3), | (4.21) |
w=(8∑k=5G6,k(t))∗Vk0+∫t08∑k=5G6,k(t−s)∗Qk(s))ds8∑k=5G7,k(t))∗Vk0+∫t08∑k=5G7,k(t−s)∗Qk(s))ds8∑k=5G8,k(t))∗Vk0+∫t08∑k=5G8,k(t−s)∗Qk(s))ds)=:(N1N2N3), | (4.22) |
where G(t)∗V0, G(t−s)∗N(V)(s) obey matrix multiplication.
Lemma 4.8. Under the assumption (4.15) and the condition (4.13), it holds that
‖(v(t),w(t))‖L1(R3)≤CE250, t∈[0,T]. |
Proof. Because we can't directly get the estimate of ‖(v(t),w(t))‖L1, we first obtain the boundedness of ‖(v(t),w(t))‖Lr for some r∈(1,2). In the following, we take r=43.
We first consider ‖(v,w)(t)‖Lr. By (4.21) and the representation (3.15) of S, we have
‖M1‖Lr≤4∑k=1‖G2,k(t))∗Vk0‖Lr+∫t04∑k=1‖G2,k(t−s)∗Qk(s))‖Lrds≤4∑k=1‖G2,kL(t))∗Vk0‖Lr+4∑k=1‖G2,kRH(t))∗Vk0‖Lr+∫t04∑k=1‖G2,kL(t−s)∗Qk(s))‖Lrds+∫t04∑k=1‖G2,kRH(t−s)∗Qk(s))‖Lrds=:4∑i=1Pi. | (4.23) |
For the estimate of the linear part, it is easy to check that
\begin{equation} P_{1}\leq C\|V_{0}\|_{L^1}\leq CE_{0},\ \ P_{2}\leq Ce^{-C_0t}\|V_{0}\|_{L^1}\leq CE_{0}. \end{equation} | (4.24) |
For the estimate of the nonlinear part, utilizing the definitions (3.4)–(3.7) of Q_i , we have
\begin{align} \begin{split} P_{3}&\leq\int_{0}^{t}\sum\limits_{k = 1}^4\|\mathbb{G}_{L}^{2,k}(t-s)*Q^k(s))\|_{L^\frac{1}{r}}ds\\ &\leq \int_{0}^{t}(1+t-s)^{-\frac{3}{2}(1-\frac{1}{r})-\frac{1}{2}}(\|n(s)\|_{L^1} +\|m(s)\|_{L^1})(\|v(s)\|_{L^{\infty}}+\|w(s)\|_{L^{\infty}})ds\\ &+\int_{0}^{t}(1+t-s)^{-\frac{3}{2}\cdot(1-\frac{1}{r})}(\|v(s)\|_{L^2} +\|w(s)\|_{L^2})(\|\nabla v(s)\|_{L^2}+\|\nabla w(s)\|_{L^2})ds\\ &+\int_{0}^{t}(1+t-s)^{-\frac{3}{2}\cdot(1-\frac{1}{r})}(\|n(s)\|_{L^2} +\|m(s)\|_{L^2})(\|\nabla n(s)\|_{L^2}+\|\nabla m(s)\|_{L^2})ds\\ &+\int_{0}^{t}(1+t-s)^{-\frac{3}{2}\cdot(1-\frac{1}{r})}(\|\nabla^2v(s)\|_{L^2} +\|\nabla^2w(s)\|_{L^2})(\|n(s)\|_{L^{\infty}}+\|m(s)\|_{L^{\infty}})ds\\ &\leq CE_{0}\int_{0}^{t}(1+t-s)^{-\frac{7}{8}}(1+s)^{-\frac{6}{5}}ds+ CE_{0}\int_{0}^{t}(1+t-s)^{-\frac{3}{8}}(1+s)^{-\frac{3}{4}}ds\\ &+CE_{0}^{\frac{2}{5}}\int_{0}^{t}(1+t-s)^{-\frac{3}{8}}(1+s)^{-\frac{6}{5}}ds\\ &\leq CE_{0}^{\frac{2}{5}}, \end{split} \end{align} | (4.25) |
and
\begin{align} \begin{split} P_{4}&\leq\int_{0}^{t}\sum\limits_{k = 1}^4\|\mathbb{G}_{RH}^{2,k}(t-s)*Q^k(s))\|_{L^\frac{1}{r}}ds\\ &\leq C\int_{0}^{t}e^{-C_0(t-s)}(\|n(s)\|_{L^1} +\|m(s)\|_{L^1})(\|v(s)\|_{L^{\infty}}+\|w(s)\|_{L^{\infty}})ds\\ &+C\int_{0}^{t}e^{-C_0(t-s)}(\|v(s)\|_{L^2} +\|w(s)\|_{L^2})(\|\nabla v(s)\|_{L^2}+\|\nabla w(s)\|_{L^2})ds\\ &+C\int_{0}^{t}e^{-C_0(t-s)}(\|n(s)\|_{L^2} +\|m(s)\|_{L^2})(\|\nabla n(s)\|_{L^2}+\|\nabla m(s)\|_{L^2})ds\\ &+C\int_{0}^{t}e^{-C_0(t-s)}(\|\nabla^2v(s)\|_{L^2} +\|\nabla^2w(s)\|_{L^2})(\|n(s)\|_{L^2}+\|m(s)\|_{L^2})ds\\ &\leq CE_{0}^{\frac{7}{5}}\int_{0}^{t}e^{-C_0(t-s)}(1+s)^{-\frac{6}{5}}ds+ CE_{0}\int_{0}^{t}e^{-C_0(t-s)}(1+s)^{-\frac{3}{4}}ds\\ &+CE_{0}\int_{0}^{t}e^{-C_0(t-s)}ds\\ &\leq CE_{0}^{\frac{2}{5}}. \end{split} \end{align} | (4.26) |
Combining (4.24)–(4.26), we obtain
\|M_{1}\|_{L^\frac{4}{3}}\leq CE_{0}^{\frac{2}{5}}. |
The same procedure gives
\sum\limits_{i = 1}^3(\|M_{i}\|_{L^\frac{4}{3}}+\|N_{i}\|_{L^\frac{4}{3}})\leq CE_{0}^{\frac{2}{5}}. |
So far, we can obtain
\|(v(t), w(t))\|_{L^\frac{4}{3}}\leq CE_{0}^{\frac{2}{5}}. |
Now, we can obtain the estimate of \|(v, w)\|_{L^1} by using \|(v, w)\|_{L^{\frac{4}{3}}}. For M_{1},
\begin{align*} \begin{split} \|M_{1}\|_{L^1}&\leq\sum\limits_{k = 1}^4\|\mathbb{G}_{L}^{2,k}(t))*V_{0}^k\|_{L^1}+\sum\limits_{k = 1}^4\|\mathbb{G}_{RH}^{2,k}(t))*V_{0}^k\|_{L^1}\\ &+\int_{0}^{t}\sum\limits_{k = 1}^4\|\mathbb{G}_{L}^{2,k}(t-s)*N^k(s))\|_{L^1}ds+\int_{0}^{t}\sum\limits_{k = 1}^4\|\mathbb{G}_{RH}^{2,k}(t-s)*N^k(s))\|_{L^1}ds\\ & = :\sum\limits_{i = 1}^4F_{i}. \\ \end{split} \end{align*} |
A simple check gives us that
F_{1}\leq C\|V_{0}\|_{L^1}\leq CE_{0}, |
F_{2}\leq Ce^{-C_0t}\|V_{0}\|_{L^1}\leq CE_{0}. |
Then, by using Lemma 4.6 and Young's inequality for convolution, we have
\begin{align} \begin{split} F_{3}&\leq\int_{0}^{t}\sum\limits_{k = 1}^4\|\mathbb{G}_{L}^{2,k}(t-s)*Q^k(s))\|_{L^1}ds\\ &\leq \int_{0}^{t}(1+t-s)^{-\frac{3}{2}(1-1)-\frac{1}{2}}(\|n(s)\|_{L^1} +\|m(s)\|_{L^1})(\|v(s)\|_{L^{\infty}}+\|w(s)\|_{L^{\infty}})ds\\ &+\int_{0}^{t}(1+t-s)^{-\frac{3}{2}\cdot(1-1)}(\|v(s)\|_{L^\frac{4}{3}} +\|w(s)\|_{L^\frac{4}{3}})(\|\nabla v(s)\|_{L^4}+\|\nabla w(s)\|_{L^4})ds\\ &+\int_{0}^{t}(1+t-s)^{-\frac{3}{2}\cdot(1-1)}(\|n(s)\|_{L^1} +\|m(s)\|_{L^1})(\|\nabla n(s)\|_{L^\infty}+\|\nabla m(s)\|_{L^\infty})ds\\ &+\int_{0}^{t}(1+t-s)^{-\frac{3}{2}\cdot(1-1)}(\|\nabla^2v(s)\|_{L^2} +\|\nabla^2w(s)\|_{L^2})(\|n(s)\|_{L^{\infty}}+\|m(s)\|_{L^{\infty}})ds\\ &\leq CE_{0}\int_{0}^{t}(1+t-s)^{-\frac{1}{2}}(1+s)^{-\frac{6}{5}}ds+CE_{0}^{\frac{2}{5}}\int_{0}^{t}(1+s)^{-\frac{11}{8}}ds+ CE_{0}\int_{0}^{t}(1+s)^{-2}ds\\ &+CE_{0}^{\frac{2}{5}}\int_{0}^{t}(1+s)^{-\frac{6}{5}}ds\\ &\leq CE_{0}^{\frac{2}{5}}, \end{split} \end{align} | (4.27) |
and
\begin{align} \begin{split} F_{4}&\leq\int_{0}^{t}\sum\limits_{k = 1}^4\|\mathbb{G}_{RH}^{2,k}(t-s)*Q^k(s))\|_{L^1}ds\\ &\leq C\int_{0}^{t}e^{-C_0(t-s)}(\|n(s)\|_{L^1} +\|m(s)\|_{L^1})(\|v(s)\|_{L^{\infty}}+\|w(s)\|_{L^{\infty}})ds\\ &+C\int_{0}^{t}e^{-C_0(t-s)}(\|v(s)\|_{L^2} +\|w(s)\|_{L^2})(\|\nabla v(s)\|_{L^2}+\|\nabla w(s)\|_{L^2})ds\\ &+C\int_{0}^{t}e^{-C_0(t-s)}(\|n(s)\|_{L^2} +\|m(s)\|_{L^2})(\|\nabla n(s)\|_{L^2}+\|\nabla m(s)\|_{L^2})ds\\ &+C\int_{0}^{t}e^{-C_0(t-s)}(\|\nabla^2v(s)\|_{L^2} +\|\nabla^2w(s)\|_{L^2})(\|n(s)\|_{L^2}+\|m(s)\|_{L^2})ds\\ &\leq CE_{0}^{\frac{7}{5}}\int_{0}^{t}e^{-C_0(t-s)}(1+s)^{-\frac{6}{5}}ds+ CE_{0}\int_{0}^{t}e^{-C_0(t-s)}(1+s)^{-\frac{3}{4}}ds\\ &+CE_{0}\int_{0}^{t}e^{-C_0(t-s)}ds\\ &\leq CE_{0}^{\frac{2}{5}}. \end{split} \end{align} | (4.28) |
Combining the estimate of each F_{i} , we obtain
\|M_{1}\|_{L^1}\leq CE_{0}^{\frac{2}{5}}. |
The same procedure gives
\sum\limits_{i = 1}^3(\|M_{i}\|_{L^1}+\|N_{i}\|_{L^1})\leq CE_{0}^{\frac{2}{5}}. |
So far, we can obtain
\|(v(t), w(t))\|_{L^1}\leq CE_{0}^{\frac{2}{5}}, |
and we complete the proof of the lemma.
Lemma 4.9. Under the assumption (4.15) and the condition (4.13), we have
\begin{align} \|\nabla^2V(t)\|_{L^{\infty}(\mathbb{R}^3)}\leq CE_{0}^{\frac{1}{8}}(1+t)^{-\frac{6}{5}},\ \ t\in [0,T]. \end{align} | (4.29) |
Proof. Applying Duhamel's principle, we have
D^2V(t) = D^2(\mathbb{G}(t)*V_{0})+\displaystyle{\int}_{0}^{t}D^2(\mathbb{G}(t-s)*N(V)(s))ds. |
Then,
\begin{equation*} \|D^2V(t)\|_{L^{\infty}(\mathbb{R}^3)}\leq\|D^2(\mathbb{G}(t)*V_{0})\|_{L^{\infty}(\mathbb{R}^3)}+\displaystyle{\int}_{0}^{t}\|D^2(\mathbb{G}(t-s)*N(V)(s))\| _{L^{\infty}(\mathbb{R}^3)}ds.\\ \end{equation*} |
Now we estimate the righthand side of the inequality. Simple computation yields
\begin{align} \begin{split} &\ \ \ \|D^2(\mathbb{G}(t)*V_{0})\|_{L^{\infty}(\mathbb{R}^3)}\\ &\leq \|D^2(\mathbb{G}_{L}(t)*V_{0})\|_{L^{\infty}(\mathbb{R}^3)}+\|D^2(\mathbb{G}_{RH}(t)*V_{0})\|_{L^{\infty}(\mathbb{R}^3)} +\|D^2(S(t)*V_{0})\|_{L^{\infty}(\mathbb{R}^3)}\\ &\leq C\|V_{0}\|_{L^1(\mathbb{R}^3)}(1+t)^{-\frac{5}{2}}+C\|D\mathbb{G}_{RH}\|_{L^2(\mathbb{R}^3)}\|DV_{0}\|_{L^2(\mathbb{R}^3)} +C\|D^2V_{0}\|_{L^{\infty}(\mathbb{R}^3)}e^{-C_0t}\\ &\leq C\|V_{0}\|_{L^1(\mathbb{R}^3)}(1+t)^{-\frac{5}{2}}+C\|V_{0}\|_{L^2(\mathbb{R}^3)}^{\frac{1}{2}}\|D^2V_{0}\|_{L^2(\mathbb{R}^3)}^ {\frac{1}{2}}e^{-C_0t}+C\|V_{0}\|_{L^2(\mathbb{R}^3)}^{\frac{1}{8}}\|D^4V_{0}\|_{L^2(\mathbb{R}^3)}^{\frac{7}{8}}e^{-C_0t}\\ &\leq C(\|V_{0}\|_{L^1(\mathbb{R}^3)}+\|V_{0}\|_{L^2(\mathbb{R}^3)}^{\frac{1}{8}})(1+t)^{-\frac{6}{5}}\\ &\leq CE_{0}^{\frac{1}{8}}(1+t)^{-\frac{6}{5}},\\ \end{split} \end{align} | (4.30) |
and
\begin{align*} \begin{split} &\ \ \ \int_{0}^{t}\|D^2(\mathbb{G}(t-s)*N(V)(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &\leq\int_{0}^{t}\|D^2(\mathbb{G}_{L}(t-s)*N(V)(s))\|_{L^{\infty}(\mathbb{R}^3)}ds+\int_{0}^{t}\|D^2(\mathbb{G}_{RH}(t-s)*N(V)(s))\| _{L^{\infty}(\mathbb{R}^3)}ds\\ &+\int_{0}^{t}\|D^2(S(t-s)*N(V)(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ & = :\sum\limits_{i = 1}^{3}O_i. \\ \end{split} \end{align*} |
Now we turn to estimate each O_i. By the definition (4.20) of N(V) , we easily check that
\begin{align} \begin{split} O_{1}&\leq \int_{0}^{t}\|D^2\mathbb{G}_{L}(t-s)\|_{L^2}(\|n(s)\|_{L^2}+\|m(s)\|_{L^2}+\|v(s)\|_{L^2}+\|w(s)\|_{L^2})\\ &(\|\nabla v(s)\|_{L^{\infty}}+\|\nabla w(s)\|_{L^{\infty}}+\|\nabla n(s)\|_{L^{\infty}}+\|\nabla m(s)\|_{L^{\infty}})ds\\ &+\int_{0}^{t}\|D^2\mathbb{G}_{L}(t-s)\|_{L^2}(\|v(s)\|_{L^2} +\|w(s)\|_{L^2})(\|\nabla v(s)\|_{L^{\infty}}+\|\nabla w(s)\|_{L^{\infty}})ds\\ &+\int_{0}^{t}\|D^2\mathbb{G}_{L}(t-s)\|_{L^2}(\|n(s)\|_{L^2}+\|m(s)\|_{L^2})\|\nabla n(s)\|_{L^{\infty}}+\|\nabla m(s)\|_{L^{\infty}})ds\\ &+\int_{0}^{t}\|D^2\mathbb{G}_{L}(t-s)\|_{L^2}(\|\nabla^2v(s)\|_{L^2} +\|\nabla^2w(s)\|_{L^2})(\|n(s)\|_{L^{\infty}}+\|m(s)\|_{L^{\infty}})ds. \end{split} \end{align} | (4.31) |
Using Sobolev's inequality, we obtain
\|(n(s), m(s))\|_{L^{\infty}}\leq CE_{0}^{\frac{2}{5}}(1+s)^{-\frac{6}{5}}, |
then for O_{1} , we have
\begin{align} \begin{split} O_{1}&\leq CE_{0}\int_{0}^{t}(1+t-s)^{-\frac{3}{4}-1}(1+s)^{-2}ds\\ &+CE_{0}^{\frac{2}{5}}\int_{0}^{t} (1+t-s)^{-\frac{3}{4}-1}(1+s)^{-\frac{6}{5}}ds\\ &\leq CE_{0}^{\frac{2}{5}}(1+t)^{-\frac{6}{5}}. \end{split} \end{align} | (4.32) |
For O_{2} , we can get
\begin{align} \begin{split} O_{2}&\leq \int_{0}^{t}\|\nabla \mathbb{G}_{RH}(t-s)\|_{L^2}[(\|\nabla^2v(s)\|_{L^2} +\|\nabla^2w(s)\|_{L^2})(\|n(s)\|_{L^{\infty}}+\|m(s)\|_{L^{\infty}})\\ &+(\|\nabla v(s)\|_{L^2} +\|\nabla w(s)\|_{L^2})(\|\nabla n(s)\|_{L^{\infty}}+\|\nabla m(s)\|_{L^{\infty}})\\ &+(\|\nabla^2n(s)\|_{L^2} +\|\nabla^2m(s)\|_{L^2})(\|v(s)\|_{L^{\infty}}+\|w(s)\|_{L^{\infty}})]ds\\ &+\int_{0}^{t}\|\nabla \mathbb{G}_{RH}(t-s)\|_{L^2}[(\|\nabla v(s)\|_{L^2} +\|\nabla w(s)\|_{L^2})(\|\nabla v(s)\|_{L^{\infty}}+\|\nabla w(s)\|_{L^{\infty}})\\ &+(\|\nabla^2v(s)\|_{L^2} +\|\nabla^2w(s)\|_{L^2})(\|v(s)\|_{L^{\infty}}+\|w(s)\|_{L^{\infty}})]ds\\ &+\int_{0}^{t}\|\nabla \mathbb{G}_{RH}(t-s)\|_{L^2}[(\|\nabla n(s)\|_{L^2} +\|\nabla m(s)\|_{L^2})(\|\nabla n(s)\|_{L^{\infty}}+\|\nabla m(s)\|_{L^{\infty}})\\ &+(\|\nabla^2n(s)\|_{L^2} +\|\nabla^2m(s)\|_{L^2})(\|n(s)\|_{L^{\infty}}+\|m(s)\|_{L^{\infty}})]ds\\ &+\int_{0}^{t}\|\nabla \mathbb{G}_{RH}(t-s)\|_{L^2}[(\|\nabla^2v(s)\|_{L^2} +\|\nabla^2w(s)\|_{L^2})(\|\nabla n(s)\|_{L^{\infty}}+\|\nabla m(s)\|_{L^{\infty}})\\ &+(\|\nabla^3v(s)\|_{L^2} +\|\nabla^3w(s)\|_{L^2})(\|n(s)\|_{L^{\infty}}+\|m(s)\|_{L^{\infty}})]ds. \end{split} \end{align} | (4.33) |
Also by Sobolev's inequality, it holds
\|(\nabla^2v(s), \nabla^2w(s))\|_{L^2}\leq C\|(v(s), w(s))\|_{L^2}^{\frac{1}{3}}\|(\nabla^3v(s), \nabla^3w(s))\|_{L^2}^{\frac{2}{3}}, |
then we have
\begin{align} \begin{split} O_{2}&\leq CE_{0}^{\frac{2}{5}}\int_{0}^{t}e^{-C_0(t-s)}(1+s)^{-\frac{6}{5}}ds+ CE_{0}^{\frac{1}{4}}\int_{0}^{t}e^{-C_0(t-s)}(1+s)^{-2}ds\\ &+CE_{0}^{\frac{1}{3}}\int_{0}^{t}e^{-C_0(t-s)}(1+s)^{-2}ds\\ &\leq CE_{0}^{\frac{1}{3}}(1+t)^{-\frac{6}{5}}. \end{split} \end{align} | (4.34) |
For O_{3} , we have
\begin{align} \begin{split} O_{3}&\leq C\int_{0}^{t}e^{-C_0(t-s)}[(\|n(s)\|_{L^{\infty}}+\|m(s)\|_{L^{\infty}}) (\|\nabla^3v(s)\|_{L^{\infty}}+\|\nabla^3w(s)\|_{L^{\infty}})\\ &+(\|\nabla n(s)\|_{L^{\infty}}+\|\nabla m(s)\|_{L^{\infty}}) (\|\nabla^2v(s)\|_{L^{\infty}}+\|\nabla^2w(s)\|_{L^{\infty}})\\ &+(\|\nabla^2n(s)\|_{L^{\infty}}+\|\nabla^2m(s)\|_{L^{\infty}}) (\|\nabla v(s)\|_{L^{\infty}}+\|\nabla w(s)\|_{L^{\infty}})\\ &+(\|\nabla^3n(s)\|_{L^{\infty}}+\|\nabla^3m(s)\|_{L^{\infty}}) (\|v(s)\|_{L^{\infty}}+\|w(s)\|_{L^{\infty}})]ds. \end{split} \end{align} | (4.35) |
Sobolev's inequality gives
\begin{align} \begin{split} &\ \ \ \|(\nabla^2n(s), \nabla^2v(s), \nabla^2m(s), \nabla^2w(s))\|_{L^{\infty}}\\ &\leq C\|(n(s), v(s), m(s), w(s))\|_{L^2}^{\frac{1}{8}}\|(\nabla^4n(s), \nabla^4v(s), \nabla^4m(s), \nabla^4w(s))\|_{L^{2}}^{\frac{7}{8}}, \end{split} \end{align} | (4.36) |
then we have
\begin{align} \begin{split} O_{3}&\leq CE_{0}^{\frac{2}{5}}\int_{0}^{t}e^{-C_0(t-s)}(1+s)^{-\frac{6}{5}}ds+ CE_{0}^{\frac{1}{8}}\int_{0}^{t}e^{-C_0(t-s)}(1+s)^{-2}ds\\ &\leq CE_{0}^{\frac{1}{8}}(1+t)^{-\frac{6}{5}}ds. \end{split} \end{align} | (4.37) |
Combining (4.32), (4.34), and (4.37), we obtain
\begin{align} \begin{split} &\ \ \ \int_{0}^{t}\|D^2(\mathbb{G}(t-s)*N(V)(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &\leq CE_{0}^{\frac{1}{8}}(1+t)^{-\frac{6}{5}}. \end{split} \end{align} | (4.38) |
Using (4.30) and (4.38), we complete the proof of the lemma.
With the above lemmas at hand, we now check the condition 1 of Lemma 4.4, that is, the following lemma.
Lemma 4.10. Under the assumption (4.15) and the condition (4.13), we have
\begin{equation*} (\|\nabla n(t)\|_{L^{\infty}(\mathbb{R}^3)}, \|\nabla v(t)\|_{L^{\infty}(\mathbb{R}^3)}, \|\nabla m(t)\|_{L^{\infty}(\mathbb{R}^3)},\|\nabla w(t)\|_{L^{\infty}(\mathbb{R}^3)})\leq \frac{\delta}{2}(1+t)^{-2}. \end{equation*} |
Proof. Duhamel's principle gives rise to
\begin{equation*} \nabla V(t) = \nabla (\mathbb{G}*V_{0})(t)+\displaystyle{\int}_{0}^{t}\nabla(\mathbb{G}(t-s)*N(V)(s))ds. \end{equation*} |
Then
\begin{equation*} \|\nabla V(t)\|_{L^{\infty}(\mathbb{R}^3)}\leq\|\nabla(\mathbb{G}*V_{0})(t)\|_{L^{\infty}(\mathbb{R}^3)} +\displaystyle{\int}_{0}^{t}\|\nabla(\mathbb{G}(t-s)*N(V)(s))\| _{L^{\infty}(\mathbb{R}^3)}ds.\\ \end{equation*} |
Similar to the proof of Lemma 4.9, it holds for the first term on the righthand side of the inequality that
\begin{equation} \|\nabla(\mathbb{G}*V_{0})(t)\|_{L^{\infty}(\mathbb{R}^3)}\leq \frac{\delta}{4}(1+t)^{-2}. \end{equation} | (4.39) |
For for the second term on the right side of the inequality, it is easy to get that
\begin{align*} \begin{split} &\ \ \ \int_{0}^{t}\|\nabla(\mathbb{G}(t-s)*N(V)(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &\leq\int_{0}^{t}\|\nabla(\mathbb{G}_{L}(t-s)*N(V)(s))\|_{L^{\infty}(\mathbb{R}^3)}ds+\int_{0}^{t}\|\nabla(\mathbb{G} _{RH}(t-s)*N(V)(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &+\int_{0}^{t}\|\nabla(S(t-s)*N(V)(s))\|_{L^{\infty}(\mathbb{R}^3)}ds. \end{split} \end{align*} |
For the nonlinear part of lower frequency, Lemma 4.5 gives
\begin{align} \begin{split} &\ \ \ \int_{0}^{t}\|\nabla(\mathbb{G}_{L}^{k,\{1\}}(t-s)*N(V)^1(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ & = \int_{0}^{\frac{t}{2}}\|\nabla(\mathbb{G}_{L}^{k,\{1\}}(t-s)*N(V)^1(s))\|_{L^{\infty}(\mathbb{R}^3)}ds+ \int_{\frac{t}{2}}^{t}\|\nabla(\mathbb{G}_{L}^{k,\{1\}}(t-s)*N(V)^1(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &\leq\int_{0}^{\frac{t}{2}}\|D^2\mathbb{G}_{L}^{k,\{1\}}(t-s)\|_{L^{\infty}(\mathbb{R}^3)} \|(n(s), m(s)\|_{L^1(\mathbb{R}^3)}\|(v(s), w(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &+\int_{\frac{t}{2}}^{t}\|\nabla \mathbb{G}_{L}^{k,\{1\}}(t-s)\|_{L^1(\mathbb{R}^3)}[(\|(n(s), m(s))\|_{L^{\infty}(\mathbb{R}^3)}\|(\nabla v(s), \nabla w(s))\|_{L^{\infty}(\mathbb{R}^3)}\\ &+\|(v(s), w(s))\|_{L^{\infty}(\mathbb{R}^3)}\|(\nabla n(s), \nabla m(s))\|_{L^{\infty}(\mathbb{R}^3)})]ds\\ &\leq CE_{0}\int_{0}^{\frac{t}{2}}(1+t-s)^{-\frac{5}{2}}(1+s)^{-\frac{6}{5}}ds\\ &+CE_{0}^{\frac{2}{5}}\int_{\frac{t}{2}}^{t} (1+t-s)^{-\frac{1}{2}}(1+s)^{-\frac{6}{5}-2}ds\\ &\leq CE_{0}^{\frac{2}{5}}((1+t)^{-\frac{5}{2}}+(1+t)^{\frac{27}{10}})\\ &\leq CE_{0}^{\frac{2}{5}}(1+t)^{-2}. \\ \end{split} \end{align} | (4.40) |
For the terms of \int_{0}^{t}\|\nabla(\mathbb{G}_{L}^{k, \{2, 3, 4\}}(t-s)*N(V)^{\{2, 3, 4\}}(s))\|_{L^{\infty}(\mathbb{R}^3)}ds, Lemma 4.5 gives
\begin{align*} \begin{split} &\ \ \ \int_{0}^{t}\|\nabla(\mathbb{G}_{L}^{k,\{2,3,4\}}(t-s)*N(V)^{\{2,3,4\}}(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &\leq\int_{0}^{\frac{t}{2}}\|\nabla \mathbb{G}_{L}^{k,\{2,3,4\}}(t-s)\|_{L^{\infty}(\mathbb{R}^3)} \|(v(s), w(s))\|_{L^1(\mathbb{R}^3)}\|(\nabla v(s), \nabla w(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &+\int_{\frac{t}{2}}^{t}\|\nabla \mathbb{G}_{L}^{k,\{2,3,4\}}(t-s)\|_{L^1(\mathbb{R}^3)} \|(v(s), w(s))\|_{L^{\infty}(\mathbb{R}^3)}\|(\nabla v(s), \nabla w(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &+\int_{0}^{\frac{t}{2}}\|\nabla \mathbb{G}_{L}^{k,\{2,3,4\}}(t-s)\|_{L^{\infty}(\mathbb{R}^3)} \|(n(s), m(s))\|_{L^1(\mathbb{R}^3)}\|(\nabla n(s), \nabla m(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &+\int_{\frac{t}{2}}^{t}\|\nabla \mathbb{G}_{L}^{k,\{2,3,4\}}(t-s)\|_{L^1(\mathbb{R}^3)} \|(n(s), m(s))\|_{L^{\infty}(\mathbb{R}^3)}\|(\nabla n(s), \nabla m(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &+\int_{0}^{\frac{t}{2}}\|\nabla \mathbb{G}_{L}^{k,\{2,3,4\}}(t-s)\|_{L^{\infty}(\mathbb{R}^3)} \|(n(s), m(s))\|_{L^1(\mathbb{R}^3)}\|(\nabla^2v(s), \nabla^2w(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &+\int_{\frac{t}{2}}^{t}\|\nabla \mathbb{G}_{L}^{k,\{2,3,4\}}(t-s)\|_{L^2(\mathbb{R}^3)} \|(\nabla n(s), \nabla m(s))\|_{L^{\infty}(\mathbb{R}^3)}\|(\nabla v(s), \nabla w(s))\|_{L^2(\mathbb{R}^3)}ds\\ &+\int_{\frac{t}{2}}^{t}\|\nabla^2\mathbb{G}_{L}^{k,\{2,3,4\}}(t-s)\|_{L^2(\mathbb{R}^3)} \|(n(s), m(s))\|_{L^2(\mathbb{R}^3)}\|(\nabla v(s), \nabla w(s))\|_{L^{\infty}(\mathbb{R}^3)}ds. \end{split} \end{align*} |
By Lemmas 3.2 and 4.5–4.9, we have
\begin{align} \begin{split} &\ \ \ \int_{0}^{t}\|\nabla(\mathbb{G}_{L}^{k,\{2,3,4\}}(t-s)*N(V)^{\{2,3,4\}}(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &\leq CE_{0}\int_{0}^{\frac{t}{2}}(1+t-s)^{-2}(1+s)^{-2}ds\\ &+CE_{0}^{\frac{2}{5}}\int_{\frac{t}{2}}^{t}(1+t-s)^{-\frac{1}{2}}(1+s)^{-2-\frac{6}{5}}ds\\ &+CE_{0}\int_{0}^{\frac{t}{2}}(1+t-s)^{-2}(1+s)^{-2}ds\\ &+CE_{0}^{\frac{9}{8}}\int_{0}^{\frac{t}{2}}(1+t-s)^{-2}(1+s)^{-\frac{6}{5}}ds\\ &+CE_{0}^{\frac{2}{5}}\int_{\frac{t}{2}}^{t}(1+t-s)^{-\frac{1}{2}-\frac{3}{4}}(1+s)^{-2-\frac{6}{5}}ds\\ &+CE_{0}\int_{\frac{t}{2}}^{t}(1+t-s)^{-\frac{3}{4}-1}(1+s)^{-2}ds\\ &\leq CE_{0}^{\frac{2}{5}}(1+t)^{-2}. \end{split} \end{align} | (4.41) |
The same procedure gives
\begin{align} \begin{split} &\ \ \ \int_{0}^{t}\|\nabla(\mathbb{G}_{L}^{k,5}(t-s)*N(V)^5(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &\leq CE_{0}^{\frac{2}{5}}(1+t)^{-2}, \end{split} \end{align} | (4.42) |
and
\begin{align} \begin{split} &\ \ \ \int_{0}^{t}\|\nabla(\mathbb{G}_{L}^{k,\{6,7,8\}}(t-s)*N(V)^{\{6,7,8\}}(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &\leq CE_{0}^{\frac{2}{5}}(1+t)^{-2}. \end{split} \end{align} | (4.43) |
Combining (4.40), (4.41), (4.42), and (4.43), we have
\begin{align} \begin{split} &\ \ \ \int_{0}^{t}\|\nabla(\mathbb{G}_{L}(t-s)*N(V)(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &\leq CE_{0}^{\frac{2}{5}}(1+t)^{-2}. \end{split} \end{align} | (4.44) |
For the nonlinear part of high frequency, Lemma 4.5 gives
\begin{align} \begin{split} &\ \ \ \int_{0}^{t}\|\nabla(\mathbb{G}_{RH}^{k,\{1,5\}}(t-s)*N(V)^{\{1,5\}}(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &\leq \int_{0}^{t}\|\nabla\mathbb{G}_{RH}^{k,\{1,5\}}(t-s)\|_{L^2(\mathbb{R}^3)} [(\|(n(s), m(s))\|_{L^2(\mathbb{R}^3)}\|(\nabla v(s), \nabla w(s))\|_{L^{\infty}(\mathbb{R}^3)}\\ &+(\|(\nabla n(s), \nabla m(s))\|_{L^{\infty}(\mathbb{R}^3)}\|(v(s), w(s))\|_{L^2(\mathbb{R}^3)}]ds\\ &\leq CE_{0}\int_{0}^{t}e^{-C_0(t-s)}(1+s)^{-2}ds\\ &\leq CE_{0}(1+t)^{-2}, \end{split} \end{align} | (4.45) |
and
\begin{align} \begin{split} &\ \ \ \int_{0}^{t}\|\nabla(\mathbb{G}_{RH}^{k,\{2,3,4,6,7,8\}}(t-s)*N(V)^{\{2,3,4,6,7,8\}}(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &\leq \int_{0}^{t}\|\nabla\mathbb{G}_{RH}^{k,\{2,3,4,6,7,8\}}(t-s)\|_{L^2(\mathbb{R}^3)}[(\|(v(s), w(s))\|_{L^2(\mathbb{R}^3)}\|(\nabla v(s), \nabla w(s))\|_{L^{\infty}(\mathbb{R}^3)}\\ &+(\|(n(s), m(s))\|_{L^2(\mathbb{R}^3)}\|(\nabla n(s), \nabla m(s))\|_{L^{\infty}(\mathbb{R}^3)}]ds\\ &+\int_{0}^{t}\|\nabla\mathbb{G}_{RH}^{k,\{2,3,4,6,7,8\}}(t-s)\|_{L^1(\mathbb{R}^3)} (\|(n(s), m(s))\|_{L^{\infty}(\mathbb{R}^3)})\|(\nabla^2v(s), \nabla^2w(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &\leq CE_{0}\int_{0}^{t}e^{-C_0(t-s)}(1+s)^{-2}ds+CE_{0}^{\frac{21}{40}}\int_{0}^{t}e^{-C_0(t-s)}(1+s)^{-\frac{6}{5}-\frac{6}{5}}ds\\ &\leq CE_{0}^{\frac{21}{40}}(1+t)^{-2}. \end{split} \end{align} | (4.46) |
(4.45) and (4.46) gives
\begin{align} \begin{split} &\ \ \ \int_{0}^{t}\|\nabla(\mathbb{G}_{RH}(t-s)*N(V)(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &\leq CE_{0}^{\frac{21}{40}}(1+t)^{-2}. \end{split} \end{align} | (4.47) |
For the nonlinear part estimate for the singular part, by Lemmas 4.6 and 4.9, we have
\begin{align} \begin{split} &\ \ \ \int_{0}^{t}\|\nabla(S(t-s)*N(V)(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &\leq C\int_{0}^{t}e^{-C_0(t-s)}[\|(n(s), m(s))\|_{L^{\infty}(\mathbb{R}^3)}\|(\nabla^2v(s), \nabla^2w(s))\|_{L^{\infty}(\mathbb{R}^3)}\\ &+\|(\nabla n(s), \nabla m(s))\|_{L^{\infty}(\mathbb{R}^3)}\|(\nabla v(s), \nabla w(s))\|_{L^{\infty}(\mathbb{R}^3)}\\ &+\|(\nabla^2n(s), \nabla^2m(s))\|_{L^{\infty}(\mathbb{R}^3)}\|(v(s), w(s))\|_{L^{\infty}(\mathbb{R}^3)}]ds\\ &\leq CE_{0}^{\frac{21}{40}}\int_{0}^{t}e^{-C_0(t-s)}(1+s)^{-\frac{6}{5}-\frac{6}{5}}ds + CE_{0}^{\frac{1}{6}}\int_{0}^{t}e^{-C_0(t-s)}(1+s)^{-2}ds\\ &\leq CE_{0}^{\frac{1}{6}}(1+t)^{-2}. \end{split} \end{align} | (4.48) |
Combining (4.43), (4.47), and (4.48), we obtain
\begin{align} \begin{split} &\ \ \ \int_{0}^{t}\|\nabla(\mathbb{G}(t-s)*N(V)(s))\|_{L^{\infty}(\mathbb{R}^3)}ds\\ &\leq CE_{0}^{\frac{1}{6}}(1+t)^{-2}. \end{split} \end{align} | (4.49) |
By (4.39) and (4.49), we complete the proof of the lemma.
Under the condition (4.13), the bootstrap argument and the local existence of the solutions for the syestem (1.1) and (1.2) gives the following result.
Proposition 4.1. Let (\rho_{1, 0}-1, u_{1, 0}, \rho_{2, 0}-1, u_{2, 0}, \nabla\Phi_{0}) \in H^{s+1}(s\geq4), and
\|\rho_{1,0}-1, u_{1,0}, \rho_{2,0}-1, u_{2,0}, \nabla\Phi_{0})\|_{L^2(\mathbb{R}^3)\bigcap L^1(\mathbb{R}^3)}\leq E_0, |
where E_0 is sufficiently small, then the Cauchy problem (1.1) and (1.2) has a global solution in time that satisfies
(\rho_1-1, u_1, \rho_2-1, u_2)\in L^{\infty}([0,\infty);H^{s+1}),\ \ \nabla (u_1, u_2)\in L^{2}([0,\infty);H^{s+1}), |
and
\begin{equation} (\|(\nabla \rho_1, \nabla u_1, \nabla \rho_2, \nabla u_2)(t)\|_{L^{\infty}(\mathbb{R}^3)}\leq CE_{0}^{\frac{1}{6}}(1+t)^{-2},\ \ t\in (0,+\infty). \end{equation} | (4.50) |
In this section, we would like to get the decay rate of solutions. The main result is stated as follows.
Proposition 5.1. Let (\rho_{1, 0}-1, u_{1, 0}, \rho_{2, 0}-1, u_{2, 0}, \nabla\Phi_{0})\in H^{s+1}(s\geq4), and
\|\rho_{1,0}-1, u_{1,0}, \rho_{2,0}-1, u_{2,0}, \nabla\Phi_{0})\|_{L^2(\mathbb{R}^3)\bigcap L^1(\mathbb{R}^3)}\leq E_0, |
where E_0 is sufficiently small, (\rho_1, u_1, \rho_2, u_2) is the solutions for the Cauchy problem (1.1) and (1.2), then when 2\leq p\leq +\infty and \alpha = (\alpha_{1}, \alpha_{2}, \alpha_{3}), \alpha_i\geq 0, |\alpha|\leq s-1, it holds
\|D^\alpha (\rho_1-1, u_1, \rho_2-1, u_2)(\cdot,t)\|_{L^p}\leq C(1+t)^{-\frac{3}{2}(1-\frac{1}{p})-\frac{|\alpha|}{2}}. |
Proof. From (3.1) and (3.2), the decay rate of (\rho_1-1, u_1, \rho_2-1, u_2) is equivalent to the decay rate of (n, v, m, w) . Therefore, we only need to consider the attenuation estimate of (n, v, m, w) . Note that if the L^2 decay rate of the higher-order spatial derivatives of the solution are obtained, then the general L^q decay rate of the solution follows by the Sobolev interpolation. For instance, using the Sobolev embedding theorem, we have
\begin{equation*} \|D^{\alpha}V(t)\|_{L^\infty}\leq \|D^{\alpha}V(t)\|_{L^2}^\frac{1}{4} \|D^{\alpha+2}V(t)\|_{L^2}^\frac{3}{4},\\ \end{equation*} |
where V(t) is defined as (3.10). So we only consider the decay of \|D^{\alpha}V(t)\|_{L^2}. Below we will prove the following assertion by induction,
\begin{equation} \|D^\alpha V(\cdot,t)\|_{L^2}\leq C(1+t)^{-\frac{3}{4}-\frac{|\alpha|}{2}},\ \ 0\leq |\alpha|\leq s+1. \end{equation} | (5.1) |
When |\alpha| = 0, we have
\begin{align*} \begin{split} &\ \ \ \|V(t)\|_{L^2}\\ &\leq\|\mathbb{G}(t)*V_{0}\|_{L^2}+\int_{0}^{t}\|\mathbb{G}(t-s)*N(V)(s)\| _{L^2}ds.\\ &\leq \|\mathbb{G}_{L}(t)\|_{L^2}\|V_{0}\|_{L^1}+\|\mathbb{G}_{RH}(t)\|_{L^2}\|V_{0}\|_{L^1}+\|S*V_{0}\|_{L^2}\\ &+\int_{0}^{t}\|\mathbb{G}_{L}(t-s)*N(V)(s))\|_{L^2}ds+\int_{0}^{t}\|\mathbb{G}_{RH}(t-s)*N(V)(s))\|_{L^2}ds\\ &+\int_{0}^{t}\|S(t-s)*N(V)(s)\|_{L^2}ds. \end{split} \end{align*} |
For the terms \int_{0}^{t}\|\mathbb{G}_{L}(t-s)*N(V)(s))\|_{L^2}ds and \int_{0}^{t}\|\mathbb{G}_{RH}(t-s)*N(V)(s))\|_{L^2}ds , we have
\begin{align*} \begin{split} &\ \ \ \int_{0}^{t}\|\mathbb{G}_{L}(t-s)*N(V)(s))\|_{L^2}+\|\mathbb{G}_{RH}(t-s)*N(V)(s))\|_{L^2}ds\\ &\leq \int_{0}^{t}(\|\mathbb{G}_{L}(t-s)\|_{L^2}+\|\mathbb{G}_{RH}(t-s)\|_{L^2})[\|(n, m)(s)\|_{L^1}\|(\nabla v, \nabla w)(s)\|_{L^{\infty}}\\ &+\|(v, w)(s)\|_{L^1}\|(\nabla n, \nabla m)(s)\|_{L^{\infty}}]ds\\ &+\int_{0}^{t}(\|\mathbb{G}_{L}(t-s)\|_{L^2}+\|\mathbb{G}_{RH}(t-s)\|_{L^2})\|(n, m)(s)\|_{L^1}\|(\nabla n, \nabla m)(s)\|_{L^{\infty}}ds\\ &+\int_{0}^{t}(\|\mathbb{G}_{L}(t-s)\|_{L^2}+\|\mathbb{G}_{RH}(t-s)\|_{L^2})\|(v, w)(s)\|_{L^1}\|(\nabla v, \nabla w)(s)\|_{L^{\infty}}ds\\ &+\int_{0}^{t}(\|\nabla\mathbb{G}_{L}(t-s)\|_{L^2}+\|\nabla\mathbb{G}_{RH}(t-s)\|_{L^2})\|(n, m)(s)\|_{L^1}\|(\nabla v, \nabla w)(s)\|_{L^{\infty}}ds\\ &+\int_{0}^{t}(\|\mathbb{G}_{L}(t-s)\|_{L^2}+\|\mathbb{G}_{RH}(t-s)\|_{L^2})\|(\nabla n, \nabla m)(s)\|_{L^2}\|(\nabla v, \nabla w)(s)\|_{L^2}ds. \end{split} \end{align*} |
Then, from Proposition (4.1), using Sobolev's inequality, we obtain
\begin{align*} \begin{split} &\ \ \ \int_{0}^{t}\|\mathbb{G}_{L}(t-s)*N(V)(s))\|_{L^2}ds\\ &\leq C\int_{0}^{t}(1+t-s)^{-\frac{3}{4}}(1+s)^{-2}ds+C\int_{0}^{t}(1+t-s)^{-\frac{3}{4}-\frac{1}{2}}(1+s)^{-2}ds\\ &+ C\int_{0}^{t}(1+t-s)^{-\frac{3}{4}}(1+s)^{-\frac{3}{4}-\frac{3}{4}}ds\\ &\leq C(1+t)^{-\frac{3}{4}}, \end{split} \end{align*} |
and
\begin{align*} \begin{split} &\ \ \ \int_{0}^{t}\|\mathbb{G}_{RH}(t-s)*N(V)(s))\|_{L^2}ds\\ &\leq C\int_{0}^{t}e^{-C_0(t-s)}(1+s)^{-2}ds + C\int_{0}^{t}e^{-C_0(t-s)}(1+s)^{-\frac{3}{4}-\frac{3}{4}}ds\\ &\leq C(1+t)^{-\frac{3}{4}}. \end{split} \end{align*} |
For the term \int_{0}^{t}\|S(t-s)*N(V)(s)\|_{L^2}ds , from Proposition (4.1), we have
\begin{align*} \begin{split} &\ \ \ \int_{0}^{t}\|S(t-s)*N(V)(s)\|_{L^2}ds\\ &\leq C\int_{0}^{t}e^{-C_0(t-s)}(\|(n, m)(s)\|_{L^2}\|(\nabla v, \nabla w)(s)\|_{L^{\infty}}+ \|(v, w)(s)\|_{L^2}\|(\nabla n, \nabla m)(s)\|_{L^{\infty}})ds\\ &\leq C\int_{0}^{t}e^{-C_0(t-s)}(1+s)^{-2}ds\\ &\leq C(1+t)^{-\frac{3}{4}}. \end{split} \end{align*} |
We assume that (5.1) holds when |\alpha| = k-1, and later we shall prove that (5.1) holds when |\alpha| = k. To get the estimate of \|D^{k}V(t)\|_{L^2}, we perform high and low frequency decomposition of the solution itself. The low frequency part of the solution V is
V_{L} = \chi(D)V, |
and the high frequency part of the solution V is
V_{H} = (1-\chi(D))V, |
where \chi(D) is a pseudo-differential operator with symbol \chi(\xi), we can see (3.14) for the definition of \chi(\xi). Then, we can decompose D^{k}V(t) as follows:
\begin{equation} D^{k}V(t)_{L} = D^k(\mathbb{G}\ast(V_{0})_{L})+\displaystyle{\int}_{0}^{t}D^k(\mathbb{G}(t-s)\ast (N(V)(s))_{L})ds, \end{equation} | (5.2) |
\begin{equation} D^{k}V(t)_{H} = D^k(\mathbf{G}\ast(V_{0})_{H})+\displaystyle{\int}_{0}^{t}D^k(\mathbf{G}(t-s)\ast (\widetilde{N(V)}(s))_{H})ds, \end{equation} | (5.3) |
where \mathbb{G} , \mathbf{G} , and N(V) are defined in (3.12), (3.13), and (4.20), respectively. We define \widetilde{N(V)} as follows:
\begin{align} \begin{split} \widetilde{N(V)} = :\begin{pmatrix} Q_{1}\\ Q_{2}\\ \widetilde{Q_{3}}\\ Q_{4}\\ \end{pmatrix}.\\ \end{split} \end{align} | (5.4) |
Here, \widetilde{Q_{3}} = Q_{3}+2\nabla\triangle^{-1}m, and Q_{1}, Q_{2}, Q_{3} , and Q_{4} are defined in (3.4)–(3.7). For D^{k}V(t) , we have
D^{k}V(t) = D^{k}V(t)_L+D^{k}V(t)_H. |
Then, (5.2) and (5.3) are equal to
\begin{equation} D^{k}V(t)_{L} = D^{k}(\mathbb{G}_{\widetilde{L}}\ast(V_{0})_{L})+\displaystyle{\int}_{0}^{t}D^{k}(\mathbb{G}_{\widetilde{L}}(t-s)\ast (N(V)(s))_{L})ds, \end{equation} | (5.5) |
\begin{equation} D^{k}V(t)_{H} = D^{k}(\mathbf{G}_{\widetilde{H}}\ast(V_{0})_{H})+\displaystyle{\int}_{0}^{t}D^{k}(\mathbf{G}_{\widetilde{H}}(t-s)\ast (\widetilde{N(V)}(s))_{H})ds, \end{equation} | (5.6) |
Now let us estimate (5.5) and (5.6) separately. For the low frequency part, we have
\begin{align*} \|D^{k}V(t)_{L}\|_{L^2}&\leq \|D^{k}(\mathbb{G}_{\widetilde{L}}\ast(V_{0})_{L})\|_{L^2} +\int_{0}^{\frac{t}{2}}\|D^{k}(\mathbb{G}_{\widetilde{L}}(t-s)\ast (N(V)(s))_{L})\|_{L^2}ds\\ &+\int_{\frac{t}{2}}^{t}\|D^{k}(\mathbb{G}_{\widetilde{L}}(t-s)\ast (N(V)(s))_{L})\|_{L^2}ds\\ & = :\sum\limits_{i = 1}^3H_i. \end{align*} |
For H_1, we have
H_1\leq C\|D^{k}\mathbb{G}_{\widetilde{L}}\|_{L^2}\|(V_{0})_{L})\|_{L^1}\leq C(1+t)^{-\frac{3}{4}-\frac{k}{2}}. |
For H_2, we first estimate \|(N(V))_L\|_{L^1},
\begin{align*} \|(N(V))_L\|_{L^1}&\leq C\|(n, v, m, w)(t)\|_{L^1}\|(\nabla n, \nabla v, \nabla m, \nabla w)(t)\|_{L^{\infty}}\\ &+\|(n, m)(t)\|_{L^1}\|(\nabla^2 v, \nabla^2 w)(t)\|_{L^{\infty}}.\\ &\leq C(1+t)^{-2}+C(1+t)^{-\frac{5}{2}}. \end{align*} |
Then, for H_2, we have
\begin{align*} H_2&\leq C\int_0^{\frac{t}{2}}\|D^{k}(\mathbb{G}_{\widetilde{L}})(t-s)\|_{L^2}\|(N(V)(s))_L\|_{L^1}ds\\ &\leq C\int_0^{\frac{t}{2}}(1+t-s)^{-\frac{3}{4}-\frac{k}{2}}((1+s)^{-2}+(1+s)^{-\frac{5}{2}})ds\\ &\leq C(1+t)^{-\frac{3}{4}-\frac{k}{2}}. \end{align*} |
For H_3, we have
\begin{align*} H_3&\leq C\int_{\frac{t}{2}}^{t}\|D\mathbb{G}_{\widetilde{L}}(t-s)\|_{L^1}\|D^{k-1}((N(V)(s))_L)\|_{L^2}ds\\ &\leq C\int_{\frac{t}{2}}^{t}\|D\mathbb{G}_{\widetilde{L}}(t-s)\|_{L^1}\|(n, v, m, w)(t)\|_{L^\infty} \|D^{k-1}(n, v, m, w)(t)\|_{L^2}ds\\ &+C\int_{\frac{t}{2}}^{t}\|D\mathbb{G}_{\widetilde{L}}(t-s)\|_{L^1}\|\nabla(v, w)(t)\|_{L^{\infty}} \|D^{k-2}(n, v, m, w)(t)\|_{L^2}ds\\ &+C\int_{\frac{t}{2}}^{t}\|D\mathbb{G}_{\widetilde{L}}(t-s)\|_{L^1}\|\nabla^2(v, w)(t)\|_{L^{\infty}}\|D^{k-3}(n, v, m, w)(t)\|_{L^2}ds\\ &\leq C\int_{\frac{t}{2}}^{t}(1+t-s)^{-\frac{1}{2}}(1+s)^{-\frac{3}{2}}(1+s)^{-\frac{3}{4}-\frac{k-1}{2}}ds\\ &+C\int_{\frac{t}{2}}^{t}(1+t-s)^{-\frac{1}{2}}(1+s)^{-2}(1+s)^{-\frac{3}{4}-\frac{k-2}{2}}ds\\ &+C\int_{\frac{t}{2}}^{t}(1+t-s)^{-\frac{1}{2}}(1+s)^{-\frac{5}{2}}(1+s)^{-\frac{3}{4}-\frac{k-3}{2}}ds\\ &\leq C(1+t)^{-\frac{3}{4}-\frac{k}{2}}, \end{align*} |
where we use the Gagliardo-Nirenberg inequality to obtain the estimate for \|(n, v, m, w)(t)\|_{L^\infty} and \|\nabla^2(v, w)(t)\|_{L^{\infty}} . Combining the estimate of H_1 , H_2 , and H_3 , we obtain the low frequency part estimate of D^{k}V(t)
\|D^{k}V(t)_L\|_{L^2}\leq C(1+t)^{-\frac{3}{4}-\frac{k}{2}}. |
For the high frequency part of D^{k}V(t) , by Lemma 3.5, we have
\begin{align} \begin{split} &\ \ \ \ \|D^{k}(V(\cdot,t))_H\|_{L^2}^2\\ & = \|D^{k}(\mathbf{G}(\cdot,t)\ast (V_{0})_H)\|_{L^2}^2\\ &+2\int_{0}^t\int_{\mathbb{R}^3}\mathbf{G}(x-\cdot,t-\tau)\ast D^{k}(V(\cdot,s))_H\cdot \mathbf{G}(x-\cdot,t-s)\ast D^{k}\widetilde{(N(V(\cdot,s)))}_Hdxds.\\ \end{split} \end{align} | (5.7) |
By the definition of \chi_2(\xi) in (3.16), we know (5.7) is equal to
\begin{align} \begin{split} &\ \ \ \ \|D^{k}(V(\cdot,t))_H\|_{L^2}^2\\ & = \|D^{k}(\mathbf{G}_{\widetilde{H}}(\cdot,t)\ast (V_{0})_H)\|_{L^2}^2\\ &+2\int_{0}^t\int_{\mathbb{R}^3}\mathbf{G}_{\widetilde{H}}(x-\cdot,t-\tau)\ast D^{k}(V(\cdot,s))_H\cdot \mathbf{G}_{\widetilde{H}}(x-\cdot,t-s)\ast D^{k}\widetilde{(N(V(\cdot,s)))}_Hdxds\\ & = \sum\limits_{i = 1}^2I_i. \end{split} \end{align} | (5.8) |
Now we turn to estimate each I_i. To start, for I_1,
\begin{align} \begin{split} I_1& = |D^{k}(\mathbf{G}_{\widetilde{H}}(\cdot,t)\ast V_{0})\|_{L^2}^2\\ &\leq C\|D\mathbb{G}_{\widetilde{HR}}\|_{L^1}^2\|D^{k-1}V_{0}\|_{L^2}^2 +Ce^{-2C_0t}\|D^{k}V_{0}\|_{L^2}^2\\ &\leq Ce^{-2C_0t}\\ &\leq C(1+t)^{-\frac{3}{2}-k}.\\ \end{split} \end{align} | (5.9) |
From the definition of N(V) and \widetilde{N(V)} in (4.20), (5.4), we have
\begin{equation} \widetilde{N(V)} = N(V)+(0,0,2\nabla\triangle^{-1}m,0)^T = :\widetilde{N(V)}_1+\widetilde{N(V)}_2. \end{equation} | (5.10) |
For the nonlinear item, we can check without difficulty that
\begin{align} \begin{split} &\ \ \ \|D^{k-1}(\widetilde{N(V)}_1)_{H}(t)\|_{L^2}\\ &\leq C\|(n, v, m, w)(t)\|_{L^{\infty}}\|D^{k}V(t)\|_{L^2}+C[\|(\nabla n, \nabla v, \nabla m, \nabla w)(t)\|_{L^{\infty}}\\ &+\|(\nabla^2v, \nabla^2w)(t)\|_{L^{\infty}}]\|D^{k-1}V(t)\|_{L^2} +\|(n, m)(t)\|_{L^{\infty}}\|D^{k+1}(v, w)(t)\|_{L^2}.\\ \end{split} \end{align} | (5.11) |
From the conclusion of Proposition (4.1), we know (4.14) holds, i.e.,
\begin{equation} (\|\nabla n(t)\|_{L^{\infty}(\mathbb{R}^3)}, \|\nabla v(t)\|_{L^{\infty}(\mathbb{R}^3)}, \|\nabla m(t)\|_{L^{\infty}(\mathbb{R}^3)},\|\nabla w(t)\|_{L^{\infty}(\mathbb{R}^3)})\leq C(1+t)^{-2}, \end{equation} | (5.12) |
and by the Sobolev embedding theorem, we have
\begin{align} (\|n(t)\|_{L^{\infty}(\mathbb{R}^3)},\|v(t)\|_{L^{\infty}(\mathbb{R}^3)},\|m(t)\|_{L^{\infty}(\mathbb{R}^3)}, \|w(t)\|_{L^{\infty}(\mathbb{R}^3)})\leq CE_{0}^{\frac{2}{5}}(1+t)^{-\frac{6}{5}}. \end{align} | (5.13) |
Now for I_2, we can obtain
\begin{align*} \begin{split} I_2&\leq C\int_{0}^t\int_{\mathbb{R}^3}\mathbf{G}_{\widetilde{H}}(x-\cdot,t-\tau)\ast D^{k}(V(\cdot,s))_H\cdot \mathbf{G}_{\widetilde{H}}(x-\cdot,t-s)\ast D^{k}(\widetilde{(N(V(\cdot,s))_1})_Hdxds\\ &+C\int_{0}^t\int_{\mathbb{R}^3}\mathbf{G}_{\widetilde{H}}(x-\cdot,t-\tau)\ast D^{k}(V(\cdot,s))_H\cdot \mathbf{G}_{\widetilde{H}}(x-\cdot,t-s)\ast D^{k}(\widetilde{(N(V(\cdot,s))_2})_Hdxds\\ &\leq C\int_{0}^t\int_{\mathbb{R}^3}\mathbf{G}_{\widetilde{RH}}(x-\cdot,t-\tau)\ast D^{k}(V(\cdot,s))_H\cdot \mathbf{G}_{\widetilde{RH}}(x-\cdot,t-s)\ast D^{k}(\widetilde{(N(V(\cdot,s))_1})_Hdxds\\ &+C\int_{0}^t\int_{\mathbb{R}^3}\mathbf{G}_{\widetilde{RH}}(x-\cdot,t-\tau)\ast D^{k}(V(\cdot,s))_H\cdot \mathbf{G}_{\widetilde{S}}(x-\cdot,t-s)\ast D^{k}(\widetilde{(N(V(\cdot,s))_1})_Hdxds\\ &+C\int_{0}^t\int_{\mathbb{R}^3}\mathbf{G}_{\widetilde{S}}(x-\cdot,t-\tau)\ast D^{k}(V(\cdot,s))_H\cdot \mathbf{G}_{\widetilde{RH}}(x-\cdot,t-s)\ast D^{k}(\widetilde{(N(V(\cdot,s))_1})_Hdxds\\ &+C\int_{0}^t\int_{\mathbb{R}^3}\mathbf{G}_{\widetilde{S}}(x-\cdot,t-\tau)\ast D^{k}(V(\cdot,s))_H\cdot \mathbf{G}_{\widetilde{S}}(x-\cdot,t-s)\ast D^{k}(\widetilde{(N(V(\cdot,s))_1})_Hdxds\\ &+C\int_{0}^t\int_{\mathbb{R}^3}\mathbf{G}_{\widetilde{RH}}(x-\cdot,t-\tau)\ast D^{k}(V(\cdot,s))_H\cdot \mathbf{G}_{\widetilde{RH}}(x-\cdot,t-s)\ast D^{k}(\widetilde{(N(V(\cdot,s))_2})_Hdxds\\ &+C\int_{0}^t\int_{\mathbb{R}^3}\mathbf{G}_{\widetilde{RH}}(x-\cdot,t-\tau)\ast D^{k}(V(\cdot,s))_H\cdot \mathbf{G}_{\widetilde{S}}(x-\cdot,t-s)\ast D^{k}(\widetilde{(N(V(\cdot,s))_2})_Hdxds\\ &+C\int_{0}^t\int_{\mathbb{R}^3}\mathbf{G}_{\widetilde{S}}(x-\cdot,t-\tau)\ast D^{k}(V(\cdot,s))_H\cdot \mathbf{G}_{\widetilde{RH}}(x-\cdot,t-s)\ast D^{k}(\widetilde{(N(V(\cdot,s))_2})_Hdxds\\ &+C\int_{0}^t\int_{\mathbb{R}^3}\mathbf{G}_{\widetilde{S}}(x-\cdot,t-\tau)\ast D^{k}(V(\cdot,s))_H\cdot \mathbf{G}_{\widetilde{S}}(x-\cdot,t-s)\ast D^{k}(\widetilde{(N(V(\cdot,s))_2})_Hdxds\\ & = :\sum\limits_{i = 1}^8K_i. \end{split} \end{align*} |
For K_{1} , we have
\begin{align*} \begin{split} K_{1}& = C\int_{0}^t\int_{\mathbb{R}^3}\mathbf{G}_{\widetilde{RH}}(x-\cdot,t-s)\ast D^{k}(V(\cdot,s))_{H}\cdot D\mathbf{G}_{\widetilde{RH}}(x-\cdot,t-s)\ast D^{k-1}(N(V(\cdot,s))_1)_Hdxds\\ &\leq C\|D^{k}V(s)\|_{L^\infty(0,t; L^2)}^2\int_{0}^te^{-2C_0(t-s)}\|(n, v, m, w)(t)\|_{L^{\infty}}ds\\ &+C\int_{0}^te^{-2C_0(t-s)}(\|(\nabla n, \nabla v, \nabla m, \nabla w)(t)\|_{L^{\infty}}\\ &+\|(\nabla^2v, \nabla^2w)(t)\|_{L^{\infty}})\|D^{k-1}u(s)\|_{L^2}\|D^{k}V(s)\|_{L^2}ds\\ &+C\int_{0}^te^{-2C_0(t-s)}\|(n, m)(t)\|_{L^{\infty}}\|D^{k+1}(v, w)(t)\|_{L^2}\|D^{k}V(s)\|_{L^2}ds.\\ \end{split} \end{align*} |
With the help of (5.11)–(5.13), we can get the estimates of K_{1} as follows. By using Lemma 3.4, we can get the estimate for K_{1} ,
\begin{align*} \begin{split} K_{1}&\leq CE_{0}^{\frac{2}{5}}\|D^{k}V(s)\|_{L^\infty(0,t;L^2)}^2\int_{0}^te^{-2C_0(t-s)}(1+s)^{-\frac{6}{5}}ds +C(E_{0}^{\frac{1}{8}})^2\|D^{k}V(s)\|_{L^\infty(0,t;L^2)}^2\\ &+C\|D^{k-1}V(s)\|_{L^\infty(0,t;L^2)}^2(\int_{0}^te^{-2C_0(t-s)}(1+s)^{-\frac{6}{5}}ds)^2\\ &+CE_{0}^{\frac{2}{5}}\|D^{k}V(s)\|_{L^\infty(0,t;L^2)}^2\\ &+CE_{0}^{\frac{2}{5}}\|D^{k+1}(v, w)(s)\|_{L^\infty(0,t;L^2)}^2(\int_{0}^te^{-2C_0(t-s)}(1+s)^{-\frac{6}{5}}ds)^2\\ &\leq CE_{0}^{\frac{1}{4}}\|D^{k}V(s)\|_{L^\infty(0,t;L^2)}^2+C(1+t)^{-1}\|D^{k-1}V(s)\|_{L^\infty(0,t;L^2)}^2\\ &+CE_{0}^{\frac{2}{5}}\|D^{k+1}(v, w)(s)\|_{L^\infty(0,t;L^2)}^2\\ &\leq CE_{0}^{\frac{1}{4}}\|D^{k}V(s)\|_{L^\infty(0,t;L^2)}^2+C(1+t)^{-\frac{3}{2}-|k|}+CE_{0}^{\frac{2}{5}}\|D^{k+1}(v, w)(s)\|_{L^\infty(0,t;L^2)}^2, \end{split} \end{align*} |
where we used the (5.1) when \alpha = k-1 in the last inequality of the above. The estimate of K_{3} is parallel to K_{1}, i.e.,
\begin{align*} K_{3}\leq CE_{0}^{\frac{1}{4}}\|D^{k}V(s)\|_{L^\infty(0,t;L^2)}^2+C(1+t)^{-\frac{3}{2}-|k|}+CE_{0}^{\frac{2}{5}}\|D^{k+1}(v, w)(s)\|_{L^\infty(0,t;L^2)}^2. \end{align*} |
By integrating by parts, the estimate of K_{2} is similar to K_{1}, then we have
\begin{align*} K_{2}\leq CE_{0}^{\frac{1}{4}}\|D^{k}V(s)\|_{L^\infty(0,t;L^2)}^2+C(1+t)^{-\frac{3}{2}-|k|}+CE_{0}^{\frac{2}{5}}\|D^{k+1}(v, w)(s)\|_{L^\infty(0,t;L^2)}^2. \end{align*} |
As for K_{4}, by the defination of \mathbf{G}_{\widetilde{S}} , and using (5.12) and (5.13), we have
\begin{align*} \begin{split} K_{4}&\leq C\int_{0}^te^{-2C_0(t-s)}(\|(\nabla v, \nabla w)(s)\|_{L^{\infty}}\|D^{k}(n, m)(s)\|_{L^2}^2\\ &+\|(n, m)(s)\|_{L^{\infty}}\|D^{k+1}(v, w)(s)\|_{L^2}\|D^{k}(n, m)(s)\|_{L^2})ds\\ &\leq CE_{0}^{\frac{1}{6}}\|D^{k}(n, m)(s)\|_{L^\infty(0,t;L^2)}^2+CE_{0}^{\frac{2}{5}}\|D^{k}(n, m)(s)\|_{L^\infty(0,t;L^2)}^2\\ &+CE_{0}^{\frac{2}{5}}\|D^{k+1}(v, m)(s)\|_{L^\infty(0,t;L^2)}^2\\ &\leq CE_{0}^{\frac{1}{6}}\|D^{k}(n, m)(s)\|_{L^\infty(0,t;L^2)}^2+CE_{0}^{\frac{2}{5}}\|D^{k+1}(v, w)(s)\|_{L^\infty(0,t;L^2)}^2\\ &\leq CE_{0}^{\frac{1}{6}}\|D^{k}V(s)\|_{L^\infty(0,t;L^2)}^2+CE_{0}^{\frac{2}{5}}\|D^{k+1}(v, w)(s)\|_{L^\infty(0,t;L^2)}^2.\\ \end{split} \end{align*} |
By the Gagliardo-Nirenberg inequality, we have
\|\nabla (n, v, m, w)(t)\|_{L^2(\mathbb{R}^3)}\leq CE_{0}^{\frac{17}{28}}(1+t)^{-\frac{3}{4}}, |
\|\nabla^{k-1}m(t)\|_{L^2(\mathbb{R}^3)}\leq C\|\nabla m(t)\|_{L^2(\mathbb{R}^3)}^{\frac{1}{k-1}} \|\nabla^{k}m(t)\|_{L^2(\mathbb{R}^3)}^{\frac{k-2}{k-1}}. |
Then, for K_5, it holds
\begin{align*} \begin{split} K_5&\leq C\int_{0}^te^{-2C_0(t-s)}\|D^kV(s)\|_{L^2}\|D^{k-1}m(s)\|_{L^2}ds\\ &\leq CE_{0}^{\frac{17}{28}\cdot\frac{1}{k-1}}\|D^km(s)\|_{L^\infty(0,t;L^2)}^{\frac{k-2}{k-1}}\|D^kV(s)\|_{L^\infty(0,t;L^2)}\\ &\leq CE_{0}^{\frac{17}{28}\cdot\frac{1}{k-1}}\|D^kV(s)\|_{L^\infty(0,t;L^2)}^{1+\frac{k-2}{k-1}}\\ &\leq CE_{0}^{\frac{17}{28}\cdot\frac{1}{k-1}}\|D^kV(s)\|_{L^\infty(0,t;L^2)}^{2}. \end{split} \end{align*} |
By the same analysis as K_{5}, it holds
K_7\leq CE_{0}^{\frac{17}{28}\cdot\frac{1}{k-1}}\|D^kV(s)\|_{L^\infty(0,t;L^2)}^{2}. |
By the definition of \mathbf{G}_{\widetilde{S}} in (3.17) and \widetilde{N(V)}_2 in (5.10), we have
K_6+K_8 = 0. |
Combining the estimate for each K_i , and from (5.7), it holds,
\begin{align} \begin{split} &\ \ \ \|D^{k}(V(\cdot,t))_H\|_{L^2}^2\\ &\leq C(E_{0}^{\frac{1}{4}}+E_{0}^{\frac{17}{28}\cdot\frac{1}{k-1}})\|D^{k}V(s)\|_{L^\infty(0,t;L^2)}^2+C(1+t)^{-\frac{3}{2}-|k|}+CE_{0}^{\frac{2}{5}}\|D^{k+1}(v, w)(s)\|_{L^\infty(0,t;L^2)}^2. \end{split} \end{align} | (5.14) |
To close the estimate, we take our attention to the estimate of \|D^{k+1}(v, m)(t)\|_{L^2}. From (4.14), we know
\begin{align*} \begin{split} D^{k+1}v& = \begin{pmatrix} \sum\limits_{j = 1}^8D^{k+1}(\mathbb{G}^{2,j}(t)*V_{0}^j)+\int_{0}^{t}\sum\limits_{j = 1}^8D^{k+1}(\mathbb{G}^{2,j}(t-s)*N(V)^j(s))ds\\ \sum\limits_{j = 1}^8D^{k+1}(\mathbb{G}^{3,j}(t)*V_{0}^j)+\int_{0}^{t}\sum\limits_{j = 1}^8D^{k+1}(\mathbb{G}^{3,j}(t-s)*N(V)^j(s))ds\\ \sum\limits_{j = 1}^8D^{k+1}(\mathbb{G}^{4,j}(t)*V_{0}^j)+\int_{0}^{t}\sum\limits_{j = 1}^8D^{k+1}(\mathbb{G}^{4,j}(t-s)*N(V)^j(s))ds\\ \end{pmatrix}.\\ \end{split} \end{align*} |
For the linear partition, we have
\begin{align*} \|D^{k+1}(\mathbb{G}^{i,j}(t)*V_{0}^j)\|_{L^2}^2\leq C(1+t)^{-\frac{3}{2}-(|k|+1)}. \end{align*} |
For the nonlinear partition, we still adopt the method of high and low frequency decomposition of the solution. Here we omit the details of the calculation since the analysis is parallel to the estimate of D^kV(t).
For \|D^{k+1}v(t)\|_{L^2}, we have
\begin{align} \|D^{k+1}v(t)\|_{L^2}^2\leq C(1+t)^{-\frac{3}{2}-(k+1)}+CE_{0}^{\frac{1}{4}}(\|D^{k}V(s)\|_{L^\infty(0,t;L^2)}^2+\|D^{k+1}v(s)\|_{L^\infty(0,t;L^2)}^2). \end{align} | (5.15) |
The same way also gives
\begin{align} \|D^{k+1}w(t)\|_{L^2}^2\leq C(1+t)^{-\frac{3}{2}-(k+1)}+CE_{0}^{\frac{1}{4}}(\|D^{k}V(s)\|_{L^\infty(0,t;L^2)}^2+\|D^{k+1}w(s)\|_{L^\infty(0,t;L^2)}^2). \end{align} | (5.16) |
Combing (5.14)–(5.16), since the E_{0} is small enough, we have
\begin{align*} &\ \ \ \|D^{k}V(\cdot,t)\|_{L^\infty(0,t;L^2)}^2+\|D^{k+1}(v, w)(t)\|_{L^\infty(0,t;L^2)}^2\\ &\leq C(1+t)^{-\frac{3}{2}-k}+C(1+t)^{-\frac{3}{2}-(k+1)}\leq C(1+t)^{-\frac{3}{2}-k}. \end{align*} |
Now we proved (5.1) when \alpha = k. The proposition is proved.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This work was partially supported by the National Natural Science Foundation of China (Nos.12271357, 12161141004) and Shanghai Science and Technology Innovation Action Plan (No.21JC1403600).
The authors declare there is no conflicts of interest.
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