In this paper, the neural network domain with the backpropagation Levenberg-Marquardt scheme (NNB-LMS) is novel with a convergent stability and generates a numerical solution of the impact of the magnetohydrodynamic (MHD) nanofluid flow over a rotating disk (MHD-NRD) with heat generation/absorption and slip effects. The similarity variation in the MHD flow of a viscous liquid through a rotating disk is explained by transforming the original non-linear partial differential equations (PDEs) to an equivalent non-linear ordinary differential equation (ODEs). Varying the velocity slip parameter, Hartman number, thermal slip parameter, heat generation/absorption parameter, and concentration slip parameter, generates a Prandtl number using the Runge-Kutta 4th order method (RK4) numerical technique, which is a dataset for the suggested (NNB-LMS) for numerous MHD-NRD scenarios. The validity of the data is tested, and the data is processed and properly tabulated to test the exactness of the suggested model. The recommended model was compared for verification, and the estimation solutions for particular instances were assessed using the NNB-LMS training, testing, and validation procedures. A regression analysis, a mean squared error (MSE) assessment, and a histogram analysis were used to further evaluate the proposed NNB-LMS. The NNB-LMS technique has various applications such as disease diagnosis, robotic control systems, ecosystem evaluation, etc. Some statistical data such as the gradient, performance, and epoch of the model were analyzed. This recommended method differs from the reference and suggested results, and has an accuracy rating ranging from 10−09to 10−12.
Citation: Yousef Jawarneh, Humaira Yasmin, Wajid Ullah Jan, Ajed Akbar, M. Mossa Al-Sawalha. A neural networks technique for analysis of MHD nano-fluid flow over a rotating disk with heat generation/absorption[J]. AIMS Mathematics, 2024, 9(11): 32272-32298. doi: 10.3934/math.20241549
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In this paper, the neural network domain with the backpropagation Levenberg-Marquardt scheme (NNB-LMS) is novel with a convergent stability and generates a numerical solution of the impact of the magnetohydrodynamic (MHD) nanofluid flow over a rotating disk (MHD-NRD) with heat generation/absorption and slip effects. The similarity variation in the MHD flow of a viscous liquid through a rotating disk is explained by transforming the original non-linear partial differential equations (PDEs) to an equivalent non-linear ordinary differential equation (ODEs). Varying the velocity slip parameter, Hartman number, thermal slip parameter, heat generation/absorption parameter, and concentration slip parameter, generates a Prandtl number using the Runge-Kutta 4th order method (RK4) numerical technique, which is a dataset for the suggested (NNB-LMS) for numerous MHD-NRD scenarios. The validity of the data is tested, and the data is processed and properly tabulated to test the exactness of the suggested model. The recommended model was compared for verification, and the estimation solutions for particular instances were assessed using the NNB-LMS training, testing, and validation procedures. A regression analysis, a mean squared error (MSE) assessment, and a histogram analysis were used to further evaluate the proposed NNB-LMS. The NNB-LMS technique has various applications such as disease diagnosis, robotic control systems, ecosystem evaluation, etc. Some statistical data such as the gradient, performance, and epoch of the model were analyzed. This recommended method differs from the reference and suggested results, and has an accuracy rating ranging from 10−09to 10−12.
In this paper we are devoted to the following Euler and magnetohydrodynamics coupled system in Ω:
{ρ+∂tu++ρ+u+⋅∇u++div(p+I)=−gρ+e3,divu+=0.inΩ+(t) | (1.1) |
and
{ρ−∂tu−+ρ−u−⋅∇u−+div(p−I−h−⊗h−)=−gρ−e3,∂th−+u−⋅∇h−−h−⋅∇u−=0,divu−=0,divh−=0,inΩ−(t) | (1.2) |
where Ω=R2×(−1,1)⊂R3 is divided into Ω− and Ω+ by a moving free surface Σ(t). As shown in the above systems (1.1) and (1.2), the "upper fluid" is called Euler fluid, which is occupying Ω+, and the "lower fluid", which is occupying Ω−, is magnetohydrodynamics fluid. We use (u±,p±,h−) to describe the fluid velocity, pressure, and magnetic field. The subscript "±"refers to "upper/lower" fluid. I is the identity matrix, ρ± denotes the densities of the respective fluids, g>0 is the gravitational constant, and e3=(0,0,1).
The conditions on Σ(t) are as follows:
{[u⋅ν]|Σ(t)=0,h−⋅ν|Σ(t)=0,[(p+gρx3)ν]|Σ(t)=(h−⊗h−)ν|Σ(t), | (1.3) |
where ν is the normal vector of Σ(t).
At the fixed boundary x3=±1, we impose the conditions:
u+(t,x1,x2,1)⋅e3=u−(t,x1,x2,−1)⋅e3=0, | (1.4) |
for any t≥0, (x1,x2)∈R2.
In order to overcome the mathematical difficulties brought about by the evolution of the free interface over time, the Lagrangian coordinates are introduced. Define the following reversible maps:
φ0±:Ω±⟶Ω±(0), | (1.5) |
satisfying Σ0=φ0±{x3=0} and {x3=±1}=φ0±{x3=±1}. φ0± are continuous across {x3=0}. Define invertible flow maps φ± which solve
{∂tφ±(t,x)=u±(t,φ±(t,x)),φ±(0,x)=φ0±(x). | (1.6) |
In this paper, (t,y) with y=φ(t,x) and (t,x)∈R+×Ω denote Eulerian coordinates and Lagrangian coordinates, respectively. Since the two-layer fluids may slip each other, the slip map must be introduced. Define S±:R2×R+→R2×{0}⊂R2×(−1,1) by
S−(t,x1,x2)=φ−1−(t,φ+(t,x1,x2,0)), | (1.7) |
Now, we define the corresponding unknown functions in the Lagrangian coordinate
{v±(t,x)=u±(t,φ±(t,x)),b−(t,x)=h−(t,φ−(t,x)),(t,x)∈R+×Ω.q±(t,x)=p±(t,φ±(t,x)), | (1.8) |
Denote by A±:=((Dφ±)−1)T, where D is the derivative of the coordinates x and superscript T is the matrix transpose. Then, the evolution equations for v±,b−,q±,φ± become
{∂tφi+=vi+,ρ+∂tvi++Aik+∂kq+=0,Ajk+∂kvj+=0,∂tφi−=vi−,ρ−∂tvi−+Aik−⋅∂kq−=bj−Aik−∂kbi−,Ajk−∂kvj−=0,∂tbi−=bj−Ajk−∂kvi−,Ajk−⋅∂kbj−=0. | (1.9) |
In the above system, we have used the Einstein summation convention. The corresponding conditions on Σ(t) are
{(v+(t,x1,x2,0)−v−(t,S−(t,x1,x2)))⋅ν(t,x1,x2,0)=0,(q+(t,x1,x2,0)−q−(t,S−(t,x1,x2)))⋅ν(t,x1,x2,0)=g[ρ]φ3+(t,x1,x2)ν(t,x1,x2,0)−(b−⊗b−)(t,S−(t,x1,x2))ν(t,x1,x2,0), | (1.10) |
where
ν=∂1φ+×∂2φ+|∂1φ+×∂2φ+|, | (1.11) |
is the unit normal vector to the interface Σ(t)=φ+(t,{x3=0}), and φ3+ is the third component of φ+. Finally, we require the impermeability conditions
v−(t,x1,x2,−1)⋅e3=v+(t,x1,x2,1)⋅e3=0. | (1.12) |
In the Lagrangian coordinates, the magnetic field b− can be expressed by virtue of φ− as in [1,2]. Applying Ail− to the seventh equation of (1.9), we achieve
Ail−∂tbi−=Ail−bj−Ajk−∂kvi−=Ail−bj−Ajk−(∂t∂kφi−)=−bi−∂tAil−. |
Thus, we have ∂t(Ail−bi−)=0, which implies Ail−bi−=Ail,0−bi,0− and
bi−=∂lφi−Ajl,0−bj,0−. | (1.13) |
Now, we check the last equation of (1.9). Applying the geometric identities, we have
J=J0 and ∂k(JAik−)=0, |
where J=|Dφ|. Utilizing Aik−∂k to (1.13), one gets
Aik−∂kbi−=JJ0Aik−∂k(∂lφi−Ajl,0−bj,0−)=1J0∂k(JAik−∂lφi−Ajl,0−bj,0−)−1J0∂k(JAik−)∂lφi−Ajl,0−bi,0−=1J0∂k(JAjk,0−bj,0−)=1J0∂k(J0Ajk,0−bj,0−)=J0J0∂k(Ajk,0−bj,0−)=∂k(Ajk,0−bj,0−)=Ajk,0−∂kbj,0−. | (1.14) |
The compatibility conditions for the initial value are imposed as follows:
Ajk,0−∂kbj,0=0. | (1.15) |
Combining (1.14), we have
Ajk−∂kbj−=0,for all0≤t≤T. | (1.16) |
For simplicity, we assume that
Ail,0−bi,0=¯Ml. | (1.17) |
By virtue of (1.13) and (1.17), we can use the forcing term by the flow map φ− to represent the Lorentz term in the fifth equation of (1.9). Thus, (1.9) becomes a two-fluids Navier-stokes system:
{∂tφi±=vi±,ρ∂tvi++Aik+∂kq+=0,ρ−∂tvi−+Aik−∂kq−−ˉMlˉMr∂2lrφi−=0,Ajk±∂kvj±=0, | (1.18) |
where the magnetic field ˉM can be considered as a vector parameter.
The conditions (1.10) can be expressed as
[q+(t,x1,x2,0)−q−(t,S−(t,x1,x2))]νi(t,x1,x2,0)=g[ρ]φ3+(t,x1,x2,0)νi(t,x1,x2,0)−ˉMlˉMm(∂φi−∂mφj−)(t,S−(t,x1,x2))νj(t,x1,x2,0). | (1.19) |
The boundary conditions are the same as (1.12).
We have known that v±=0, φ±=Id, q±=const are steady -state solutions to the systems (1.18), (1.19), and (1.12). Then, ν=e3, A=Id, S−=Id{x3=0}. The linearized equation system near the steady-state solution is
{∂tφ±=v±,ρ+∂tv++∇q+=0,ρ−∂tv−+∇q−−ˉMlˉMm∂2lmφ−=0,divv±=0. | (1.20) |
The corresponding jump and fixed boundary conditions are
[[v⋅e3]]=0,[[q]]e3=g[ρ]φ3e3−ˉM3ˉMl∂lφ, | (1.21) |
v−(t,x1,x2,−1)⋅e3−v+(t,x1,x2,1)⋅e3=0, | (1.22) |
where [[⋅]] denotes the interfacial jump quantity on the boundary {x3=0}. Our aim is to study the Rayleigh-Taylor (RT) problem, so we suppose
ρ+>ρ−⇔[ρ]>0. | (1.23) |
RT instability is a ubiquitous phenomenon in nature, widely existing in various research fields such as astrophysics, atmospheric and oceanic science, laser fusion, and magnetic confinement fusion [3,4,5,6]. Before further discussion, we first review some results with regard to the RT instability problems. The studies on the RT instability can be traced back to the pioneering work due to Rayleigh [7] and Taylor [8]. From then on, many interesting physical phenomena and numerical simulations come from both physical and numerical experiments. Li and Luo [9] studied the effect of a vertical magnetic field on the RT instability of 2d nonideal magnetic fluids by constructing numerical solutions. We refer to [10] and references therein for a general research of the physics about RT instability. However, there are only very few analytical results from the mathematical point of view. Recently, Guo and Tice [11,12] studied the linear and nonlinear RT instability for Euler and Navier-Stokes fluids by the variational method or the modified variational method. In these papers, they discovered that the viscosity and surface tension have an impact on the RT instability. When considering the magnetic field, the RT instability appears by the Lorentz force. The theoretical discussion about the influence of magnetic fields was proposed by Kruskal and Schwarzchild in [13]. They found that the horizontal magnetic field can affect the development of RT instability but cannot suppress the growth of instability. Jiang et al. [1,14,15,16] used the similar method as [11,12] and employed the new techniques to discuss the RT instability for magnetohydrodynamics (MHD) fluids, as well as revealed the magnetic effect to the instability. In this paper we consider the mechanism for the effect of the magnetic field in the ideal fluid and magnetohydrodynamic coupled through the free interface.
We first introduce some definitions that are applicable throughout the paper. Define the horizontal Fourier transform for a function g∈L2(Ω) as follows:
ˆg(ξ1,ξ2,x3)=∫R2g(x1,x2,x3)e−i(x1ξ1+x2ξ2)dx1dx2. | (2.1) |
Due to the Fubini and Parseval theorems, one has that
∫Ω‖g(x)‖2dx=14π2∫Ω‖ˆg(ξ,x3)‖2dξdx3. | (2.2) |
Define the piecewise Sobolev space Hs(Ω) for any s∈R as follows:
Hs(Ω)={g|g+∈Hs(Ω+),g−∈Hs(Ω−)} |
equipped with the following norm:
‖g‖2Hs(Ω)=‖g‖2Hs(Ω+)+‖g‖2Hs(Ω−), |
and
‖g‖2Hk(Ω±):=k∑j=0∫R2×I±(1+|ξ|2)k−j|∂jx3ˆg±(ξ,x3)|2dξdx3=k∑j=0∫R2(1+|ξ|2)k−j‖∂jx3ˆg±(ξ,x3)‖2L2(I±)dξ, | (2.3) |
for I−=(−1,0) and I+=(0,1).
Next, we will give the main theorems. The first one is concerned with the linearized systems (1.20)–(1.22).
Theorem 2.1. Give a constant vector ˉM=(M,0,0), then for any k, the linear systems (1.20)–(1.22) are ill-posed in Hk(Ω). To be precise, for any fixed k,j∈N with j≥k, T0>0, and α>0, (1.20)–(1.22) have the solutions {(φn,vn,qn)}∞n=1 which satisfy
‖φn(0)‖Hj+‖vn(0)‖Hj+‖qn(0)‖Hj≤1n, | (2.4) |
but
‖vn(t)‖Hk≥‖φn(t)‖Hk≥α,for allt≥T0. | (2.5) |
Remark 2.2. The ill-posedness in the above theorem implies that the solutions to the linear systems (1.20)–(1.22) established in Theorem 3.6 depend disconstinuously on the initial conditions.
With the linear instability in hand, there is the nonlinear instability as follows:
Theorem 2.3. For any k≥4, the perturbed problem (4.2)–(4.6) does not have the property EE(k).
Remark 2.4. We can extend the conclusions in Theorems 2.1 and 2.3 to the general horizontal magnetic field ˉM=(M1,M2,0). In practice, since the L2− norm of the velocity remains unchanged under the horizontal rotation, one may rotate the coordinates so that ˉM=(M,0,0) with M=√M21+M22.
The paper is arranged as follows. In Section 1, we introduce the Lagrangian coordinates and linearize the nonlinear system. Some notations and main results are given in Section 2. In Section 3 we establish the growing mode solution to the linearized system and prove the uniqueness of the solution and discontinuous dependence on the initial value. In the last section, we investigate the ill-posedness of the nonlinear system.
When discussing the posedness of linearized Eqs (1.20)–(1.22), studying normal mode solutions is a standard practice. To this end, for some λ>0, suppose a normal mode ansatz as follows:
v±(t,x)=eλtw±(x),q±(t,x)=eλt˜q±(x),φ±(t,x)=eλt˜φ±(x). | (3.1) |
Substituting the above ansatz into the systems (1.20)–(1.22) and eliminating the unknown ˜φ± by using (1.20)1 and (1.20)3, we arrive at the following system:
{λρ+w++∇˜q+=0,λρ−w−+∇˜q−−1λˉMlˉMm∂2lmw−=0,divw±=0. | (3.2) |
At the same time, the jump and boundary conditions become
[[w3]]=0,[[˜q]]e3=1λg[ρ]w3e3−1λˉM3ˉMl∂lw, | (3.3) |
and
w3+(x1,x2,1)=w3−(x1,x2,−1)=0. | (3.4) |
Since the coefficients in (3.2) depend only on the x3 variable, we can adopt the horizontal Fourier transformation to (3.2) to reduce them into ordinary differential equations (ODEs) in terms of x3 with each spatial frequency as parameters. Define
κ±,ψ±,θ±,π±:(−1,1)→R, |
so that
κ±(x3)=iˆw1±(ξ1,ξ2,x3), |
ψ±(x3)=iˆw2±(ξ1,ξ2,x3), |
θ±(x3)=ˆw3±(ξ1,ξ2,x3), |
and
π±(x3)=ˆ˜q(ξ1,ξ2,x3). |
Then, we have
F(divw±)=ξ1ϕ±+ξ2ψ±+θ′±, | (3.5) |
where F means the Fourier transformation and ′=ddx3.
Note that we only consider ˉM=(M,0,0), and make the Fourier transform for (3.2), then we achieve the following system of ODEs:
{λρ+κ+−ξ1π+=0,λρ+ψ+−ξ2π+=0,λρ+θ++π′+=0,λ2ρ−κ−−λξ1π−+M2ξ21κ−=0,λ2ρ−ψ−−λξ2π−+M2ξ21ψ−=0,λ2ρ−θ−+λπ′−+M2ξ21θ−=0,ξ1κ±+ξ2ψ±+θ′±=0, | (3.6) |
subject to the jump conditions
[[θ]]=0,[[λπ]]=g[ρ]θ(0), | (3.7) |
and corresponding fixed boundary conditions
θ−(−1)=0,θ+(1)=0. | (3.8) |
Eliminating π± from the Eq (3.6), one has
{λ2ρ+(|ξ|2θ+−θ″+)=0,λ2ρ−(|ξ|2θ−−θ″−)=B2ξ21(|ξ|2θ−−θ″−). | (3.9) |
Equations (3.7) and (3.8) become
[[θ]]=0,λ2[[ρθ′]]−B2ξ21θ′−+g[ρ]|ξ|2θ=0, | (3.10) |
θ−(−1)=0,θ+(1)=0. | (3.11) |
In what follows, we will devote ourselves to build a solution for (3.9)–(3.11) based on the variational method, which deduces a solution for the system (3.6)–(3.8). Then, we will derive an exponential growth solution of time for the system (1.20)–(1.22).
Multiply θ+, θ− to (3.9)1 and (3.9)2, add the resulting equations, and integrate over (0,1) and (−1,0), respectively. After integration by parts, we get
−12λ2∫1−1ρ(|ξ|2|θ|2+|θ′|2)dx3=12[∫0−1B2ξ21(|ξ|2|θ−|2+|θ′−|2)dx3−g[ρ]|ξ|2θ2(0)], | (3.12) |
where we used boundary and jump conditions. We would like to find a growing mode solution to the system (3.2), which requires that there exists λ>0. One can utilize the variational method to look for the smallest value μ as follows:
μ=μ(|ξ|)=inf{12[∫0−1B2ξ21(|ξ|2|θ−|2+|θ′−|2)dx3−g[ρ]|ξ|2θ2(0)]|∫1−1ρ(|ξ|2|θ|2+|θ′|2)dx3=2}. | (3.13) |
Define
E(θ)=12[∫0−1B2ξ21(|ξ|2|θ−|2+|θ′−|2)dx3−g[ρ]|ξ|θ2(0)], | (3.14) |
and
J(θ)=12∫1−1ρ(|ξ|2|θ|2+|θ′|2)dx3. | (3.15) |
It is convenient to introduce the set A
A={θ∈H10(−1,1)|J(θ)=1}. |
For any |ξ|>0, let
−λ2=infθ∈AE(θ)<0, |
which is equivalent to
−λ2=infθ∈H10(−1,1)E(θ)J(θ). | (3.16) |
We want to find the minimizer of E on the set A and show the existence and negativity of the infimum.
Proposition 3.1. E can obtain the infimum on A for any fixed |ξ|≥0. If θ is a minimizer and −λ2:=E(θ), then (θ,λ2) solves (3.9) with (3.10) and (3.11). Moreover, θ is smooth when limited to (−1,0) or (0,1).
Proof. For any θ∈A, we estimate E(θ) as follows:
E(θ)≥−12g[ρ]|ξ|2|θ(0)|2=−12|ξ|g[ρ]|ξ|∫0−1∂x3|θ−|2dx3≥−|ξ|g[ρ]12∫0−1(|ξ|2|θ−|2+|θ′−|2)dx3≥−g[ρ]ρ−|ξ|. | (3.17) |
Therefore, E has a lower bound on A. Take θn∈A as a minimizing sequence, then we get the boundedness of θn in H10(−1,1), which implies that there exists θ∈H10(−1,1) to guarantee that θn is weakly convergent to θ in H10(−1,1) and strongly convergent in L2(−1,1). Thus, we have
E(θ)≤lim infn→∞E(θn)=infAE. | (3.18) |
Thus, E takes the infimum over A and θ is a minimizer.
For s∈R and any θ0∈H10(−1,1), define θ(s)=θ+sθ0, then
E(θ(s))+λ2J(θ(s))≥0, | (3.19) |
follows from (3.16). Let L(s)=E(θ(s))+λ2J(θ(s)), then there is L(s)≥0 for any s∈R and L(0)=0. This leads to L′(0)=0. By virtue of (3.14) and (3.15), we derive
L′(0)=∫0−1B2ξ21(|ξ|2θ−⋅(θ0)−+θ′−⋅(θ0)′−)dx3−g[ρ]|ξ|2θ(0)θ0(0)+λ2∫1−1ρ(|ξ|2θ⋅θ0+θ′⋅θ′0)dx3=0. | (3.20) |
By selecting θ0 with compact support in either (−1,0) or (0,1), one can get that θ solves Eq (3.9) in a weak sense. By standard bootstrap arguments, we may demonstrate that θ−∈Hk(−1,0)(resp., θ−∈Hk(0,1)) for all k≥0 and, hence, it is smooth when limited to the respective interval. This means that θ± are classical solutions to the Eq (3.9). The remainder is to show that (3.10) is established. For each θ0∈C∞c(−1,1), we obtain
(λ2[[ρθ′]]−B2ξ21θ′−+g[ρ]|ξ|2θ)θ0(0)=0. | (3.21) |
Since θ0(0) can be chosen arbitrary, we yield the second jump condition in (3.10). The conditions [[θ]]=0 and θ−(−1)=θ+(1)=0 are satisfied trivially since θ∈H10(−1,1)↪C0,120(−1,1).
Remark 3.2. (3.17) implies −λ2=infθ∈AE(θ)≥−g[ρ]ρ−|ξ| and, hence,
λ≤√g[ρ]ρ−|ξ|. | (3.22) |
Corollary 3.3. For any |ξ|>0, system (3.6) has a solution (κ±,ψ±,θ±,π±) with λ=λ(|ξ|)>0. Moreover, this solution satisfies (3.7) and (3.8) and is smooth when limited to (−1,0) or (0,1).
Proof. By solving (3.6), we get
π+=−λρ+θ′+|ξ|2,π−=−(λ2ρ−+M2ξ21)θ′−λ|ξ|2,κ±=−ξ1θ′±|ξ|2,ψ±=ξ2θ′±|ξ|2. | (3.23) |
From Proposition 3.1, it is obvious that π±=π±(ξ,x3),θ±=θ±(ξ,x3), and ψ±=ψ±(ξ,x3) are smooth over the interval (−1,0) or (0,1). Furthermore, the jump and boundary conditions (3.7) and (3.8) are satisfied.
Lemma 3.4. Let R1,ξ1 satisfy
e2R1−1e2R1+1≥12,and|ξ1|<g[ρ]4M2<R1, | (3.24) |
then the eigenvalue λ=λ(|ξ|) satisfies
λ≥√g[ρ]ρ++ρ−|ξ|. | (3.25) |
Proof. Denote ˉθ by
ˉθ(x3)={e|ξ|x3−e|ξ|(2−x3)x3∈[0,1),e−|ξ|x3−e|ξ|(2+x3)x3∈(−1,0), | (3.26) |
then
E(ˉθ)=12|ξ|[M2ξ21(e4|ξ|−1)−g[ρ]|ξ|(1−e2|ξ|)2], |
J(ˉθ)=12(ρ++ρ−)(e4|ξ|−1)|ξ|, |
so
E(ˉθ)J(ˉθ)=|ξ|(M2ξ21(ρ++ρ−)|ξ|−g[ρ](e2|ξ|−1)(ρ++ρ−)(e2|ξ|+1))≤|ξ|1ρ++ρ−(g[ρ]4−g[ρ]2)=−g[ρ]4(ρ++ρ−)|ξ|. |
Since −λ2=infθ∈H10(−1,1)E(θ)J(θ), the result follows.
Define
D:={ξ=(ξ1,ξ2)||ξ1|<g[ρ]4M2,|ξ|>R1}. | (3.27) |
Obviously, D is a symmetrical domain.
Lemma 3.5. Let ξ∈D, κ±,ψ±,θ±, and π± be the solutions to (3.6) constructed in Corollary 3.3, then for each k≥0, the following inequalities are valid:
||θ(ξ)||Hk(−1,1)≤Akk∑j=0|ξ|j−Δ(j), | (3.28) |
||κ(ξ)||Hk(−1,1)+||ψ(ξ)||Hk(−1,1)+||π(ξ)||Hk(−1,1)≤Bkk∑j=0|ξ|j, | (3.29) |
where
Δ(j)={0,ifj=0,1,ifj≠0. |
Moreover,
√||κ||2L2(−1,1)+||ψ||2L2(−1,1)+||θ||2L2(−1,1)≥D, | (3.30) |
where Ak,Bk,D>0 are constants depending on ρ,M,R1, and g.
Proof. θ(ξ)∈A implies that there are constants A0,A1>0 so that
||θ||L2(−1,1)≤A0,||θ||H1(−1,1)≤A1. |
By (3.9), we have
|ξ|2θ±=θ″±. | (3.31) |
Thus,
‖θ″‖2L2(−1,1)=|ξ|‖|ξ|θ‖2L2(−1,1)≤A2|ξ|, | (3.32) |
where we used θ∈A. Combining (3.31) and (3.32), we arrive at
||θ(k+1)||2L2(−1,1)≤Ak+1|ξ|k, for anyk≥0, |
which verifies (3.28). Employing (3.23) with |ξ|≥R1, we get
||θ(k)||L2(−1,1)+||ψ(k)||L2(−1,1)≤2|ξ|||θ(k)||L2(−1,1)≤Bk|ξ|k, | (3.33) |
for any k≥0. By virtue of the expression of π on (3.23), (3.22), and (3.25), with |ξ|≥R1, one has
||π(k)−||L2(−1,0)+||π(k)+||L2(0,1)=λρ+|ξ|2||θ(k+1)+||L2(0,1)+λ2ρ−+M2ξ21λ|ξ|2||θ(k+1)−||L2(−1,0)≤√g[ρ]ρ−ρ+|ξ|32||θ(k+1)+||L2(0,1)+(√g[ρ]ρ−ρ+|ξ|32+2M2√g[ρ]ρ++ρ−|ξ|12)||θ(k+1)−||L2(−1,0)≤Bk|ξ|k. | (3.34) |
Combining (3.33) and (3.34), one can achieve (3.29).
Equation (3.30) follows from that for any fixed |ξ|>0, θ(|ξ|)∈A, and (3.23).
In Corollary 3.3, we have achieved the solution to (1.20) for the fixed spatial frequency ξ∈R2. In rest of this section, we will establish the solution to (1.20) by using Fourier synthesis.
Theorem 3.6. Let 1≤R1≤R2<R3<∞ with R1 satisfy (3.24). Suppose a real-valued and radial symmetric function f∈C∞0(R2) and B(0,R2)⊂supp(f)⊂B(0,R3). For ξ∈R2, define
ˆw(ξ,x3)=−iκ(ξ,x3)e1−iψ(ξ,x3)e2+θ(ξ,x3)e3, | (3.35) |
where κ,ψ,θ,π are the solutions constructed in Proposition 3.1 and Corollary 3.3 with λ(ξ)>0.
Denote
φ(t,x)=14π2∫R2f(ξ)ˆw(ξ,x3)eλ(ξ)teix′ξdξ, | (3.36) |
v(t,x)=14π2∫R2λ(ξ)ˆw(ξ,x3)eλ(ξ)teix′ξdξ, | (3.37) |
q(t,x)=14π2∫R2λ(ξ)f(ξ)π(ξ,x3)eλ(ξ)teix′ξdξ, | (3.38) |
where x′⋅ξ=x1ξ1+x2ξ2, then (φ,v,q) is a real-valued solution to the linearized problem (1.20) with the corresponding conditions. For any k∈N, the following inequality is valid:
||φ(0)||Hk+||v(0)||Hk+||q(0)||Hk≤~Ck(∫R2(1+|ξ|2)k+1|f(ξ)|2dξ)12<∞, | (3.39) |
in which the positive constant ~Ck depends on ρ,|M|,R1, and g. Moreover, φ(t),v(t),q(t)∈Hk(Ω±) for every t>0 satisfies the following estimates:
{et√ˉc2R2||φ(0)||Hk≤||φ(t)||Hk≤et√ˉc1R3||φ(0)||Hk,et√ˉc2R2||v(0)||Hk≤||v(t)||Hk≤et√ˉc1R3||v(0)||Hk,et√ˉc2R2||q(0)||Hk≤||q(t)||Hk≤et√ˉc1R3||q(0)||Hk, | (3.40) |
where ¯c1=g[ρ]ρ−,¯c2=g[ρ]4(ρ++ρ−).
Proof. Fix ξ∈R, and
φ(t,x)=f(ξ)ˆw(ξ,x3)eλ(ξ)teix′⋅ξ,v(t,x)=λ(ξ)f(ξ)ˆw(ξ,x3)eλ(ξ)teix′⋅ξ,q(t,x)=λ(ξ)f(ξ)π(ξ,x3)eλ(ξ)teix′⋅ξ, |
are solutions to (1.20). Due to B(0,R2)⊂supp(f)⊂B(0,R3), the following inequalities follow from Lemma 3.5:
supξ∈supp(f)||∂kx3ˆw(ξ,⋅)||L∞<∞, |
and
supξ∈supp(f)||∂kx3π(ξ,⋅)||L∞<∞, |
for every k∈N.
Meanwhile, λ(ξ)≤√g[ρ]ρ−|ξ|. This boundedness indicates that the functions given by (3.36)–(3.38) are also a solution to (1.20).
For any k≥0, by applying Lemma 3.5, and where f is compactly supported, we easily achieve the estimate (3.39). According to (3.22) and (3.25), one has
0<√ˉc2R2≤√g[ρ]4(ρ++ρ−)|ξ|≤λ(|ξ|)≤√g[ρ]ρ−|ξ|≤√ˉc1R3, |
which derives the bounds (3.40).
Now, we will study the ill-posedness for the linearized problem. Suppose that (φ,v,q) is the solution to (1.20). Further, assume that the solution is band-limited at radius R>0, that is,
∪x3∈(−1,1)supp(|ˆφ(⋅,x3)|+|ˆv(⋅,x3)|+|ˆq(⋅,x3)|)⊂B(0,R). |
Since in the lower fluid the equation in (1.20) has Lorenz force, it is appropriate to use the second-derivative of velocity. Differentiating the second equation and the fifth Eq (1.20) in time and removing φ− by using the fourth equation, one arrives at
{ρ+∂ttv++∇∂tq+=0,ρ−∂ttv−+∇∂tq−−M2∂211v−=0,divv±=div(∂tv±)=0, | (3.41) |
where we used ˉM=(M,0,0). We impose the conditions:
[[v3]]=[[∂tv3]]=0,[[∂tq]]e3=g[ρ]v3e3, | (3.42) |
∂tv3−(t,x1,x2,−1)=∂tv3+(t,x1,x2,1)=0. | (3.43) |
∂tv(0) satisfies
{ρ+∂tv+(0)=−∇q+(0),ρ−∂tv−(0)=−∇q−(0)+M2∂211φ−(0). | (3.44) |
The first result is about the estimate of energy in terms of v for the evolution Eq (3.41).
Lemma 3.7. For solutions to (3.41)–(3.43), we have
12ddt(∫Ωρ|∂tv|2dx+∫R2×(−1,0)M2|∂1v−|2dx−∫R2g[ρ]|v3(x′,0)|2dx′)=0. | (3.45) |
Proof. Multiply (3.41)1 and (3.41)2 by ∂tv±(t) and integrate over Ω±, respectively. After integration by parts and employing (3.41)3, we arrive at
12ddt∫Ω+ρ+|∂tv+|2dx−∫R2∂tq+∂tv3+|x3=0dx′=0, | (3.46) |
12ddt∫Ω−ρ−|∂tv−|2dx−∫R2∂tq−∂tv3−|x3=0dx′+12ddt∫Ω−M2|∂1v−|2dx=0. | (3.47) |
Adding (3.46) and (3.47), using (3.42)2, we yield (3.7).
Lemma 3.8. If v satisfies that v∈H1(Ω) is band-limited at radius R>0, divv=0, and v3(t,x1,x2,±1)=0, then we arrive at
∫R2g[ρ]|v3(x′,0)|2dx′≤(R2+1)g[ρ]∫Ω|v|2dx. | (3.48) |
Proof. Utilizing the horizontal Fourier transform to divv=0 and denoting
κ(x3)=i^v1(ξ1,ξ2,x3),ψ(x3)=i^v2(ξ1,ξ2,x3)andθ(x3)=^v3(ξ1,ξ2,x3), | (3.49) |
we have
ξ1κ+ξ2ψ+θ′=0. | (3.50) |
From (2.2), (3.49), and (3.50), one has
∫R2g[ρ]|v3(x′,0)|2dx′=14π2∫R2g[ρ]|θ(0)|2dξ=g[ρ]4π2∫R2∫10∂x3|θ(0)|2dx3dξ≤g[ρ]4π2∫R2∫1−1(|θ|2+|θ′|2)dx3dξ=g[ρ]4π2∫R2∫1−1(|θ|2+|ξ1κ|2+|ξ2ψ|2)dx3dξ≤g[ρ]4π2∫R2∫1−1(|θ|2+R2|κ|2+R2|ψ|2)dx3dξ=g[ρ]∫R2∫1−1(|v3|2+R2|v1|2+R2|v2|2)dx≤(R2+1)g[ρ]∫R|v|2dx, |
which gives (3.48).
We may now derive growth estimates for v(t) and ∂tv(t).
Proposition 3.9. If v is a solution to (3.41) and is also band-limited at radius R>0, the following estimate holds:
||v(t)||2L2(Ω)+||∂tv(t)||2L2(Ω)≤ce((R2+1)g[ρ]ρ−+1)t(||v(0)||H1(Ω)+||∂tv(0)||L2(Ω)), | (3.51) |
where c depends on ρ,M,g,R.
Proof. Integrate (3.7) with regard to time from 0 to t to achieve
∫Ωρ|∂tv|2dx+∫R2×(−1,0)M2|∂1v−|2dx−∫R2|v3(t,x′,0)|2dx′=∫Ωρ|∂tv(0)|2dx+∫R2×(−1,0)M2|∂1v−(0)|2dx−∫R2g[ρ]|v3(0,x′,0)dx′. |
Thus, we have
∫Ωρ|∂tv|2dx≤A+∫R2g[ρ]|v3(t,x′,0)|2dx′, | (3.52) |
where
A=∫Ωρ|∂tv(0)|2dx+∫R2×(−1,0)M2|∂1v−(0)|2dx≤ρ+||∂tv(0)||L2(Ω)+M2||∂1v(0)||2L2. | (3.53) |
We apply (3.48) to (3.52) to get the inequality
∫Ωρ|∂tv|2dx≤A+(R2+1)g[ρ]∫Ω|v|2dx, |
which implies
||∂tv(t)||2≤Aρ−+(R2+1)g[ρ]ρ−∫Ω|v|2dx. | (3.54) |
By virtue of the Cauchy-Schwartz inequality, one can show that
∂t||v(t)||2=2⟨∂tv(t),v(t)⟩≤||∂tv(t)||L2(Ω)≤Aρ−+((R2+1)g[ρ]ρ−+1)||v(t)||2L2(Ω), | (3.55) |
where ⟨⋅,⋅⟩ denotes the L2 inner product. Applying the Gronwall inequality to (3.55), we derive
||v(t)||2L2(Ω)≤e((R2+1)g[ρ]ρ−+1)t(||v(0)||2L2(Ω)+A(R2+1)g[ρ]). | (3.56) |
Combining (3.55) and (3.56), we obtain
||∂tv(t)||2L2(Ω)+||v(t)||2L2(Ω)≤ce((R2+1)g[ρ]ρ−+1)t(||∂tv(0)||2L2(Ω)+||v(0)||2H1(Ω)), |
where c depends ρ±,g,R,M.
We have shown the existence of the solutions to the Eq (1.20). To investigate the ill-posedness, we turn to verify the uniqueness and discontinuous dependence on the initial conditions of the solutions. To do this, we first build a projection operator related to the horizontal spatial frequency. Let Φ be a function that is infinitely differentiable and a compact support in R2, satisfies Φ∈[0,1],supp(Φ)⊂B(0,1), and Φ(x)=1 for x∈B(0,12), then define ΦR(x)=Φ(xR) for R>0. For f∈L2(Ω), the projection operator PR is defined by
PRf=F−1(ΦRFf). | (3.57) |
It is easy to show that PR verifies the following properties [11]:
(1) PRf is band-limited at radius R;
(2) PR is a bounded linear operator on Hk(Ω) for all k≥0;
(3)PR commutes with partial differential and multiplication by functions depending only on x3;
(4) PRf=0 for all R>0 if, and only if, f=0.
Theorem 3.10. Solutions to (1.20) are unique.
Proof. We only need to prove that when the initial data is zero, the solutions to (1.20) are also zero. Assume that η±,v±,q± solve (1.20) with zero initial conditions. For any fixed R>0, define ηR=PRη,vR=PRv,qR=PRq, then ηR,vR, and qR also solve (1.20). Moreover, vR also solves (3.41) with zero initial value. By virtue of (3.51), for any t≥0, we derive
||vR(t)||L2(Ω)=||∂tvR(t)||L2(Ω)=0. | (3.58) |
Thus, there is φR(t)=qR(t)=vR(t)=0 for all t≥0. Due to the arbitrariness of R, we have that φ(t)=v(t)=q(t)=0 for all t≥0.
Lastly, we will show that the solution to the problem (1.20) is discontinuously dependent on the initial data.
Now, let us complete the proof of Theorem 2.1.
Proof. Fix j≥k≥0,α>0,T0>0. Let positive constants ˜Ck,R1,D come from Lemma 3.5 and Theorem 3.6. For every n∈N, take R(n) large enough such that R(n)>R1,√g[ρ]4(ρ++ρ−)R(n)>1, and
exp(T0√g[ρ]4(ρ++ρ−)R(n))(1+(R(n)+1)2)j−k+1≥α2n2ˉC2jD2. | (3.59) |
Choose a family of real-valued, radial, and compact supported functions fn as f in (3.36)–(3.38) so that B(0,R(n))⊂supp(fn)⊂B(0,R(n)+1) and
∫R2(1+|ξ|2)j+1|fn(ξ)|2dξ=1ˉC2jn2. | (3.60) |
Take R2=R(n) and R3=R(n)+1 in Theorem 3.6 to get φn(t),vn(t),qn(t)∈Hj(Ω) that solves (1.20) for all t≥0. By virtue of (3.39) and (3.60), we have that (2.4) holds for all n. Due to the definition (2.2), there is
‖φn(T0)‖2Hk(Ω±)=k∑j=0∫R2(1+|ξ|2)k−j|∂jx3ˆφn(T0,ξ)|2dξ=k∑j=0∫R2(1+|ξ|2)k−j‖∂jx3ˆφn(T0,ξ,⋅)‖2L2(−1,1)dξ≥∫R2(1+|ξ|2)k‖ˆφn(T0,ξ,⋅)‖2L2(−1,1)dξ=∫R2(1+|ξ|2)k‖φn(T0,ξ,⋅)‖2L2(−1,1)dξ=∫R2(1+|ξ|2)k|fn(ξ)|2e2T0λ(ξ)‖ˆw(ξ,⋅)‖2L2(−1,1)dξ≥exp(T0√g[ρ]4(ρ++ρ−)R(n))(1+(R(n)+1)2)j−k+1∫R2(1+|ξ|2)j+1|fn(ξ)|2e2T0λ(ξ)‖ˆw(ξ,⋅)‖2L2(−1,1)dξ≥α2n2ˉC2jD2∫R2(1+|ξ|2)j+1|fn(ξ)|2D2dξ=α2, |
where we used (3.25), (3.40), and (3.30).
Since λ(|ξ|)≥√g[ρ]4(ρ++ρ−)R(n)≥1 on the support of fn, we also arrive at
‖vn(t)‖2Hk≥‖φn(t)‖2Hk≥‖φn(T0)‖2Hk,fort≥T0. |
We finish the proof of Theorem 2.1.
We focus on showing the ill-posedness of the nonlinear system. Since A=Id,S−=Id{x3=0}, v±=0,φ±=Id,q±=const are the steady-state solutions to (1.18), one can rewrite (1.18) using the perturbation equations near the steady-state solutions. Let
φ±=Id+˜φ±,φ−1±=Id−ζ±,q±=const+σ±,A±=I−G±, | (4.1) |
where GT±=∑∞n=1(−1)n−1(D˜φ±)n.
Substituting (4.1) into (1.18) with ˉM=(M,0,0), we yield the following equations about ˜φ±,v±,σ±
{∂t˜φ±=v±,ρ+∂tv++(I−G+)∇σ+=0,ρ−∂tv−+(I−G−)∇σ−−M2∂11˜φ−=0,divv±−tr(G±∇v±)=0, | (4.2) |
where tr(⋅) is the matrix trace. The following compatibility conditions are required:
ζ±=˜φ±∘(Id−ζ±). |
We impose the corresponding jump conditions as follows:
(v+(t,x1,x2,0)−v−(t,S−(x1,x2)))⋅ν(t,x1,x2,0)=0, | (4.3) |
(σ+(t,x1,x2,0)−σ−(t,S−(x1,x2))⋅ν(t,x1,x2,0)=g[ρ]˜φ3+(t,x1,x2,0)ν(t,x1,x2,0)−M2(e1+∂1˜φ−)(e1+∂1˜φj−)(t,S−(t,x1,x2))νj(t,x1,x2,0), | (4.4) |
where
S−=(IdR2−ζ−)∘(IdR2+˜φ+)=IdR2+˜φ+−ζ−∘(IdR2+˜φ+), | (4.5) |
v−(t,x1,x2,−1)⋅e3=v+(t,x1,x2,1)⋅e3=0. | (4.6) |
We collect the equation, jump, and boundary Eqs (4.2)–(4.6) as "the perturbed problem". For k≥0, we use the following abbreviation:
‖(˜φ,v,σ)(t)‖Hk=‖˜φ(t)‖Hk+‖v(t)‖Hk+‖σ(t)‖Hk. | (4.7) |
Before proving it, we give an importance definition.
Definition 4.1. (Property EE(k)) For any δ,t0,C>0, and the initial data ˜φ0,v0,σ0 meeting
‖(˜φ0,v0,σ0)‖Hk<δ, | (4.8) |
there exists (˜φ,v,σ)∈L∞((0,t0);H3(Ω)), which solves the perturbed problems (4.2)–(4.6) on Ω×(0,t0) and satisfies:
(1) φ(t)=Id+˜φ(t) is reversible and φ−1(t)=Id−ζ(t) for 0≤t<t0, and
(2)
sup0≤t<t0‖(˜φ,v,σ)(t)‖H3≤Q(‖(˜φ0,v0,σ0)‖Hk), | (4.9) |
where Q:[0,δ)→R+ and Q(y)≤Cy for z∈[0,δ). We say the perturbed problems (4.2)–(4.6) has property EE(k).
Next, we will use the proof by contradiction to prove Theorem 2.3.
Proof. For some k≥4, we assume that the problems (4.2)–(4.6) has the property EE(k) of the above definition. For n∈N, let T=t02,k≥4, and α=1 in Theorem 2.1. Then, ˉφ,ˉv,ˉσ solves (1.20) with ˉM=(M,0,0) and the initial data satisfying
‖(ˉφ,ˉv,ˉσ)(0)‖Hk<1n, |
but
‖ˉv(t)‖H4≥‖ˉφ(t)‖H4≥1,fort≥T. | (4.10) |
For any ε>0, denote
ˉφε0=εˉφ(0),ˉvε0=εˉv(0),ˉσε0=εˉσ(0), |
then we have
‖(ˉφε0,ˉvε0,ˉσε0)‖Hk<εn. |
Select n such that n>C,εn<δ, where C,δ are the constants in the above property EE(k).
Due to EE(k), the perturbed problem exists a solution (˜φε,vε,σε)∈L∞((0,t0);H4(Ω)) with (ˉφε0,ˉvε0,ˉσε0) as the initial data. In addition,
sup0≤t<t0‖(˜φε,vε,σε,∂tσε)(t)‖H4≤Q(‖(ˉφε0,ˉvε0,ˉσε0‖Hk)≤Cε‖(ˉφ,ˉv,ˉσ)(0)‖Hk<ε. | (4.11) |
Defining
ˉφε=˜φεε,ˉvε=vεε,ˉσε=σεε, | (4.12) |
and inputting them into (4.11), we derive
sup0≤t<t0‖(ˉφε,ˉvε,ˉσε,∂tˉσε)‖H4≤1, | (4.13) |
and
(ˉφε,ˉvε,ˉσε)(0)=(ˉφ,ˉv,ˉσ)(0). | (4.14) |
We next demonstrate that
limε→0(ˉφε,ˉvε,ˉσε)=(ˉφ,ˉv,ˉσ), |
where (ˉφ,ˉv,ˉσ) solves the linearized system (1.20) with ˉM=(M,0,0). Substitute (4.12) into (4.2), then we have
{∂tˉφε±=ˉvε±,ρ+∂tˉvε++(I−εˉGε+)∇ˉσε+=0,ρ−∂tˉvε−+(I−εˉGε−)∇ˉσε−−M2∂11ˉφε−=0,divˉvε±−tr(ˉGε±∇ˉvε±)=0, | (4.15) |
where
ˉGε±:=I−(I+εDˉφT±)−1ε, | (4.16) |
then ˉGε± is well-defined. Thus,
‖ˉGε±‖H2=‖∞∑n=1(−ε)n−1(Dˉφε±)n‖H2≤∞∑n=1εn−1‖(Dˉφε±)n‖H2≤∞∑n=1(εK1)n−1‖Dˉφε±)‖nH2≤∞∑n=112n−1‖ˉφε±)‖nH4<∞∑n=112n−1=2, | (4.17) |
where the positive constant K1 is the optimal constant in the inequality ‖FH‖H2≤K1‖F‖H2‖H‖H2. Take ε small enough so that ε<12K1, then ˉGε± is uniform boundness in L∞(0,t0;H2(Ω)).
Now we will show some convergence results. From (4.15)1, one gets
sup0≤t<t0‖∂tˉφε±(t)‖H4=sup0≤t<t0‖ˉvε±(t)‖H4≤1. | (4.18) |
Expanding (4.15)2, we have
ρ+∂tˉvε++∇ˉσε+−εˉGε+∇σε+=0, | (4.19) |
whence
limε→0sup0≤t<t0‖ρ+∂tˉvε++∇ˉσε+‖H3=0, | (4.20) |
and
sup0≤t<t0‖∂tˉvε+‖H3≤K3for some constantK3>0. | (4.21) |
By virtue of (4.15)3, we achieve
ρ−∂tˉvε−+∇ˉσε−−εˉGε−∇σε−−M2∂11ˉφε−=0, | (4.22) |
which implies
limε→0sup0≤t<t0‖ρ−∂tˉvε−+∇ˉσε−−M2∂11ˉφε−‖H2=0, | (4.23) |
Thus, we have
sup0≤t<t0‖∂tˉvε−‖H2≤K4for some constantK4>0, | (4.24) |
(4.15)4 implies
limε→0sup0≤t<t0‖divˉvε±‖H3=0. | (4.25) |
The convergence results about the jump conditions are as follows. Due to the invertibility of Id+εˉφε, denote ˉζε by
(Id+εˉφε)−1=Id−εˉζε, |
which means
ˉζε=ˉφε∘(Id−εˉζε), |
then Sε−:R2×R+→R2×{0} can be expressed by
Sε−=IdR2+εˉφε+−εˉζε∘(IdR2+εˉφε+). | (4.26) |
Hence,
sup0≤t≤t0‖Sε−(t)−IdR2‖L∞≤2εsup0≤t<t0‖ˉφε(t)‖L∞≤2εK2sup0≤t<t0‖ˉφε‖H4<2εaK2, | (4.27) |
where the positive constant K2 is the Sobolev embedding constant in the trace mapping H4(Ω)↪L∞(R2×{0}). Define ˉSε−=Sε−−IdR2ε, then ˉSε− is uniform boundness in L∞((0,t0);L∞(R2×{0})) by (4.27). Denote the normal at the interface by νε=Nε|Nε| with
Nε=(e1+ε∂x1ˉφε+)×(e2+ε∂x2ˉφε+)=e3+ε(e1×∂x2ˉφε++∂x1ˉφε+×e2+ε∂x1ˉφε+×∂x2ˉφε+):=e3+εˉNε. | (4.28) |
As ε→0, one gets |Nε|>0. The jump condition (4.3) can be rewritten as follows:
(ˉvε+−ˉvε−∘(IdR2+εˉSεa−))⋅(e3+εˉNε)=0. | (4.29) |
It is obvious that sup0≤t<t0‖ˉNε(t)‖L∞ is uniformly bounded since
sup0≤t<t0‖ˉvε−∘(IdR2+εˉSε−)−ˉvε−‖L∞≤sup0≤t<t0‖Dˉvε(t)‖L∞sup0≤t<t0‖εˉSε−(t)‖L∞→0asε→0. |
Therefore,
sup0≤t<t0‖e3⋅(ˉvε+(t)−ˉvε−(t))‖L∞→0asε→0. | (4.30) |
For the jump condition (4.4), we can rewrite it as
[ˉσε+−ˉσε−∘(IdR2+εˉSε−)−g[ρ]ˉφε,3+](e3+εNε)=−M2(ˉN1,ε+∂1ˉφ3,ε−)∘(IdR2+εˉSε−)e1−εM2ˉFε, |
where
ˉQε=[(∂1ˉφε−⋅ˉNε)e1+(ˉN1,ε+∂1ˉφ3,ε−)∂1ˉφε−+ε(∂1ˉφ−⋅ˉNε)∂1ˉφε−]∘(IdR2+εˉSε−). |
Obviously, sup0≤t<t0‖ˉQε(t)‖L∞ is uniformly bounded. Thus, we achieve
sup0≤t<t0‖(ˉσv+−ˉσε−−g[ρ]ˉφ3,ε+)e3+M2∂1ˉφ3,ε−)e1‖L∞→0,asε→0. | (4.31) |
Collecting (4.13), (4.18), (4.21), and (4.24), there exists (ˉφ0,ˉv0,ˉσ0,∂tˉσ0,∂tˉφ0)∈L∞(0,t0;H4(Ω)) so that
(ˉφε,ˉvε,ˉσε,∂tˉσε,∂tˉφε)→(ˉφ0,ˉv0,ˉσ0,∂tˉσ0,∂tˉφ0)weak-* in L∞(0,t0;H4(Ω)), |
and
∂tˉvε→∂tv0weakly-∗ inL∞(0,t0;H2(Ω)). | (4.32) |
By virtue of the lower semi-continuity, we derive
sup0≤t<t0‖(ˉφ0,ˉv0,ˉσ0)(t)‖H4≤1. | (4.33) |
Combining (4.13), (4.18), (4.21), and (4.24), one has
lim supε→0sup0≤t<t0‖(∂tˉφε,∂tˉvε,∂tˉσε)‖H2<∞. |
By virtue of a conclusion in [17], (ˉφε,ˉvε,ˉσε) is strongly pre-compact in L∞(0,t0;H114(Ω)). So, we have
(ˉφε,ˉvε,ˉσε)strongly→(ˉφ0,ˉv0,ˉσ0)inL∞(0,t0;H114(Ω)). | (4.34) |
Following from the above strong convergence, the convergence results (4.20) and (4.23), and the equation ∂tˉφε=ˉvε, we arrive at
∂tˉφεstrongly→∂tˉφ0 inL∞(0,t0;H114(Ω)),∂tˉvεstrongly→∂tˉv0inL∞(0,t0;L2(Ω)), |
and
{∂tˉφ0±=ˉv0±,ρ+∂tˉv0++∇ˉσ0+=0,ρ−∂tˉv0−+∇ˉσ0−−M2∂211ˉφ0−=0,divˉv0±=0. | (4.35) |
Taking the limit for (3.44), there is
(ˉφ0,ˉv0,ˉσ0)(0)=(ˉφ,ˉv,ˉσ)(0). | (4.36) |
Combining (4.30) and (4.31), we infer
ˉv0+⋅e3=0on{x3=1},ˉv0−⋅e3=0on{x3=−1},(ˉv0+−ˉv0−)⋅e3=0on{x3=0}, | (4.37) |
and
(ˉσ0+−ˉσ0−−g[ρ]ˉφ3,0+)e3+M2∂1ˉφ3,0−e1=0on{x3=0}. | (4.38) |
(4.35)–(4.38) imply that (ˉφ0,ˉv0,ˉσ0) solves (1.20)–(1.23) with ˉM=(M,0,0) and meets the initial conditions (4.24). Thereby,
(ˉφ0,ˉv0,ˉσ0)=(ˉφ,ˉv,ˉσ)on[0,t0)×Ω, |
follows from Theorem 3.10. Furthermore, collect the inequality (4.33) and (4.10) to yield
2≤supt02≤t<t0‖(ˉφ0,ˉv0,ˉσ0)(t)‖H4≤sup0≤t<t0‖(ˉφ0,ˉv0,ˉσ0)(t)‖H4≤1, |
which is a contradiction. Therefore, for any k≥4, the property EE(k) is not valid for the perturbed problem. Theorem 2.3 has been proven.
We investigated the RT instability problem of the two-phase flow coupled with ideal fluid and magnetohydrodynamic. We obtained the RT instability of linear problems by establishing a growth mode solution to the linearization problem near the steady-state solution. By virtue of the instability of linearization problems, we ultimately obtained the RT instability of nonlinear problems.
The author would like to thank the editor and the referees for their careful reading and helpful comments. This work was partially supported by National Natural Science Foundation of China (Grant No. 11901249).
The author declares that there are no conflicts of interest regarding the publication of this paper.
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