The time-fractional partial differential equations were solved by the fractional natural transform decomposition method. Fractional derivatives are Caputo sense. The Fornberg-Whitham equation is a generalization of the Korteweg-de Vries (KdV) equation, which describes the propagation of long waves in shallow water. It includes higher-order dispersion terms, making it applicable to a wider range of dispersive media the fractional natural transform decomposition method (FNTDM) was also used to examine applications, and the solutions obtained by this method have been compared to those obtained by the variational iteration method, fractional variational iteration method, and homotopy perturbation method. In addition, the MAPLE package drew graphs of the solutions of nonlinear time-fractional partial differential equations, taking into account physics. The method described in this study exhibited a notable degree of computational precision and straightforwardness when used to the analysis and resolution of intricate phenomena pertaining to fractional nonlinear partial differential equations within the domains of science and technology.
Citation: Aslı Alkan, Halil Anaç. The novel numerical solutions for time-fractional Fornberg-Whitham equation by using fractional natural transform decomposition method[J]. AIMS Mathematics, 2024, 9(9): 25333-25359. doi: 10.3934/math.20241237
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The time-fractional partial differential equations were solved by the fractional natural transform decomposition method. Fractional derivatives are Caputo sense. The Fornberg-Whitham equation is a generalization of the Korteweg-de Vries (KdV) equation, which describes the propagation of long waves in shallow water. It includes higher-order dispersion terms, making it applicable to a wider range of dispersive media the fractional natural transform decomposition method (FNTDM) was also used to examine applications, and the solutions obtained by this method have been compared to those obtained by the variational iteration method, fractional variational iteration method, and homotopy perturbation method. In addition, the MAPLE package drew graphs of the solutions of nonlinear time-fractional partial differential equations, taking into account physics. The method described in this study exhibited a notable degree of computational precision and straightforwardness when used to the analysis and resolution of intricate phenomena pertaining to fractional nonlinear partial differential equations within the domains of science and technology.
In this paper, we study the 2D steady compressible Prandtl equations in {x>0,y>0}:
{u∂xu+v∂yu−1ρ∂2yu=−∂xP(ρ)ρ,∂x(ρu)+∂y(ρv)=0,u|x=0=u0(y),limy→∞u=U(x),u|y=0=v|y=0=0, | (1.1) |
where (u,v) is velocity field, ρ(x) and U(x) are the traces at the boundary {y=0} of the density and the tangential velocity of the outer Euler flow. The states ρ,U satisfy the Bernoulli law
U∂xU+∂xP(ρ)ρ=0. | (1.2) |
The pressure P(ρ) is a strictly increasing function of ρ with 0<ρ0≤ρ≤ρ1 for some positive constants ρ0 and ρ1.
In this paper, we assume that the pressure satisfies the favorable pressure gradient ∂xP≤0, which implies that
∂xρ≤0. |
The boundary layer is a very important branch in fluid mechanics. Ludwig Prandtl [14] first proposed the related theory of the boundary layer in 1904. Since then, many scholars have devoted themselves to studying the mathematical theory of the boundary layer [1,7,8,9,11,12,17,18,19,21,22,23,24,26,27]. For more complex fluids, such as compressible fluids, one can refer to [19,20,28] and the references therein for more details. Here, for our purposes, we only list some relevant works.
There are three very natural problems about the steady boundary layer: (ⅰ) Boundary layer separation, (ⅱ) whether Oleinik's solutions are smooth up to the boundary for any x>0 and (ⅲ) vanishing viscosity limit of the steady Navier-Stokes system. Next, we will introduce the relevant research progress in these three aspects. The separation of the boundary layer is one of the very important research contents in the boundary layer theory. [17]. The earliest mathematical theory in this regard was proposed by Caffarelli and E in an unpublished paper [25]. Their results show that the existence time x∗ of the solutions to the steady Prandtl equations in the sense of Oleinik is finite under the adverse pressure gradient. Moreover, the family uμ(x,y)=μ−12u(x∗−μx,μ14y) is compact in C0(R2+). Later, Dalibard and Masmoudi [4] proved the solution behaves near the separation as ∂yu(x,0)∼(x∗−x)12 for x<x∗. Shen, Wang and Zhang [18] found that the solution near the separation point behaves like ∂yu(x,y)∼(x∗−x)14 for x<x∗. The above work further illustrates that the boundary layer separation is a very complex phenomenon. Recently, there were also some results about the steady compressible boundary layer separation [28]. The authors found that if the heat transfer in the boundary layer disappeared, then the singularity would be the same as that in the incompressible case. There is still relatively little mathematical theory on the separation of unsteady boundary layers. This is because back-flow and separation no longer occur simultaneously. When the boundary layer back-flow occurs, the characteristics of the boundary layer will continue to maintain for a period of time. Therefore, it is very important to study the back-flow point for further research on separation. Recently, Wang and Zhu [21] studied the back-flow problem of the two-dimensional unsteady boundary layer, which is a important work. It is very interesting to further establish the mathematical theory of the unsteady boundary layer separation.
Due to degenerate near the boundary, the high regularity of the solution of the boundary layer equation is a very difficult and meaningful work. In a local time 0<x<x∗≪1, Guo and Iyer [6] studied the high regularity of of the Prandtl equations. Oleinik and Samokhin [13] studied the existence of solutions of steady Prandtl equations and Wang and Zhang [23] proved that Oleinik's solutions are smooth up to the boundary y=0 for any x>0. The goal of this paper is to prove the global C∞ regularity of the two-dimensional steady compressible Prandtl equations. Recently, Wang and Zhang [24] found the explicit decay for general initial data with exponential decay by using the maximum principle.
In addition, in order to better understand the relevant background knowledge, we will introduce some other related work. As the viscosity goes to zero, the solutions of the three-dimensional evolutionary Navier-Stokes equations to the solutions of the Euler equations are an interesting problem. Beirão da Veiga and Crispo [2] proved that in the presence of flat boundaries convergence holds uniformly in time with respect to the initial data's norm. For the non-stationary Navier-Stokes equations in the 2D power cusp domain, the formal asymptotic expansion of the solution near the singular point is constructed and the constructed asymptotic decomposition is justified in [15,16] by Pileckas and Raciene.
Before introducing the main theorem, we introduce some preliminary knowledge. To use the von Mises transformation, we set
˜u(x,y)=ρ(x)u(x,y),˜v(x,y)=ρ(x)v(x,y),˜u0(y)=ρ(0)u0(y), |
then we find that (˜u,˜v) satisfies:
{˜u∂x˜u+˜v∂y˜u−∂2y˜u−∂xρρ˜u2=−ρ∂xP(ρ),∂x˜u+∂y˜v=0,˜u|x=0=˜u0(y),limy→∞˜u=ρ(x)U(x),˜u|y=0=˜v|y=0=0. | (1.3) |
By the von Mises transformation
x=x,ψ(x,y)=∫y0˜u(x,z)dz,w=˜u2, | (1.4) |
∂x˜u=∂xω2√ω+∂ψω∂xψ2√ω,∂y˜u=∂ψω2,∂2y˜u=√ω∂2ψω2, | (1.5) |
and (1.3)–(1.5), we know that w(x,ψ) satisfies:
∂xw−√w∂2ψw−2∂xρρw=−2ρ∂xP(ρ), | (1.6) |
with
w(x,0)=0,w(0,ψ)=w0(ψ),limψ→+∞w=(ρ(x)U(x))2. | (1.7) |
In addition, we have
2∂y˜u=∂ψw,2∂2y˜u=√w∂2ψw. | (1.8) |
In [5], Gong, Guo and Wang studied the existence of the solutions of system (1.1) by using the von Mises transformation and the maximal principle proposed by Oleinik and Samokhin in [13]. Specifically, they proved that:
Theorem 1.1. If the initial data u0 satisfies the following conditions:
u∈C2,αb([0,+∞))(α>0),u(0)=0,∂yu(0)>0,∂yu(y)≥0fory∈[0,+∞),limy→+∞u(y)=U(0)>0,ρ−1(0)∂2yu(y)−ρ−1(0)∂xP(0)=O(y2) | (1.9) |
and ρ∈C2([0,X0]), then there exists 0<X≤X0 such that system (1.1) admits a solution u∈C1([0,X)×R+). The solution has the following properties:
(i) u is continuous and bounded in [0,X]×R+; ∂yu,∂2yu are continuous and bounded in [0,X)×R+; v,∂yv,∂xu are locally bounded in [0,X)×R+.
(ii) u(x,y)>0 in [0,X)×R+ and for any ˉx<X, there exists y0,m>0 such that for all (x,y)∈[0,ˉx]×[0,y0],
∂yu(x,y)≥m>0. |
(iii) if ∂xP≤0(∂xρ≤0), then
X=+∞. |
Remarks 1.2. u∈C2,αb([0,+∞))(α>0) means that u is Hölder continuity and bounded.
This theorem shows that under the favorable pressure gradient, the solution is global-in-x. However, if the pressure is an adverse pressure gradient, then boundary layer separation will occur. Xin and Zhang [26] studied the global existence of weak solutions of unsteady Prandtl equations under the favorable pressure gradient. For the unsteady compressible Prandtl equation, similar results are obtained in [3]. Recently, Xin, Zhang and Zhao [27] proposed a direct proof of the existence of global weak solutions of the Prandtl equation. The key content of this paper is that they have studied the uniqueness and regularity of weak solutions. This method can be applied to the compressible Prandtl equation.
Our main results are as follows:
Theorem 1.3. If u is a solution for equation (1.1) in Theorem 1.1, assume u0 satisfies the condition (1.9) and the known function ρ and ∂xP are smooth. Then, there exists a constant C>0 depending only on ε,X,u0,P(ρ),k,m such that for any (x,y)∈[ε,X]×[0,+∞),
|∂kx∂myu(x,y)|≤C, |
where X,ε are positive constants with ε<X and m,k are any positive integers.
Remarks 1.4. Our methods may be used to other related models. There are similar results for the magnetohydrodynamics boundary layer and the thermal boundary layer. This work will be more difficult due to the influence of temperature and the magnetic field.
Due to the degeneracy near the boundary ψ=0, the proof of the main result is divided into two parts, Theorem 1.5 and Theorem 1.6. This is similar to the result of the incompressible boundary layer, despite the fluid being compressible and the degeneracy near the boundary. Different from the incompressible case [23], we have no divergence-free conditions, which will bring new terms. It is one of the difficulties in this paper to deal with these terms. Now, we will briefly introduce our proof framework. First, we prove the following theorem in the domain [ε,X]×[0,Y1] for a small Y1. The key ingredients of proof is that we employ interior priori estimates and the maximum principle developed by Krylov [10].
Theorem 1.5. If u is a solution for equation (1.1) in Theorem 1.1, assume u0 satisfies the condition (1.9) and the known function ρ and ∂xP are smooth. Then, there exists a small constant Y1>0 and a large constant C>0 depending only on ε,X,Y1,u0,P(ρ),k,m such that for any (x,y)∈[ε,X]×[0,Y1],
|∂kx∂myu(x,y)|≤C, |
where X,ε are positive constants with ε<X and m,k are any positive integers.
Next, we prove the following theorem in the domain [ε,X]×[Y2,+∞) for a small positive constant Y2. The key of proof is that we prove (1.6) is a uniform parabolic equation in the domain [ε,X]×[Y2,+∞) in Section 4. Once we have (1.6) is a uniform parabolic equation, the global C∞ regularity of the solution is a direct result of interior Schauder estimates and classical parabolic regularity theory. The proof can be given similarly to the steady incompressible boundary layer. For the sake of simplicity of the paper, more details can be found in [23] and we omit it here.
Theorem 1.6. If u is a solution for equation (1.1) in Theorem 1.1, assume u0 satisfies the condition (1.9) and the known function ρ and ∂xP are smooth. Then, there exists a constant Y0>0 such that for any constant Y2∈(0,Y0), there exists a constant C>0 depending only on ε,X,Y2,u0,P(ρ),k,m such that for any (x,y)∈[ε,X]×[Y2,+∞),
|∂kx∂myu(x,y)|≤C, |
where X,ε are positive constants with ε<X and m,k are any positive integers.
Therefore, Theorem 1.3 can be directly proven by combining Theorem 1.5 with Theorem 1.6.
The organization of this paper is as follows. In Section 2, we study lower order and higher order regularity estimates. In Section 3, we prove Theorem 1.5 in the domain near y=0 by transforming back to the original coordinates (x,y). In Section 4, we prove (1.6) is a uniform parabolic equation by using the maximum principle and we also prove the Theorem 1.3.
In this subsection, we study the lower order regularity estimates using the standard interior a priori estimates developed by Krylov [10].
Lemma 2.1. If u is a solution for equation (1.1) in Theorem 1.1, assume u0 satisfies the condition (1.9)and the known function ρ and ∂xP are smooth. Assume 0<ε<X, then there exists some positive constants δ1>0 and C independent of ψ such that for any (x,ψ)∈[ε,X]×[0,δ1],
|∂xw(x,ψ)|≤Cψ. |
Proof. Due to Lemma 2.1 in [5] (or Theorem 2.1.14 in [13]), there exists δ1>0 for any (x,ψ)∈[0,X]×[0,δ1], such that for some α∈(0,12) and positive constants m,M (we assume δ1<1),
|∂xw|≤Cψ12+α,0<m<∂ψw<M,mψ<w<Mψ. | (2.1) |
By (1.6), we obtain
∂x∂xw−√w∂2ψ∂xw=(∂xw)22w+2ρ∂xP∂xw2w+∂xρρ∂xw+2∂x(∂xρρ)w−2∂x[ρ∂xP]. |
Take a smooth cutoff function 0≤ϕ(x)≤1 in [0,X] such that
ϕ(x)=1,x∈[ε,X],ϕ(x)=0,x∈[0,ε2], |
then
∂x[∂xwϕ(x)]−√w∂2ψ[∂xwϕ(x)]=(∂xw)22wϕ(x)+2ρ∂xP∂xw2wϕ(x)+∂xρρ∂xwϕ(x)+2∂x(∂xρρ)wϕ(x)−2∂x(ρ∂xP)ϕ(x)+∂xw∂xϕ(x):=W. |
Combining with (2.1), we know
|W|≤Cψ2α+Cψα−12+Cψα+12+Cψ+C≤Cψα−12. | (2.2) |
We take φ(ψ)=μ1ψ−μ2ψ1+β with constants μ1,μ2, then by (2.1) and (2.2), we get
∂x[∂xwϕ(x)−φ]−√w∂2ψ[∂xwϕ(x)−φ]≤|W|−μ2√wβ(1+β)ψβ−1≤Cψα−12−μ2√mβ(1+β)ψβ−12. |
By taking μ2 sufficiently large and α=β, for (x,ψ)∈(0,X]×(0,δ1), we have
∂x[∂xwϕ(x)−φ]−√w∂2ψ[∂xwϕ(x)−φ]<0. |
For any ψ∈[0,δ1], let μ1≥μ2, and we have
(∂xwϕ−φ)(0,ψ)≤0, |
and take μ1 large enough depending on M,δ1,μ2 such that
(∂xwϕ−φ)(x,δ1)≤Mδ12+α1−μ1δ1+μ2δ1+β1≤0. |
Since w(x,0)=0, we know that for any x∈[0,X],
(∂xwϕ−φ)(x,0)=0. |
By the maximum principle, it holds in [0,X]×[0,δ1] that
(∂xwϕ−φ)(x,ψ)≤0. |
Let δ1 be chosen suitably small, for (x,ψ)∈[ε,X]×[0,δ1], and we obtain
∂xw(x,ψ)≤μ1ψ−μ2ψ1+β≤μ12ψ. |
Considering −∂xwϕ−φ, the result −∂xw≤μ12ψ in [ε,X]×[0,δ1] can be proved similarly. This completes the proof of the lemma.
Lemma 2.2. If u is a solution for equation (1.1) in Theorem 1.1, assume u0 satisfies the condition (1.9) and the known function ρ and ∂xP are smooth. Assume 0<ε<X, then there exists some positive constants δ2>0 and C independent of ψ such that for any (x,ψ)∈[ε,X]×[0,δ2],
|∂ψ∂xw(x,ψ)|≤C,|∂2xw(x,ψ)|≤Cψ−12,|∂2ψ∂xw(x,ψ)|≤Cψ−1. |
Proof. From Lemma 2.1, there exists δ1>0 such that for any (x,ψ)∈[ε2,X]×[0,δ1],
|∂xw(x,ψ)|≤Cψ. |
Let Ψ0=min{23δ1,ε2}, for any (x0,ψ0)∈[ε,X]×(0,Ψ0], and we denote
Ω={(x,ψ)|x0−ψ320≤x≤x0,12ψ0≤ψ≤32ψ0}. |
By the definition of Ψ0, we know Ω⊆[ε2,X]×[0,δ1], then it holds in Ω that
|∂xw|≤Cψ. | (2.3) |
The following transformation f is defined:
Ω→˜Ω:=[−1,0]˜x×[−12,12]˜ψ,(x,ψ)↦(˜x,˜ψ), |
where x−x0=ψ320˜x,ψ−ψ0=ψ0˜ψ.
Since ∂˜x=ψ320∂x,∂˜ψ=ψ0∂ψ, it holds in Ω that
∂˜x(ψ−10w)−ψ−120√w∂2˜ψ(ψ−10w)−2∂˜xρρ(ψ−10w)=−2ρ∂˜xPψ−10. |
Combining with (2.1), we get 0<c≤ψ−120√w≤C,|ψ−10w|≤C, and for any ˜z1,˜z2∈˜Ω,
|ψ−120√w(˜z1)−ψ−120√w(˜z2)|=ψ−120|w(˜z1)−w(˜z2)|√w(˜z1)+√w(˜z2)≤Cψ0|˜z1−˜z2|ψ0=C|˜z1−˜z2|. |
This means that for any α∈(0,1), we have
|ψ−120√w|Cα(˜Ω)≤C. |
Since P and ρ are smooth, we have
|ρ−1∂˜xρ|C0,1([−1,0]˜x)+|ρ∂˜xPψ−10|C0,1([−1,0]˜x)≤C. |
By standard interior priori estimates (see Theorem 8.11.1 in [10] or Proposition 2.3 in [23]), we have
|wψ−10|Cα([−12,0]˜x×[−14,14]˜ψ)+|∂2˜ψwψ−10|Cα([−12,0]˜x×[−14,14]˜ψ)≤C. | (2.4) |
Let f:=∂xwψ−10, which satisfies
∂˜xf−√wψ120∂2˜ψf−∂2˜ψw2√wψ120f−2∂˜xρρf=−2∂x[ρ∂˜xP]ψ−10+2∂x(∂˜xρρ)(ψ−10w). |
By (2.3), we have |f|≤C in ˜Ω. Due to
|ψ120w−12(˜z1)−ψ120w−12(˜z2)|=ψ120|w(˜z1)−w(˜z2)w(˜z1)w(˜z2)|w−12(˜z1)+w−12(˜z2)≤C|˜z1−˜z2|, |
we have
|ψ120w−12|Cα(˜Ω)≤C. | (2.5) |
Since
∂2˜ψw2√wψ120=∂2˜ψwψ−10ψ1202√w, |
which along with (2.4) and (2.5) gives
|∂2˜ψw2√wψ120|Cα([−12,0]˜x×[−14,14]˜ψ)≤C. |
As before, by (2.4) and the density ρ and P are smooth, via the standard interior a priori estimates, it yield that
|∂˜xf|L∞([−14,0]˜x×[−18,18]˜ψ)+|∂˜ψf|L∞([−14,0]˜x×[−18,18]˜ψ)+|∂2˜ψf|L∞([−14,0]˜x×[−18,18]˜ψ)≤C. |
Therefore, we obtain
|∂2xw(x0,ψ0)|≤Cψ−120,|∂ψ∂xw(x0,ψ0)|≤C,|∂2ψ∂xw(x0,ψ0)|≤Cψ−10. |
This completes the proof of the lemma.
In this subsection, we study the higher order regularity estimates using the maximum principle. The two main results of this subsection are Lemma 2.3 and Lemma 2.7.
Lemma 2.3. If u is a solution for equation (1.1) in Theorem 1.1, assume u0 satisfies the condition (1.9) and the known function ρ and ∂xP are smooth. Assume 0<ε<X and k≥2, then there exists some positive constants δ>0 and C independent of ψ such that for any (x,ψ)∈[ε,X]×[0,δ],
|∂kxw|≤Cψ,|∂ψ∂kxw|≤C,|∂2ψ∂kxw|≤Cψ−1. |
Proof. By Lemma 2.1 and Lemma 2.2, we may inductively assume that for 0≤j≤k−1, there holds that in [ε2,X]×[0,δ3] (assume δ3≪1),
|∂ψ∂jxw|≤C,|∂2ψ∂jxw|≤Cψ−1,|∂jxw|≤Cψ,|∂jx√w|≤Cψ12,|∂kxw|≤Cψ−12. | (2.6) |
We will prove that there exists δ4<δ3 so that in [ε,X]×[0,δ4],
|∂ψ∂kxw|≤C,|∂2ψ∂kxw|≤Cψ−1,|∂kxw|≤Cψ,|∂kx√w|≤Cψ12,|∂k+1xw|≤Cψ−12. | (2.7) |
The above results are deduced from the following Lemma 2.4, Lemma 2.5 and Lemma 2.6.
Lemma 2.4. If u is a solution for equation (1.1) in Theorem 1.1, assume u0 satisfies the condition (1.9) and the known function ρ and ∂xP are smooth. Assume that (2.6) holds, then there is a positive constant M1 for any (x,ψ)∈[7ε8,X]×[0,δ3] and 0<β≪1,
|∂kxw|<M1ψ1−β,|∂kx√w|≤M1ψ12−β. |
Proof. Take a smooth cutoff function 0≤ϕ(x)≤1 in [0,X] such that
ϕ(x)=1,x∈[7ε8,X],ϕ(x)=0,x∈[0,5ε8]. |
As in [23], fix any h<ε8. Set
Ω={(x,ψ)|0<x≤X,0<ψ<δ3}, |
and let
(ⅰ) f=∂k−1xw(x−h,ψ)−∂k−1xw(x,ψ)−hϕ+Mψlnψ, (x,ψ)∈[5ε8,X]×[0,+∞),
(ⅱ) f=Mψlnψ, (x,ψ)∈[0,5ε8)×[ψ,+∞),
so we get f(x,0)=0, f(0,ψ)≤0. We know
f(x,δ3)≤C(δ3)−12+Mδ3lnδ3≤0, |
where M is large enough. Then, by choosing the appropriate M, we know that the positive maximum of f cannot be achieved in the interior. Finally, the lemma can be proven by the arbitrariness of h.
Assume that there exists a point
p0=(x0,ψ0)∈Ω, |
such that
f(p0)=maxˉΩf>0. |
It is easy to know that
x0>5ε8,∂k−1xw(x0−h,ψ0)<∂k−1xw(x0,ψ0). |
By (2.1), denote ξ=√m, we have
−√w∂2ψ(Mψlnψ)=−M√wψ−1≤−ξMψ−12. | (2.8) |
By (1.6), a direct calculation gives
∂x∂k−1xw−√w∂2ψ∂k−1xw=−2∂k−1x(ρ∂xP)+k−2∑m=1Cmk−1(∂k−1−mx√w)∂2ψ∂mxw+(∂k−1x√w)∂2ψw+2k−1∑m=0Cmk−1∂k−1−mx(∂xρρ)∂mxw=−2∂k−1x(ρ∂xP)+k−2∑m=1Cmk−1(∂k−1−mx√w)∂2ψ∂mxw+∂k−1xw2√w∂xw√w+(∂k−1xw2√w)2ρ∂xP√w−(∂k−1xw2√w)2∂xρρw√w+2k−1∑m=0Cmk−1∂k−1−mx(∂xρρ)∂mxw+k−2∑m=0Cmk−2∂2ψw∂m+1xw∂k−2−mx12√w:=4∑i=1Ii |
and
I1=−2∂k−1x(ρ∂xP)+k−2∑m=1Cmk−1(∂k−1−mx√w)∂2ψ∂mxw+∂k−1xw2√w∂xw√w,I2=ρ∂xPw∂k−1xw,I3=−∂xρρ∂k−1xw+2k−1∑m=0Cmk−1∂k−1−mx(∂xρρ)∂mxw,I4=k−2∑m=0Cmk−2∂2ψw∂m+1xw∂k−2−mx12√w. |
For x≥5ε8, we consider the following equality
∂xf1−√w(p1)∂2ψf1=√w(p1)−√w(p)−h∂2ψ∂k−1xw(p)+4∑i=11−h(Ii(p1)−Ii(p)), | (2.9) |
where
f1=1−h(∂k−1xw(p1)−∂k−1xw(p)), |
with p1=(x−h,ψ),p=(x,ψ).
For any x≥5ε8, by (2.6), it is easy to conclude that
|1−h(√w(p1)−√w(p))∂2ψ∂k−1xw(p)|≤Cψ−12,|1−h(I1(p1)−I1(p))|≤Cψ−12,|4∑i=31−h(Ii(p1)−Ii(p))|≤Cψ−12, | (2.10) |
where C is dependent on the parameter h.
Since
1−h(I2(p1)−I2(p))=f1⋅[ρ∂xPw(p1)]+∂k−1xw(p)1−h[ρ∂xPw(p1)−ρ∂xPw(p)], |
combining with (2.6), f1(p0)>0 and ∂xP≤0, it holds at p=p0 that
1−h(I2(p1)−I2(p0))≤C. | (2.11) |
Summing up (2.10) and (2.11), we conclude that at p=p0,
∂xf1−√w∂2ψf1≤C0ψ−12. |
This along with (2.8) shows that for x≥5ε8, it holds at p=p0 that
∂xf−√w∂2ψf≤Cψ−12−ξMψ−12. | (2.12) |
By taking M large enough, we have ∂xf(p0)−√w∂2ψf(p0)<0. By the definition of p0, we obtain
∂xf(p0)−√w∂2ψf(p0)≥0, |
which leads to a contradiction. Therefore, for M chosen as above and independent of h, we have
maxˉΩf≤0. |
We can similarly prove that minˉΩf≥0 by replacing Mψlnψ in f with −Mψlnψ. By the arbitrariness of h, for any (x,ψ)∈(7ε8,X]×(0,δ3] we have
|∂kxw|≤−Mψlnψ. |
Due to
2√w∂kx√w+k−1∑m=1Cmk(∂mx√w∂k−mx√w)=∂kx(√w√w)=∂kxw, | (2.13) |
which along with (2.6) shows that in (78ε,X]×(0,δ3],
|√w∂kx√w|≤−Cψlnψ. |
Lemma 2.5. If u is a solution for equation (1.1) in Theorem 1.1, assume u0 satisfies the condition (1.9) and the known function ρ and ∂xP are smooth. Assume that (2.6) holds, then for any (x,ψ)∈[1516ε,X]×[0,δ3],
|∂kxw|≤Cψ,|∂kx√w|≤Cψ12. |
Proof. Take a smooth cutoff function ϕ(x) so that
ϕ(x)=1,x∈[15ε16,X],ϕ(x)=0,x∈[0,7ε8]. |
Set
f=∂kxwϕ−μ1ψ+μ2ψ32−β |
with constants μ1,μ2. Let β be small enough in Lemma 2.4. Then it holds in [7ε8,X]×[0,δ3] that
|∂kxw|≤Cψ1−β,|∂kx√w|≤Cψ12−β. | (2.14) |
We denote
Ω={(x,ψ)|0<x≤X,0<ψ<δ3}. |
As in [23], we have f(x,0)=0, f(0,ψ)≤0 and f(x,δ3)≤0 by taking μ1 large depending on μ2. We claim that the maximum of f cannot be achieved in the interior.
By (1.6), we have
∂x∂kxw−√w∂2ψ∂kxw=−2∂kx(ρ∂xP)+k−1∑m=0Cmk(∂k−mx√w)∂2ψ∂mxw+2k∑m=0Cmk∂k−mx(∂xρρ)∂mxw, |
and
∂2ψ∂mxw=∂mx∂2ψw=∂mx(∂xw√w+2ρ∂xP√w−2∂xρρ√w). |
For any x≥7ε8, 0≤j≤k−1 and 0≤m≤k−1, from (2.6) and (2.14), we get
|∂jxw|≤Cψ,|∂kxw|≤Cψ1−β,|∂k−mx√w|≤Cψ12−β. |
Then let β≪12, for 0≤m≤k−1 and x≥7ε8, we obtain
|∂2ψ∂mxw|≤Cψ12−β+Cψ−12+Cψ12−β≤Cψ−12. |
Therefore, we conclude that for x≥7ε8,
∂x∂kxw−√w∂2ψ∂kxw≤C+Cψ−β+Cψ1−β≤Cψ−β. |
By the above inequality and (2.1), it holds at p=p0 that
∂xf−√w∂2ψf=∂x∂kxw−√w∂2ψ∂kxw+∂kxw∂xϕ−√w∂2ψ(−μ1ψ+μ2ψ32−β)≤C2ψ−β−ξμ2ψ−β, |
where ξ=(32−β)(12−β)√m>0. Then we have ∂xf−√w∂2ψf<0 in Ω by taking μ2 large depending on C2. This means that the maximum of f cannot be achieved in the interior. Therefore, we have
maxˉΩf≤0. |
In the same way, we can prove that
maxˉΩ−∂kxwϕ−μ1ψ+μ2ψ32−β≤0. |
So, for any (x,ψ)∈[1516ε,X]×[0,δ3], we have
|∂kxw|≤μ1ψ−μ2ψ32−β≤μ1ψ. |
Combining with (2.6) and (2.13), it holds in [1516ε,X]×[0,δ3] that
|∂kx√w|≤Cψ12. |
This completes the proof of the lemma.
Lemma 2.6. If u is a solution for equation (1.1) in Theorem 1.1, assume u0 satisfies the condition (1.9) and the known function ρ and ∂xP are smooth. Assume that (2.6) holds, then for any (x,ψ)∈[ε,X]×[0,δ4],
|∂ψ∂kxw|≤C,|∂2ψ∂kxw|≤Cψ−1,|∂k+1xw|≤Cψ−12. |
Proof. By Lemma 2.5 and (2.6), for any (x,ψ)∈[1516ε,X]×[0,δ3],
|∂jxw|≤Cψ,|∂jx√w|≤Cψ12,0≤j≤k. | (2.15) |
Set Ψ0=min{23δ3,ε16}, for (x0,ψ0)∈[ε,X]×(0,Ψ0], we denote
Ω={(x,ψ)|x0−ψ320≤x≤x0,12ψ0≤ψ≤32ψ0}. |
A direct calculation gives
∂x∂kxw−√w∂2ψ∂kxw=−2∂kx(ρ∂xP)+∂kx√w∂2ψw+k−2∑m=1Cmk(∂k−mx√w)∂2ψ∂mxw+Ck−1k∂xw2√w∂2ψ∂k−1xw+2k∑m=0Cmk∂k−mx(∂xρρ)∂mxw. |
By (1.6), we obtain
∂2ψ∂mxw=∂mx∂2ψw=∂mx(∂xw√w+2ρ∂xP√w−2∂xρρ√w)=∂m+1xw√w+m∑l=1Clm∂m−l+1xw∂lx1√w+∂mx(2ρ∂xP√w)−∂mx(2∂xρρ√w), |
and
∂kx√w=∂k−1x∂xw2√w=∂kxw2√w+k−1∑l=1Clk−1∂k−1−l+1xw∂lx12√w, |
then
∂x∂kxw−√w∂2ψ∂kxw=−2∂kx(ρ∂xP)+k−2∑m=1Cmk(∂k−mx√w)∂2ψ∂mxw+∂kxw2√w∂2ψw+k−1∑l=1Clk−1∂k−1−l+1xw∂lx(12√w)∂2ψw+Ck−1k∂xw∂kxw2w+2k∑m=0Cmk∂k−mx(∂xρρ)∂mxw+Ck−1k∂xw2√w[k−1∑l=1Clk−1∂k−lxw∂lx1√w+∂k−1x(2ρ∂xP√w)−∂k−1x(2∂xρρ√w)]. |
The following transformation f is defined:
Ω→˜Ω:=[−1,0]˜x×[−12,12]˜ψ,(x,ψ)↦(˜x,˜ψ), |
where x−x0=ψ320˜x,ψ−ψ0=ψ0˜ψ.
Let f=∂kxwψ−10, we get
∂˜xf−√wψ120∂2˜ψf−12√w∂2ψwψ320f−∂xw2wψ320f=−2ψ120∂kx(ρ∂xP)+ψ120k−2∑m=1Cmk(∂k−mx√w)∂2ψ∂mxw+ψ120k−1∑l=1Clk−1∂k−lxw(∂lx12√w)∂2ψw+2ψ120k∑m=0Cmk∂k−mx(∂xρρ)∂mxw+ψ120∂xw2√w[k−1∑l=1Clk−1∂k−lxw∂lx1√w+∂k−1x(2ρ∂xP√w)−∂k−1x(2∂xρρ√w)]:=F. |
From the proof of Lemma 2.2 and Lemma 2.6, we know that in ˜Ω for α∈(0,1),
|f|≤C,0<c≤ψ−120√w≤C,|ψ−120√w|Cα(˜Ω)≤C. |
By (2.6), (2.15) and the equality
∂ψ(∂2ψ∂mxw)=∂ψ∂m+1xw√w−∂ψw∂m+1xw2(√w)3+m∑l=1Clm∂m−l+1x∂ψw∂lx1√w+m∑l=1Clm∂m−l+1xw∂lx∂ψw−2(√w)3+∂mx(ρ∂xP∂ψw−(√w)3)−∂mx(∂xρρ∂ψw√w), |
we can conclude that for j≤k−1 and m≤k−2,
|∇˜x,˜ψ∂jx√w|≤Cψ120,|∇˜x,˜ψ∂jx(1√w)|≤Cψ−120,|∇˜x,˜ψ∂2ψ∂mxw|≤Cψ−120. |
Combining (2.4) with (2.5), we can obtain
|12√w∂2ψwψ320+∂xw2wψ320|Cα(˜Ω)+|F|Cα(˜Ω)≤C. |
By the standard interior priori estimates, we obtain
|∂˜xf|L∞([−14,0]˜x×[−18,18]˜ψ)+|∂˜ψf|L∞([−14,0]˜x×[−18,18]˜ψ)+|∂2˜ψf|L∞([−14,0]˜x×[−18,18]˜ψ)≤C. |
Therefore, this means that
|∂k+1xw(x0,ψ0)|≤Cψ−120,|∂ψ∂kxw(x0,ψ0)|≤C,|∂2ψ∂kxw(x0,ψ0)|≤Cψ−10. |
Since (x0,ψ0) is arbitrary, this completes the proof of the lemma.
Lemma 2.7. If u is a solution for equation (1.1) in Theorem 1.1, assume u0 satisfies the condition (1.9) and the known function ρ and ∂xP are smooth. Assume 0<ε<X and integer m,k≥0, then there exists a positive constant δ>0 such that for any (x,ψ)∈[ε,X]×[0,δ],
|∂mψ∂kxw|≤Cψ1−m. | (2.16) |
Proof. From Lemma 2.1, (2.1), Lemma 2.2 and Lemma 2.3, a direct calculation can prove that
|∂kx1√w|≤Cψ−12,|∂kx∂ψ1√w|≤Cψ−32,|∂kx∂2ψ1√w|≤Cψ−52, |
and (2.16) holds for m=0,1,2. Then for 0≤m≤j with j≥1, we inductively assume that
|∂mψ∂kxw|≤Cψ1−m,|∂kx∂mψ1√w|≤Cψ−12−m. | (2.17) |
In the next part, we will prove that (2.17) still holds for m=j+1.
By (1.6), we obtain
∂j+1ψ∂kxw=∂j−1ψ∂kx∂2ψw=∂kx∂j−1ψ(∂xw√w+2ρ∂xP√w−2∂xρρw√w)=∂kx(j−1∑i=0Cij−1∂j−1−iψ∂xw∂iψ1√w+2ρ∂xP∂j−1ψ1√w−2∂xρρj−1∑i=0Cij−1∂j−1−iψw∂iψ1√w). |
Combining with (2.17), we get
|∂j+1ψ∂kxw|≤Cψ32−j+Cψ12−j+Cψ32−j≤Cψ12−j. | (2.18) |
By straight calculations, we get
0=∂kx∂j+1ψ(1√w1√ww)=∂kx[2√w∂j+1ψ1√w+j∑i=1j+1−i∑l=0Cij+1Clj+1−i(∂iψ1√w)(∂lψ1√w)∂j+1−l−iψw+j∑l=0Clj+11√w(∂lψ1√w)∂j+1−lψw]. |
Combining the above equality with (2.17), we can conclude that
|∂kx∂j+1ψ1√w|≤Cψ−32−j. |
This completes the proof of the lemma.
In this section, we will prove the regularity of the solution u in the domain
{(x,ψ)|ε≤x≤X,0≤y≤Y1}. |
Proof of Theorem 1.5:
Proof. For the convenience of proof, we denote
(˜x,ψ)=(x,∫y0˜udy). |
A direct calculation gives (see P13 in [23])
∂y=√w∂ψ,∂x=∂˜x+∂xψ(x,y)∂ψ,∂xψ=12√w∫ψ0w−32∂˜xwdψ. |
By (2.1) and Lemma 2.3, we have |∂xψ|≤Cψ. Due to ∂y=√w∂ψ, we obtain
∂kx2∂y˜u=(∂˜x+∂xψ∂ψ)k∂ψw,∂kx2∂2y˜u=(∂˜x+∂xψ∂ψ)k(∂˜xw+2ρ∂xP−2∂xρρw)=(∂˜x+∂xψ∂ψ)k(∂˜xw)+2∂k˜x(ρ∂xP)−2(∂xρρ)(∂˜x+∂xψ∂ψ)kw−2∂k˜x(∂xρρ)w. |
By |∂xψ|≤Cψ and Lemma 2.7, we obtain that Theorem 1.5 holds for m=0,1,2,
|∂kx∂y˜u|+|∂kx∂2y˜u|≤C. | (3.1) |
We inductively assume that for any integer k and m≥1,
|∂kx∂jy˜u|≤C,j≤m. | (3.2) |
A direct calculation gives
∂kx∂m+1y˜u=∂kx∂m−1y∂2y˜u=∂kx∂m−1y(˜u∂x˜u−∂y˜u∫y0∂x˜udy−∂xρρ˜u2)=∂kx(m−1∑i=0Cim−1∂m−1−iy˜u∂iy∂x˜u−m−2∑i=0Ci+1m−1∂m−1−iy˜u∂iy∂x˜u−∂my˜u∫y0∂x˜udy−∂xρρ∂m−1y˜u2), |
and we can deduce from (3.1) and (3.2) that
|∂kx∂jy˜u|≤C,j≤m+1.⇒|∂kx∂jyu|≤C,j≤m+1. |
This completes the proof of the theorem.
In this section, we prove our main theorem. The key point is to prove that (1.6) is a uniform parabolic equation. The proof is based on the classical parabolic maximum principle. The specific proof details are as follows.
Proof. By (1.2) and ∂xP≤0, we obtain
C≥U2(x)=U2(0)−2∫x0∂xP(ρ)ρdx≥U2(0). |
By (1.7) and w increasing in ψ (see below), we know that there exists some positive constants Ψ and C0 such that for any (x,ψ)∈[0,X]×[Ψ,+∞),
w≥C0U2(0). | (4.1) |
From Theorem 1.1, we know that there exists positive constants y0,M,m such that for any (x,ψ)∈[0,X]×[0,y0] (we can take y0 to be small enough),
M≥∂y˜u(x,y)≥m. | (4.2) |
The fact that ψ∼y2 is near the boundary y=0 (see Remark 4.1 in [23]), for some small positive constant 0<κ<1, we get
κ2y20≤ψ≤κy20⇒σy0≤y≤y02, | (4.3) |
for some constant σ>0 depends on κ,m,M.
We denote
Ω={(x,ψ)|0≤x≤X,κ2y20≤ψ≤+∞}. |
By (4.2) and (4.3), we get ˜u(x,σy0)≥mσy0, then for any x∈[0,X], we have
w(x,κ2y20)≥m2σ2y20. | (4.4) |
Since the initial data u0 satisfies the condition (1.9) and w=˜u2, we know w(0,ψ)>0 for ψ>0 and there exists a positive constant ζ, such that for ψ∈[κ2y20,Ψ],
w(0,ψ)>ζ. | (4.5) |
Then, we only consider
Ω1={(x,ψ)|0≤x≤X,κ2y20≤ψ≤Ψ}. |
We denote H(x,ψ):=e−λx∂ψw(x,ψ), which satisfies the following system in the region Ω0={(x,ψ)|0≤x<X,0<ψ<+∞}:
{∂xH−∂ψw2√w∂ψH−√w∂2ψH+(λ−2∂xρρ)H=0,H|x=0=∂ψw0(ψ),H|ψ=0=2e−λx∂y˜u|y=0,H|ψ=+∞=0. | (4.6) |
Then, we choose λ properly large such that λ−2∂xρρ≥0. Due to
H|x=0=∂ψw0(ψ)≥0,H|ψ=0=2e−λx∂y˜u|y=0>0,H|ψ=+∞=0, |
it follows that
H(x,ψ)=e−λxF(x,ψ)=e−λx∂ψw≥0,(x,ψ)∈[0,X∗)×R+, |
which means ∂ψw≥0 in [0,X)×R+. Hence, w is increasing in ψ. Therefore, we know that there exists a positive constant λ≥m2σ2y20 such that for any x∈[0,X],
w(x,Ψ)≥λ. | (4.7) |
By (1.6), for any ε>0, we know W:=w+εx satisfies the following system in Ω1:
{∂xW−√w∂2ψW−2∂xρρW=F,W|x=0=W0>ζ,W|ψ=κ2y20=W1≥m2σ2y20,W|ψ=Ψ=W2≥λ, |
where
F=−2ρ∂xP+ε−2εx∂xρρ. |
Since ∂xP≤0, we know the diffusive term F>0. Therefore, the minimum cannot be reached inside Ω1. Set
η0=min{W0,W1,W2}, |
then by the maximum principle, we obtain W=w+εx≥η0. Let ε→0, we have w≥η0 in Ω1. Then we denote
η=min{η0,C0U2(0)}>0, |
combining with (4.1), we have w≥η in Ω. Therefore, there exists some positive constant c such that c≤w in Ω. From Theorem 1.1, we have w≤C in Ω. In sum, there exists positive constants c,C such that c≤w≤C in Ω. This further means that
0<√c≤√w≤√C, | (4.8) |
where C depends on X. Therefore, we prove (1.6) is a uniform parabolic equation. Furthermore, by Theorem 1.1, we know ∂y˜u,∂2y˜u are continuous and bounded in [0,X)×R+. Combining ρ, ∂xP are smooth, (4.8) with
2∂y˜u=∂ψw,2∂2y˜u=√w∂2ψw=∂xw−2∂xρρw+2ρ∂xP(ρ), |
we obtain
‖√w‖Cα(Ω)≤C. |
Once we have the above conclusion, the proof of Theorem 1.6 can be given in a similar fashion to [23]. Here, we provide a brief explanation for the reader's convenience. More details can be found in [23].
Step 1: For any (x1,ψ1)∈[ε,X]×[κy20,+∞), we denote
Ωx1,ψ1={(x,ψ)|x1−ε2≤x≤x1,ψ1−κ2y20≤ψ≤ψ1+κ2y20}. |
Step 2: Note that the known function ρ, ∂xP is smooth, we can repeat interior Schauder estimates in Ωx1,ψ1 to achieve uniform estimates independent of choice of (x1,ψ1) for any order derivatives of w. Since the width and the length of Ωx1,ψ1 are constants and the estimates employed are independent of (x1,ψ1), restricting the estimates to the point (x1,ψ1), we can get for any m<+∞,|∇mw(x1,ψ)|≤CX,m,y0,ε.
Step 3: Since (x1,ψ1) is arbitrary, we have for any m<+∞,|∇mw(x1,ψ)|≤CX,m,y0,ε in [ε,X]×[κy20,+∞). Then, as in Section 3, we can prove Theorem 1.6.
Finally, Theorem 1.3 is proven by combining Theorem 1.5 and Theorem 1.6.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The research of Zou was supported by the Fundamental Research Funds for the Central Universities (Grant No. 202261101).
The authors declare there is no conflict of interest.
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