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Research article

Global regularity of solutions to the 2D steady compressible Prandtl equations

  • Received: 16 August 2023 Revised: 22 October 2023 Accepted: 23 October 2023 Published: 27 October 2023
  • 35Q30, 76D10, 76N20

  • In this paper, we study the global C regularity of solutions to the boundary layer equations for two-dimensional steady compressible flow under the favorable pressure gradient. To our knowledge, the difficulty of the proof is the degeneracy near the boundary. By using the regularity theory and maximum principles of parabolic equations together with the von Mises transformation, we give a positive answer to it. When the outer flow and the initial data satisfied appropriate conditions, we prove that Oleinik type solutions smooth up the boundary y=0 for any x>0.

    Citation: Yonghui Zou. Global regularity of solutions to the 2D steady compressible Prandtl equations[J]. Communications in Analysis and Mechanics, 2023, 15(4): 695-715. doi: 10.3934/cam.2023034

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  • In this paper, we study the global C regularity of solutions to the boundary layer equations for two-dimensional steady compressible flow under the favorable pressure gradient. To our knowledge, the difficulty of the proof is the degeneracy near the boundary. By using the regularity theory and maximum principles of parabolic equations together with the von Mises transformation, we give a positive answer to it. When the outer flow and the initial data satisfied appropriate conditions, we prove that Oleinik type solutions smooth up the boundary y=0 for any x>0.



    In this paper, we study the 2D steady compressible Prandtl equations in {x>0,y>0}:

    {uxu+vyu1ρ2yu=xP(ρ)ρ,x(ρu)+y(ρv)=0,u|x=0=u0(y),limyu=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

    UxU+xP(ρ)ρ=0. (1.2)

    The pressure P(ρ) is a strictly increasing function of ρ with 0<ρ0ρρ1 for some positive constants ρ0 and ρ1.

    In this paper, we assume that the pressure satisfies the favorable pressure gradient xP0, which implies that

    xρ0.

    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)=μ12u(xμx,μ14y) is compact in C0(R2+). Later, Dalibard and Masmoudi [4] proved the solution behaves near the separation as yu(x,0)(xx)12 for x<x. Shen, Wang and Zhang [18] found that the solution near the separation point behaves like yu(x,y)(xx)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<x1, 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:

    {˜ux˜u+˜vy˜u2y˜uxρρ˜u2=ρxP(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=xω2ω+ψωxψ2ω,y˜u=ψω2,2y˜u=ω2ψω2, (1.5)

    and (1.3)–(1.5), we know that w(x,ψ) satisfies:

    xww2ψw2xρρw=2ρxP(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

    2y˜u=ψw,22y˜u=w2ψw. (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:

    uC2,αb([0,+))(α>0),u(0)=0,yu(0)>0,yu(y)0fory[0,+),limy+u(y)=U(0)>0,ρ1(0)2yu(y)ρ1(0)xP(0)=O(y2) (1.9)

    and ρC2([0,X0]), then there exists 0<XX0 such that system (1.1) admits a solution uC1([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 xP0(xρ0), then

    X=+.

    Remarks 1.2. uC2,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,+),

    |kxmyu(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],

    |kxmyu(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,+),

    |kxmyu(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

    xxww2ψxw=(xw)22w+2ρxPxw2w+xρρxw+2x(xρρ)w2x[ρxP].

    Take a smooth cutoff function 0ϕ(x)1 in [0,X] such that

    ϕ(x)=1,x[ε,X],ϕ(x)=0,x[0,ε2],

    then

    x[xwϕ(x)]w2ψ[xwϕ(x)]=(xw)22wϕ(x)+2ρxPxw2wϕ(x)+xρρxwϕ(x)+2x(xρρ)wϕ(x)2x(ρxP)ϕ(x)+xwxϕ(x):=W.

    Combining with (2.1), we know

    |W|Cψ2α+Cψα12+Cψα+12+Cψ+CCψα12. (2.2)

    We take φ(ψ)=μ1ψμ2ψ1+β with constants μ1,μ2, then by (2.1) and (2.2), we get

    x[xwϕ(x)φ]w2ψ[xwϕ(x)φ]|W|μ2wβ(1+β)ψβ1Cψα12μ2mβ(1+β)ψβ12.

    By taking μ2 sufficiently large and α=β, for (x,ψ)(0,X]×(0,δ1), we have

    x[xwϕ(x)φ]w2ψ[xwϕ(x)φ]<0.

    For any ψ[0,δ1], let μ1μ2, and we have

    (xwϕφ)(0,ψ)0,

    and take μ1 large enough depending on M,δ1,μ2 such that

    (xwϕφ)(x,δ1)Mδ12+α1μ1δ1+μ2δ1+β10.

    Since w(x,0)=0, we know that for any x[0,X],

    (xwϕφ)(x,0)=0.

    By the maximum principle, it holds in [0,X]×[0,δ1] that

    (xwϕφ)(x,ψ)0.

    Let δ1 be chosen suitably small, for (x,ψ)[ε,X]×[0,δ1], and we obtain

    xw(x,ψ)μ1ψμ2ψ1+βμ12ψ.

    Considering xwϕφ, the result xwμ12ψ 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],

    |ψxw(x,ψ)|C,|2xw(x,ψ)|Cψ12,|2ψxw(x,ψ)|Cψ1.

    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)|x0ψ320xx0,12ψ0ψ32ψ0}.

    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 xx0=ψ320˜x,ψψ0=ψ0˜ψ.

    Since ˜x=ψ320x,˜ψ=ψ0ψ, it holds in Ω that

    ˜x(ψ10w)ψ120w2˜ψ(ψ10w)2˜xρρ(ψ10w)=2ρ˜xPψ10.

    Combining with (2.1), we get 0<cψ120wC,|ψ10w|C, and for any ˜z1,˜z2˜Ω,

    |ψ120w(˜z1)ψ120w(˜z2)|=ψ120|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

    |ψ120w|Cα(˜Ω)C.

    Since P and ρ are smooth, we have

    |ρ1˜xρ|C0,1([1,0]˜x)+|ρ˜xPψ10|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

    |wψ10|Cα([12,0]˜x×[14,14]˜ψ)+|2˜ψwψ10|Cα([12,0]˜x×[14,14]˜ψ)C. (2.4)

    Let f:=xwψ10, which satisfies

    ˜xfwψ1202˜ψf2˜ψw2wψ120f2˜xρρf=2x[ρ˜xP]ψ10+2x(˜xρρ)(ψ10w).

    By (2.3), we have |f|C in ˜Ω. Due to

    |ψ120w12(˜z1)ψ120w12(˜z2)|=ψ120|w(˜z1)w(˜z2)w(˜z1)w(˜z2)|w12(˜z1)+w12(˜z2)C|˜z1˜z2|,

    we have

    |ψ120w12|Cα(˜Ω)C. (2.5)

    Since

    2˜ψw2wψ120=2˜ψwψ10ψ1202w,

    which along with (2.4) and (2.5) gives

    |2˜ψw2wψ120|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)|Cψ120,|ψxw(x0,ψ0)|C,|2ψxw(x0,ψ0)|Cψ10.

    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 k2, 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,|ψkxw|C,|2ψkxw|Cψ1.

    Proof. By Lemma 2.1 and Lemma 2.2, we may inductively assume that for 0jk1, there holds that in [ε2,X]×[0,δ3] (assume δ31),

    |ψjxw|C,|2ψjxw|Cψ1,|jxw|Cψ,|jxw|Cψ12,|kxw|Cψ12. (2.6)

    We will prove that there exists δ4<δ3 so that in [ε,X]×[0,δ4],

    |ψkxw|C,|2ψkxw|Cψ1,|kxw|Cψ,|kxw|Cψ12,|k+1xw|Cψ12. (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<β1,

    |kxw|<M1ψ1β,|kxw|M1ψ12β.

    Proof. Take a smooth cutoff function 0ϕ(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<xX,0<ψ<δ3},

    and let

    (ⅰ) f=k1xw(xh,ψ)k1xw(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+Mδ3lnδ30,

    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,k1xw(x0h,ψ0)<k1xw(x0,ψ0).

    By (2.1), denote ξ=m, we have

    w2ψ(Mψlnψ)=Mwψ1ξMψ12. (2.8)

    By (1.6), a direct calculation gives

    xk1xww2ψk1xw=2k1x(ρxP)+k2m=1Cmk1(k1mxw)2ψmxw+(k1xw)2ψw+2k1m=0Cmk1k1mx(xρρ)mxw=2k1x(ρxP)+k2m=1Cmk1(k1mxw)2ψmxw+k1xw2wxww+(k1xw2w)2ρxPw(k1xw2w)2xρρww+2k1m=0Cmk1k1mx(xρρ)mxw+k2m=0Cmk22ψwm+1xwk2mx12w:=4i=1Ii

    and

    I1=2k1x(ρxP)+k2m=1Cmk1(k1mxw)2ψmxw+k1xw2wxww,I2=ρxPwk1xw,I3=xρρk1xw+2k1m=0Cmk1k1mx(xρρ)mxw,I4=k2m=0Cmk22ψwm+1xwk2mx12w.

    For x5ε8, we consider the following equality

    xf1w(p1)2ψf1=w(p1)w(p)h2ψk1xw(p)+4i=11h(Ii(p1)Ii(p)), (2.9)

    where

    f1=1h(k1xw(p1)k1xw(p)),

    with p1=(xh,ψ),p=(x,ψ).

    For any x5ε8, by (2.6), it is easy to conclude that

    |1h(w(p1)w(p))2ψk1xw(p)|Cψ12,|1h(I1(p1)I1(p))|Cψ12,|4i=31h(Ii(p1)Ii(p))|Cψ12, (2.10)

    where C is dependent on the parameter h.

    Since

    1h(I2(p1)I2(p))=f1[ρxPw(p1)]+k1xw(p)1h[ρxPw(p1)ρxPw(p)],

    combining with (2.6), f1(p0)>0 and xP0, it holds at p=p0 that

    1h(I2(p1)I2(p0))C. (2.11)

    Summing up (2.10) and (2.11), we conclude that at p=p0,

    xf1w2ψf1C0ψ12.

    This along with (2.8) shows that for x5ε8, it holds at p=p0 that

    xfw2ψfCψ12ξMψ12. (2.12)

    By taking M large enough, we have xf(p0)w2ψf(p0)<0. By the definition of p0, we obtain

    xf(p0)w2ψf(p0)0,

    which leads to a contradiction. Therefore, for M chosen as above and independent of h, we have

    maxˉΩf0.

    We can similarly prove that minˉΩf0 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

    2wkxw+k1m=1Cmk(mxwkmxw)=kx(ww)=kxw, (2.13)

    which along with (2.6) shows that in (78ε,X]×(0,δ3],

    |wkxw|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,|kxw|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=kxwϕμ1ψ+μ2ψ32β

    with constants μ1,μ2. Let β be small enough in Lemma 2.4. Then it holds in [7ε8,X]×[0,δ3] that

    |kxw|Cψ1β,|kxw|Cψ12β. (2.14)

    We denote

    Ω={(x,ψ)|0<xX,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

    xkxww2ψkxw=2kx(ρxP)+k1m=0Cmk(kmxw)2ψmxw+2km=0Cmkkmx(xρρ)mxw,

    and

    2ψmxw=mx2ψw=mx(xww+2ρxPw2xρρw).

    For any x7ε8, 0jk1 and 0mk1, from (2.6) and (2.14), we get

    |jxw|Cψ,|kxw|Cψ1β,|kmxw|Cψ12β.

    Then let β12, for 0mk1 and x7ε8, we obtain

    |2ψmxw|Cψ12β+Cψ12+Cψ12βCψ12.

    Therefore, we conclude that for x7ε8,

    xkxww2ψkxwC+Cψβ+Cψ1βCψβ.

    By the above inequality and (2.1), it holds at p=p0 that

    xfw2ψf=xkxww2ψkxw+kxwxϕw2ψ(μ1ψ+μ2ψ32β)C2ψβξμ2ψβ,

    where ξ=(32β)(12β)m>0. Then we have xfw2ψf<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

    maxˉΩf0.

    In the same way, we can prove that

    maxˉΩkxwϕμ1ψ+μ2ψ32β0.

    So, for any (x,ψ)[1516ε,X]×[0,δ3], we have

    |kxw|μ1ψμ2ψ32βμ1ψ.

    Combining with (2.6) and (2.13), it holds in [1516ε,X]×[0,δ3] that

    |kxw|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],

    |ψkxw|C,|2ψkxw|Cψ1,|k+1xw|Cψ12.

    Proof. By Lemma 2.5 and (2.6), for any (x,ψ)[1516ε,X]×[0,δ3],

    |jxw|Cψ,|jxw|Cψ12,0jk. (2.15)

    Set Ψ0=min{23δ3,ε16}, for (x0,ψ0)[ε,X]×(0,Ψ0], we denote

    Ω={(x,ψ)|x0ψ320xx0,12ψ0ψ32ψ0}.

    A direct calculation gives

    xkxww2ψkxw=2kx(ρxP)+kxw2ψw+k2m=1Cmk(kmxw)2ψmxw+Ck1kxw2w2ψk1xw+2km=0Cmkkmx(xρρ)mxw.

    By (1.6), we obtain

    2ψmxw=mx2ψw=mx(xww+2ρxPw2xρρw)=m+1xww+ml=1Clmml+1xwlx1w+mx(2ρxPw)mx(2xρρw),

    and

    kxw=k1xxw2w=kxw2w+k1l=1Clk1k1l+1xwlx12w,

    then

    xkxww2ψkxw=2kx(ρxP)+k2m=1Cmk(kmxw)2ψmxw+kxw2w2ψw+k1l=1Clk1k1l+1xwlx(12w)2ψw+Ck1kxwkxw2w+2km=0Cmkkmx(xρρ)mxw+Ck1kxw2w[k1l=1Clk1klxwlx1w+k1x(2ρxPw)k1x(2xρρw)].

    The following transformation f is defined:

    Ω˜Ω:=[1,0]˜x×[12,12]˜ψ,(x,ψ)(˜x,˜ψ),

    where xx0=ψ320˜x,ψψ0=ψ0˜ψ.

    Let f=kxwψ10, we get

    ˜xfwψ1202˜ψf12w2ψwψ320fxw2wψ320f=2ψ120kx(ρxP)+ψ120k2m=1Cmk(kmxw)2ψmxw+ψ120k1l=1Clk1klxw(lx12w)2ψw+2ψ120km=0Cmkkmx(xρρ)mxw+ψ120xw2w[k1l=1Clk1klxwlx1w+k1x(2ρxPw)k1x(2xρρw)]:=F.

    From the proof of Lemma 2.2 and Lemma 2.6, we know that in ˜Ω for α(0,1),

    |f|C,0<cψ120wC,|ψ120w|Cα(˜Ω)C.

    By (2.6), (2.15) and the equality

    ψ(2ψmxw)=ψm+1xwwψwm+1xw2(w)3+ml=1Clmml+1xψwlx1w+ml=1Clmml+1xwlxψw2(w)3+mx(ρxPψw(w)3)mx(xρρψww),

    we can conclude that for jk1 and mk2,

    |˜x,˜ψjxw|Cψ120,|˜x,˜ψjx(1w)|Cψ120,|˜x,˜ψ2ψmxw|Cψ120.

    Combining (2.4) with (2.5), we can obtain

    |12w2ψwψ320+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)|Cψ120,|ψkxw(x0,ψ0)|C,|2ψkxw(x0,ψ0)|Cψ10.

    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,k0, then there exists a positive constant δ>0 such that for any (x,ψ)[ε,X]×[0,δ],

    |mψkxw|Cψ1m. (2.16)

    Proof. From Lemma 2.1, (2.1), Lemma 2.2 and Lemma 2.3, a direct calculation can prove that

    |kx1w|Cψ12,|kxψ1w|Cψ32,|kx2ψ1w|Cψ52,

    and (2.16) holds for m=0,1,2. Then for 0mj with j1, we inductively assume that

    |mψkxw|Cψ1m,|kxmψ1w|Cψ12m. (2.17)

    In the next part, we will prove that (2.17) still holds for m=j+1.

    By (1.6), we obtain

    j+1ψkxw=j1ψkx2ψw=kxj1ψ(xww+2ρxPw2xρρww)=kx(j1i=0Cij1j1iψxwiψ1w+2ρxPj1ψ1w2xρρj1i=0Cij1j1iψwiψ1w).

    Combining with (2.17), we get

    |j+1ψkxw|Cψ32j+Cψ12j+Cψ32jCψ12j. (2.18)

    By straight calculations, we get

    0=kxj+1ψ(1w1ww)=kx[2wj+1ψ1w+ji=1j+1il=0Cij+1Clj+1i(iψ1w)(lψ1w)j+1liψw+jl=0Clj+11w(lψ1w)j+1lψw].

    Combining the above equality with (2.17), we can conclude that

    |kxj+1ψ1w|Cψ32j.

    This completes the proof of the lemma.

    In this section, we will prove the regularity of the solution u in the domain

    {(x,ψ)|εxX,0yY1}.

    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=wψ,x=˜x+xψ(x,y)ψ,xψ=12wψ0w32˜xwdψ.

    By (2.1) and Lemma 2.3, we have |xψ|Cψ. Due to y=wψ, we obtain

    kx2y˜u=(˜x+xψψ)kψw,kx22y˜u=(˜x+xψψ)k(˜xw+2ρxP2xρρw)=(˜x+xψψ)k(˜xw)+2k˜x(ρxP)2(xρρ)(˜x+xψψ)kw2k˜x(xρρ)w.

    By |xψ|Cψ and Lemma 2.7, we obtain that Theorem 1.5 holds for m=0,1,2,

    |kxy˜u|+|kx2y˜u|C. (3.1)

    We inductively assume that for any integer k and m1,

    |kxjy˜u|C,jm. (3.2)

    A direct calculation gives

    kxm+1y˜u=kxm1y2y˜u=kxm1y(˜ux˜uy˜uy0x˜udyxρρ˜u2)=kx(m1i=0Cim1m1iy˜uiyx˜um2i=0Ci+1m1m1iy˜uiyx˜umy˜uy0x˜udyxρρm1y˜u2),

    and we can deduce from (3.1) and (3.2) that

    |kxjy˜u|C,jm+1.|kxjyu|C,jm+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 xP0, we obtain

    CU2(x)=U2(0)2x0xP(ρ)ρdxU2(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,+),

    wC0U2(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),

    My˜u(x,y)m. (4.2)

    The fact that ψy2 is near the boundary y=0 (see Remark 4.1 in [23]), for some small positive constant 0<κ<1, we get

    κ2y20ψκy20σy0yy02, (4.3)

    for some constant σ>0 depends on κ,m,M.

    We denote

    Ω={(x,ψ)|0xX,κ2y20ψ+}.

    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)|0xX,κ2y20ψΨ}.

    We denote H(x,ψ):=eλxψw(x,ψ), which satisfies the following system in the region Ω0={(x,ψ)|0x<X,0<ψ<+}:

    {xHψw2wψHw2ψH+(λ2xρρ)H=0,H|x=0=ψw0(ψ),H|ψ=0=2eλxy˜u|y=0,H|ψ=+=0. (4.6)

    Then, we choose λ properly large such that λ2xρρ0. Due to

    H|x=0=ψw0(ψ)0,H|ψ=0=2eλxy˜u|y=0>0,H|ψ=+=0,

    it follows that

    H(x,ψ)=eλxF(x,ψ)=eλxψw0,(x,ψ)[0,X)×R+,

    which means ψw0 in [0,X)×R+. Hence, w is increasing in ψ. Therefore, we know that there exists a positive constant λm2σ2y20 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:

    {xWw2ψW2xρρW=F,W|x=0=W0>ζ,W|ψ=κ2y20=W1m2σ2y20,W|ψ=Ψ=W2λ,

    where

    F=2ρxP+ε2εxxρρ.

    Since xP0, 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+εxη0. Let ε0, we have wη0 in Ω1. Then we denote

    η=min{η0,C0U2(0)}>0,

    combining with (4.1), we have wη in Ω. Therefore, there exists some positive constant c such that cw in Ω. From Theorem 1.1, we have wC in Ω. In sum, there exists positive constants c,C such that cwC in Ω. This further means that

    0<cwC, (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

    2y˜u=ψw,22y˜u=w2ψw=xw2xρρw+2ρxP(ρ),

    we obtain

    wCα(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)|x1ε2xx1,ψ1κ2y20ψψ1+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.



    [1] R. Alexandre, Y. Wang, C. Xu, T. Yang, Well-posedness of the Prandtl equation in Sobolev spaces, J. Amer. Math. Soc., 28 (2015), 745–784. https://doi.org/10.1090/S0894-0347-2014-00813-4 doi: 10.1090/S0894-0347-2014-00813-4
    [2] H. Beirão da Veiga, F. Crispo, A survey on some vanishing viscosity limit results, Adv. Nonlinear Anal., 12 (2023), 20220309. https://doi.org/10.1515/anona-2022-0309 doi: 10.1515/anona-2022-0309
    [3] M. Ding, S. Gong, Global existence of weak solution to the compressible Prandtl equations, J. Math. Fluid Mech., 19 (2017), 239–254. https://doi.org/10.1007/s00021-016-0274-5 doi: 10.1007/s00021-016-0274-5
    [4] A. L. Dalibard, N. Masmoudi, Separation for the stationary Prandtl equation, Publ. math. IHES, 130 (2019), 187–297. https://doi.org/10.1007/s10240-019-00110-z doi: 10.1007/s10240-019-00110-z
    [5] S. Gong, Y. Guo, Y. Wang, Boundary layer problems for the two-dimensional compressible Navier-Stokes equations, Anal. Appl. (Singap.), 14 (2016), 1–37. https://doi.org/10.1142/S0219530515400011 doi: 10.1142/S0219530515400011
    [6] Y. Guo, S. Iyer, Regularity and expansion for steady Prandtl equations, Commun. Math. Phys., 382 (2021), 1403–1447. https://doi.org/10.1007/s00220-021-03964-9 doi: 10.1007/s00220-021-03964-9
    [7] J. Han, R. Xu, C. Yang. Continuous dependence on initial data and high energy blowup time estimate for porous elastic system, Commun. Anal. Mech., 15 (2023), 214–244. https://doi.org/10.3934/cam.2023012 doi: 10.3934/cam.2023012
    [8] S. Iyer, N. Masmoudi, Reversal in the stationary Prandtl equations, preprint, arXiv: 2203.02845.
    [9] S. Iyer, N. Masmoudi, Higher regularity theory for a Mixed-Type parabolic equation, preprint, arXiv: 2212.08735.
    [10] N. V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces, American Mathematical Soc., 1996. https://doi.org/10.1090/gsm/012
    [11] C. Liu, F. Xie, T. Yang, MHD boundary layers theory in Sobolev spaces without monotonicity Ⅰ: Well-posedness theory, Comm. Pure Appl. Math., 72 (2019), 63–121. https://doi.org/10.1002/cpa.21763 doi: 10.1002/cpa.21763
    [12] N. Masmoudi, T. K. Wong, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods, Comm. Pure Appl. Math., 68 (2015), 1683–1741. https://doi.org/10.1002/cpa.21595 doi: 10.1002/cpa.21595
    [13] O. A. Oleinik, V. N. Samokhin, Mathematical models in boundary layer theory. Routledge, 2018. https://doi.org/10.1201/9780203749364
    [14] L. Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Reibung, Verhandl. Ⅲ, Internat. Math.-Kong., Heidelberg, Teubner, Leipzig, 1904, 1961,575–584.
    [15] K. Pileckas, A. Raciene, Non-stationary Navier–Stokes equations in 2D power cusp domain: Ⅰ. Construction of the formal asymptotic decomposition, Adv. Nonlinear Anal., 10 (2021), 982–1010. https://doi.org/10.1515/anona-2020-0164 doi: 10.1515/anona-2020-0164
    [16] K. Pileckas, A. Raciene, Non-stationary Navier–Stokes equations in 2D power cusp domain: Ⅱ. Existence of the solution, Adv. Nonlinear Anal., 10 (2021), 1011–1038. https://doi.org/10.1515/anona-2020-0165 doi: 10.1515/anona-2020-0165
    [17] H. Schlichting, K. Gersten, Boundary-Layer Theory, Enlarged Edition. New York: Springer-Verlag, 2000. https://doi.org/10.1007/978-3-662-52919-5
    [18] W. Shen, Y. Wang, Z. Zhang, Boundary layer separation and local behavior for the steady Prandtl equation, Adv. Math., 389 (2021), 107896. https://doi.org/10.1016/j.aim.2021.107896 doi: 10.1016/j.aim.2021.107896
    [19] Y. Wang, M. Williams, The inviscid limit and stability of characteristic boundary layers for the compressible Navier-Stokes equations with Navier-friction boundary conditions, Ann. Inst. Fourier (Grenoble), 62 (2012), 2257–2314. https://doi.org/10.5802/aif.2749 doi: 10.5802/aif.2749
    [20] Y. Wang, F. Xie, T. Yang, Local well-posedness of Prandtl equations for compressible flow in two space variables. SIAM J. Math. Anal., 47 (2015), 321–346. https://doi.org/10.1137/140978466 doi: 10.1137/140978466
    [21] Y. Wang, S. Zhu, Back flow of the two-dimensional unsteady Prandtl boundary layer under an adverse pressure gradient, SIAM J. Math. Anal., 52 (2020), 954–966. https://doi.org/10.1137/19M1270355 doi: 10.1137/19M1270355
    [22] Y. Wang, S. Zhu, On back flow of boundary layers in two-dimensional unsteady incompressible heat conducting flow, J. Math. Phys., 63 (2022), 081504. https://doi.org/10.1063/5.0088618 doi: 10.1063/5.0088618
    [23] Y. Wang, Z. Zhang, Global C regularity of the steady Prandtl equation with favorable pressure gradient, Ann. Inst. H. Poincaré C Anal. Non Linéaire., 38 (2021), 1989–2004. https://doi.org/10.1016/J.ANIHPC.2021.02.007 doi: 10.1016/J.ANIHPC.2021.02.007
    [24] Y. Wang, Z. Zhang, Asymptotic behavior of the steady Prandtl equation, Math. Ann., 387 (2023), 1289–1331. https://doi.org/10.1007/s00208-022-02486-6 doi: 10.1007/s00208-022-02486-6
    [25] E. Weinan, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation, Acta Math. Sinica, 16 (2000), 207–218. https://doi.org/10.1007/s101140000034 doi: 10.1007/s101140000034
    [26] Z. Xin, L. Zhang, On the global existence of solutions to the Prandtl's system, Adv. Math., 181 (2004), 88–133. https://doi.org/10.1016/S0001-8708(03)00046-X doi: 10.1016/S0001-8708(03)00046-X
    [27] Z. Xin, L. Zhang, J. Zhao, Global well-posedness and eegularity of weak solutions to the Prandtl's system, preprint, arXiv: 2203.08988.
    [28] Y. Zou, J. Wang, Boundary layer separation for the steady compressible Prandtl equation, J. Math. Anal. Appl., 527 (2023), 127379. https://doi.org/10.1016/j.jmaa.2023.127379 doi: 10.1016/j.jmaa.2023.127379
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