二维可压缩Prandtl方程倒流点的存在性

Abstract

Abstract:

In this paper, we study the back-flow problem of the two-dimensional unsteady compressible Prandtl boundary layer equations. By using the maximum principle, we obtain that a first back-flow point should appear on the boundary $\left\{y=0\right\}$ if back-flow occurs under Oleinik's monotonicity assumption. Moreover, when the pressure gradient of the outer flow is adverse and the initial velocity satisfies certain growth condition, we obtain the existence of a back-flow point of the compressible Prandtl boundary layer by Lyapunov functional method. Finally, an example of the existence of the back-flow point is given.

Key words: Compressible Prandtl equation, Back-flow point, The maximum principle, Adverse pressure gradient, Lyapunov functional

CLC Number:

O175.29

Zou Yonghui,Xu Xin. Existence of Back-Flow Point for the Two-Dimensional Compressible Prandtl Equation[J].Acta mathematica scientia,Series A, 2023, 43(3): 691-701.

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References

[1]	Alexandre R, Wang Y, Xu C, et al. Well-posedness of the Prandtl equation in Sobolev spaces. J Amer Math Soc, 2015, 28(3): 745-784
[2]	Chen D, Wang Y, Zhang Z. Well-posedness of the Prandtl equation with monotonicity in Sobolev spaces. J Differential Equations, 2018, 264(9): 5870-5893
[3]	Dalibard A L, Masmoudi N. Separation for the stationary Prandtl equation. Publ Math Inst Hautes Etudes Sci, 2019, 130: 187-297
[4]	Ding M, Gong S. Global existence of weak solution to the compressible Prandtl equations. J Math Fluid Mech, 2017, 19(2): 239-254
[5]	E W N. Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation. Acta Math Sin (Engl Ser), 2000, 16(2): 207-218
[6]	E W N, Engquist B. Blowup of solutions of the unsteady Prandtl's equation. Comm Pure Appl Math, 1997, 50: 1287-1293
[7]	Fan L, Ruan L, Yang A. Local well-posedness of solutions to the boundary layer equations for 2D compressible flow. J Math Anal Appl, 2021, 493(2): 124565
[8]	Gong S, Guo Y, Wang Y. Boundary layer problems for the two-dimensional compressible Navier-Stokes equations. Anal Appl (Singap), 2016, 14(1): 1-37
[9]	Gong S, Wang X. On a global weak solution and back flow of the mixed Prandtl-Hartmann boundary layer problem. J Math Fluid Mech, 2021, 23(11): 1-16
[10]	Gong S, Wang X, Wang Y. Stability and back flow of boundary layers for wind-driven oceanic current. Commun Math Sci, 2020, 18(3): 593-612
[11]	Kukavica I, Vicol V. On the local existence of analytic solutions to the Prandtl boundary layer equations. Commun Math Sci, 2013, 11(1): 269-292
[12]	Kukavica I, Vicol V, Wang F. The van Dommelen and Shen singularity in the Prandtl equations. Adv Math, 2017, 307: 288-311
[13]	Li W, Masmoudi N, Yang T. Well-Posedness in Gevrey Function Space for 3D Prandtl Equations without structural assumption. Comm Pure Appl Math, 2022, 75(8): 1755-1797
[14]	Liu C, Wang Y, Yang T. On the ill-posedness of the Prandtl equations in three space dimensions. Arch Rational Mech Anal, 2016, 220: 83-108
[15]	Liu C, Wang Y, Yang T. A well-posedness theory for the Prandtl equations in three space variables. Adv Math, 2017, 308: 1074-1126
[16]	Liu C, Xie F, Yang T. MHD boundary Layers theory in Sobolev spaces without monotonicity I: Well-Posedness theory. Comm Pure Appl Math, 2019, 72(1): 63-121
[17]	Masmoudi N, Wong T K. Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods. Comm Pure Appl Math, 2015, 68(10): 1683-1741
[18]	Moore F K. Three-dimensional boundary layer theory. Adv Appl Mech, 1956, 4: 159-228
[19]	Oleinik O A. On the system of Prandtl equations in boundary-layer theory. Dokl Akad Nauk SSSR, 1963, 150: 28-31
[20]	Oleinik O A, Samokhin V N. Mathematical models in boundary layer theory. Routledge, 2018
[21]	Prandtl L. über Flüssigkeitsbewegung bei sehr kleiner Reibung. Verhandl. III, Internat Math-Kong, Heidelberg, Teubner, Leipzig, 1904, 452: 575-584
[22]	Paicu M, Zhang P. Global existence and the decay of solutions to the Prandtl system with small analytic data. Arch Ration Mech Anal, 2021, 241(1): 403-446
[23]	Sammartino M, Caflisch R E. Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations. Comm Math Phys, 1998, 192(2): 433-461
[24]	Sammartino M, Caflisch R E. Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution. Comm Math Phys, 1998, 192(2): 463-491
[25]	Schlichting H, Gersten K. Boundary-Layer Theory, Enlarged Edition. New York: Springer-Verlag, 2000
[26]	Sears W R, Telionis D P. Boundary-layer separation in unsteady flow. SIAM J Math Anal, 1975, 28(1): 215-235
[27]	Shen W, Wang Y, Zhang Z. Boundary layer separation and local behavior for the steady Prandtl equation. Adv Math, 2021, 389: 107896
[28]	Wang Y, Williams M. The inviscid limit and stability of characteristic boundary layers for the compressible Navier-Stokes equations with Navier-friction boundary conditions. Ann Inst Fourier (Grenoble), 2013, 62(6): 2257-2314
[29]	Wang Y, Xie F, Yang T. Local well-posedness of Prandtl equations for compressible flow in two space variables. SIAM J Math Anal, 2015, 47(1): 321-346
[30]	Wang Y, Zhu S. Well-posedness of thermal boundary layer equation in two-dimensional incompressible heat conducting flow with analytic datum. Math Methods Appl Sci, 2020, 43(7): 4683-4716
[31]	Wang Y, Zhu S. Back flow of the two-dimensional unsteady Prandtl boundary layer under an adverse pressure gradient. SIAM J Math Anal, 2020, 52(1): 954-966
[32]	Wang Y, Zhu S. On back flow of boundary layers in two-dimensional unsteady incompressible heat conducting flow. J Math Phys, 2022, 63(8): 081504
[33]	Xin Z, Zhang L. On the global existence of solutions to the Prandtl's system. Adv Math, 2004, 181: 88-133
[34]	Xin Z, Zhang L, Zhao J. Global well-posedness and regularity of weak solutions to the Prandtl's system. arXiv:2203.08988v1, 2022
[35]	Xu C, Zhang X. Long time well-posedness of Prandtl equations in Sobolev space. J Differential Equations, 2017, 263(12): 8749-8803
[36]	Zhang P, Zhang Z. Long time well-posedness of Prandtl system with small and analytic initial data. J Funct Anal, 2016, 270(7): 2591-2615

Existence of Back-Flow Point for the Two-Dimensional Compressible Prandtl Equation

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