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doi: 10.3934/era.2021070
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Local well-posedness of solutions to the boundary layer equations for compressible two-fluid flow

1. 

The Hubei Key Laboratory of Mathematical Physics, School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

2. 

School of Mathematics and Statistics, Shanxi Datong University, Datong 037009, China

3. 

School of Mathematical Sciences, Institute of Natural Sciences, Center of Applied Mathematics

4. 

MOE-LSC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, China

5. 

Hong Kong Institute for Advanced Study, City University of Hong Kong, Hong Kong, China

6. 

The Hubei Key Laboratory of Mathematical Physics, School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

* Corresponding author: Lizhi Ruan, rlz@mail.ccnu.edu.cn

Received  May 2021 Early access September 2021

In this paper, we consider the two-dimensional (2D) two-fluid boundary layer system, which is a hyperbolic-degenerate parabolic-elliptic coupling system derived from the compressible isentropic two-fluid flow equations with nonslip boundary condition for the velocity. The local existence and uniqueness is established in weighted Sobolev spaces under the monotonicity assumption on tangential velocity along normal direction based on a nonlinear energy method by employing a nonlinear cancelation technic introduced in [R. Alexandre, Y.-G. Wang, C.-J. Xu and T. Yang, J. Amer. Math. Soc., 28 (2015), 745-784; N. Masmoudi and T.K. Wong, Comm. Pure Appl. Math., 68(2015), 1683-1741] and developed in [C.-J. Liu, F. Xie and T. Yang, Comm. Pure Appl. Math., 72(2019), 63-121].

Citation: Long Fan, Cheng-Jie Liu, Lizhi Ruan. Local well-posedness of solutions to the boundary layer equations for compressible two-fluid flow. Electronic Research Archive, doi: 10.3934/era.2021070
References:
[1]

R. AlexandreY.-G. WangC.-J. Xu and T. Yang, Well-posedness of the Prandtl equation in Sobolev spaces, J. Amer. Math. Soc., 28 (2015), 745-784.  doi: 10.1090/S0894-0347-2014-00813-4.  Google Scholar

[2]

D. Bresch, B. Desjardins, J.-M. Ghidaglia, E. Grenier and M. Hilliairet, Multifluid models including compressible fluids. Handbook of mathematical analysis in mechanics of viscous fluids, Eds. Giga Y. et Novotny A., (2018), 2927–2978. doi: 10.1007/978-3-319-13344-7_74.  Google Scholar

[3]

R. E. Caflisch and M. Sammartino, Existence and singularities for the Prandtl boundary layer equations, ZAMM Z. Angew. Math. Mech., 80 (2000), 733-744.  doi: 10.1002/1521-4001(200011)80:11/12<733::AID-ZAMM733>3.0.CO;2-L.  Google Scholar

[4]

M. CannoneM. C. Lombardo and M. Sammartino, Well-posedness of Prandtl equations with non-compatible data, Nonlinearity, 26 (2013), 3077-3100.  doi: 10.1088/0951-7715/26/12/3077.  Google Scholar

[5]

W. E and B. Engquist, Blow up of solutions of the unsteady Prandtl's equation, Comm. Pure Appl. Math., 50 (1997), 1287-1293.  doi: 10.1002/(SICI)1097-0312(199712)50:12<1287::AID-CPA4>3.0.CO;2-4.  Google Scholar

[6]

L. Fan, L. Ruan and A. Yang, Local well-posedness of solutions to the boundary layer equations for 2D compressible flow, J. Math. Anal. Appl., 493 (2021), 124565, 25 pp. doi: 10.1016/j.jmaa.2020.124565.  Google Scholar

[7]

D. Gérard-Varet and E. Dormy, On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc., 23 (2010), 591-609.  doi: 10.1090/S0894-0347-09-00652-3.  Google Scholar

[8]

D. Gérard-Varet and T. Nguyen, Remarks on the ill-posedness of the Prandtl equation, Asymptot. Anal., 77 (2012), 71-88.  doi: 10.3233/ASY-2011-1075.  Google Scholar

[9]

E. Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math., 53 (2000), 1067-1091.  doi: 10.1002/1097-0312(200009)53:9<1067::AID-CPA1>3.0.CO;2-Q.  Google Scholar

[10]

S. GongY. Guo and Y.-G. Wang, Boundary layer problems for the two-dimensional compressible Navier-Stokes equations, Anal. Appl. (Singap.), 14 (2016), 1-37.  doi: 10.1142/S0219530515400011.  Google Scholar

[11]

Y. Guo and T. Nguyen, A note on Prandtl boundary layers, Commun. Pure Appl. Math., 64 (2011), 1416-1438.  doi: 10.1002/cpa.20377.  Google Scholar

[12]

Y. HuangC.-J. Liu and T. Yang, Local-in-time well-posedness for compressible MHD boundary layer, J. Differential Equations, 266 (2019), 2978-3013.  doi: 10.1016/j.jde.2018.08.052.  Google Scholar

[13]

M. Ishii and T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow, Springer-Verlag, New York, 2006. doi: 10.1007/978-0-387-29187-1.  Google Scholar

[14]

N. I. Kolev, Multiphase Flow Dynamics. Vol. 1. Fundamentals, Springer-Verlag, Berlin, 2005. Google Scholar

[15]

N. I. Kolev, Multiphase Flow Dynamics. Vol. 2. Thermal and Mechanical Interactions, Springer-Verlag, Berlin, 2005.  Google Scholar

[16]

W.-X. Li, N. Masmoudi and T. Yang, Well-posedness in Gevrey function space for 3D Prandtl equations without structural assumption, to appear in Comm. Pure Appl. Math.. Google Scholar

[17]

W.-X. Li and T. Yang, Well-posedness in Gevrey space for the Prandtl equations with nondegenerate points, J. Eur. Math. Soc. (JEMS), 22 (2020), 717-775.  doi: 10.4171/jems/931.  Google Scholar

[18]

X. Lin and T. Zhang, Almost global existence for 2D magnetohydrodynamics boundary layer system, Math. Methods Appl. Sci., 41 (2018), 7530-7553.  doi: 10.1002/mma.5217.  Google Scholar

[19]

X. Lin and T. Zhang, Almost global existence for the 3D Prandtl boundary layer equations, Acta Appl. Math., 169 (2020), 383-410.  doi: 10.1007/s10440-019-00303-y.  Google Scholar

[20]

C.-J. Liu, D. Wang, F. Xie and T. Yang, Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces, J. Funct. Anal., 279 (2020), 108637, 45 pp. doi: 10.1016/j.jfa.2020.108637.  Google Scholar

[21]

C.-J. LiuY.-G. Wang and T. Yang, Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 2011-2029.  doi: 10.3934/dcdss.2016082.  Google Scholar

[22]

C.-J. LiuY.-G. Wang and T. Yang, On the ill-posedness of the Prandtl equations in three space dimensions, Arch. Ration. Mech. Anal., 220 (2016), 83-108.  doi: 10.1007/s00205-015-0927-1.  Google Scholar

[23]

C.-J. LiuY.-G. Wang and T. Yang, A well-posedness theory for the Prandtl equations in three space variables, Adv. Math., 308 (2017), 1074-1126.  doi: 10.1016/j.aim.2016.12.025.  Google Scholar

[24]

C.-J. LiuF. Xie and T. Yang, A note on the ill-posedness of shear flow for the MHD boundary layer equations, Sci. China Math., 61 (2018), 2065-2078.  doi: 10.1007/s11425-017-9306-0.  Google Scholar

[25]

C.-J. LiuF. Xie and T. Yang, MHD boundary layers theory in Sobolev spaces without monotonicity Ⅰ: Well-posedness theory, Comm. Pure Appl. Math., 72 (2019), 63-121.  doi: 10.1002/cpa.21763.  Google Scholar

[26]

C.-J. LiuF. Xie and T. Yang, Justification of Prandtl ansatz for MHD boundary layer, SIAM J. Math. Anal., 51 (2019), 2748-2791.  doi: 10.1137/18M1219618.  Google Scholar

[27]

N. Masmoudi and 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.  doi: 10.1002/cpa.21595.  Google Scholar

[28]

O. A. Oleinik, On the mathematical theory of boundary layer for an unsteady flow of incompressible fluid, J. Appl. Math. Mech., 30 (1966), 951-974.  doi: 10.1016/0021-8928(66)90001-3.  Google Scholar

[29]

O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, Applied Mathematics and Mathematical Computation, 15., Chapman & Hall/CRC, Boca Raton, Fla., 1999.  Google Scholar

[30]

M. Paicu, P. Zhang and Z. Zhang, On the hydrostatic approximation of the Navier-Stokes equations in a thin strip, Adv. Math., 372 (2020), 107293, 42 pp. doi: 10.1016/j.aim.2020.107293.  Google Scholar

[31]

L. Prandtl, Über Flüssigkeitsbewegungen bei sehr Kleiner Reibung, In "Verh. Int. Math. Kongr., Heidelberg 1904, " Teubner, 1905. Google Scholar

[32]

X. QinT. YangZ. Yao and W. Zhou, A study on the boundary layer for the planar magnetohydrodynamics system, Acta Math. Sci. Ser. B (Engl. Ed.), 35 (2015), 787-806.  doi: 10.1016/S0252-9602(15)30022-9.  Google Scholar

[33]

X. QinT. YangZ. Yao and W. Zhou, Vanishing shear viscosity limit and boundary layer study for the planar MHD system, Math. Models Methods Appl. Sci., 29 (2019), 1139-1174.  doi: 10.1142/S0218202519500180.  Google Scholar

[34]

M. Sammartino and R. E. Caflisch, 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., 192 (1998), 433-461.  doi: 10.1007/s002200050304.  Google Scholar

[35]

M. Sammartino and R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half-space. Ⅱ. Construction of the Navier-Stokes solution, Comm. Math. Phys., 192 (1998), 463-491.  doi: 10.1007/s002200050305.  Google Scholar

[36]

Y.-G. Wang and 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.  doi: 10.5802/aif.2749.  Google Scholar

[37]

Y.-G. WangF. Xie and T. Yang, Local well-posedness of Prandtl equations for compressible flow in two space variables, SIAM J. Math. Anal., 47 (2015), 321-346.  doi: 10.1137/140978466.  Google Scholar

[38]

F. Xie and T. Yang, Global-in-time stability of 2D MHD boundary layer in the Prandtl-Hartmann regime, SIAM J. Math. Anal., 50 (2018), 5749-5760.  doi: 10.1137/18M1174969.  Google Scholar

[39]

F. Xie and T. Yang, Lifespan of solutions to MHD boundary layer equations with analytic perturbation of general shear flow, Acta Math. Appl. Sin. Engl. Ser., 35 (2019), 209-229.  doi: 10.1007/s10255-019-0805-y.  Google Scholar

[40]

Z. Xin and L. Zhang, On the global existence of solutions to the Prandtl's system, Adv. Math., 181 (2004), 88-133.  doi: 10.1016/S0001-8708(03)00046-X.  Google Scholar

[41]

C.-J. Xu and X. Zhang, Long time well-posedness of Prandtl equations in Sobolev space, J. Differential Equations, 263 (2017), 8749-8803.  doi: 10.1016/j.jde.2017.08.046.  Google Scholar

[42]

P. Zhang and Z. Zhang, Long time well-posednessof Prandtl system with small and analytic initial data, J. Funct. Anal., 270 (2016), 2591-2615.  doi: 10.1016/j.jfa.2016.01.004.  Google Scholar

show all references

References:
[1]

R. AlexandreY.-G. WangC.-J. Xu and T. Yang, Well-posedness of the Prandtl equation in Sobolev spaces, J. Amer. Math. Soc., 28 (2015), 745-784.  doi: 10.1090/S0894-0347-2014-00813-4.  Google Scholar

[2]

D. Bresch, B. Desjardins, J.-M. Ghidaglia, E. Grenier and M. Hilliairet, Multifluid models including compressible fluids. Handbook of mathematical analysis in mechanics of viscous fluids, Eds. Giga Y. et Novotny A., (2018), 2927–2978. doi: 10.1007/978-3-319-13344-7_74.  Google Scholar

[3]

R. E. Caflisch and M. Sammartino, Existence and singularities for the Prandtl boundary layer equations, ZAMM Z. Angew. Math. Mech., 80 (2000), 733-744.  doi: 10.1002/1521-4001(200011)80:11/12<733::AID-ZAMM733>3.0.CO;2-L.  Google Scholar

[4]

M. CannoneM. C. Lombardo and M. Sammartino, Well-posedness of Prandtl equations with non-compatible data, Nonlinearity, 26 (2013), 3077-3100.  doi: 10.1088/0951-7715/26/12/3077.  Google Scholar

[5]

W. E and B. Engquist, Blow up of solutions of the unsteady Prandtl's equation, Comm. Pure Appl. Math., 50 (1997), 1287-1293.  doi: 10.1002/(SICI)1097-0312(199712)50:12<1287::AID-CPA4>3.0.CO;2-4.  Google Scholar

[6]

L. Fan, L. Ruan and A. Yang, Local well-posedness of solutions to the boundary layer equations for 2D compressible flow, J. Math. Anal. Appl., 493 (2021), 124565, 25 pp. doi: 10.1016/j.jmaa.2020.124565.  Google Scholar

[7]

D. Gérard-Varet and E. Dormy, On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc., 23 (2010), 591-609.  doi: 10.1090/S0894-0347-09-00652-3.  Google Scholar

[8]

D. Gérard-Varet and T. Nguyen, Remarks on the ill-posedness of the Prandtl equation, Asymptot. Anal., 77 (2012), 71-88.  doi: 10.3233/ASY-2011-1075.  Google Scholar

[9]

E. Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math., 53 (2000), 1067-1091.  doi: 10.1002/1097-0312(200009)53:9<1067::AID-CPA1>3.0.CO;2-Q.  Google Scholar

[10]

S. GongY. Guo and Y.-G. Wang, Boundary layer problems for the two-dimensional compressible Navier-Stokes equations, Anal. Appl. (Singap.), 14 (2016), 1-37.  doi: 10.1142/S0219530515400011.  Google Scholar

[11]

Y. Guo and T. Nguyen, A note on Prandtl boundary layers, Commun. Pure Appl. Math., 64 (2011), 1416-1438.  doi: 10.1002/cpa.20377.  Google Scholar

[12]

Y. HuangC.-J. Liu and T. Yang, Local-in-time well-posedness for compressible MHD boundary layer, J. Differential Equations, 266 (2019), 2978-3013.  doi: 10.1016/j.jde.2018.08.052.  Google Scholar

[13]

M. Ishii and T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow, Springer-Verlag, New York, 2006. doi: 10.1007/978-0-387-29187-1.  Google Scholar

[14]

N. I. Kolev, Multiphase Flow Dynamics. Vol. 1. Fundamentals, Springer-Verlag, Berlin, 2005. Google Scholar

[15]

N. I. Kolev, Multiphase Flow Dynamics. Vol. 2. Thermal and Mechanical Interactions, Springer-Verlag, Berlin, 2005.  Google Scholar

[16]

W.-X. Li, N. Masmoudi and T. Yang, Well-posedness in Gevrey function space for 3D Prandtl equations without structural assumption, to appear in Comm. Pure Appl. Math.. Google Scholar

[17]

W.-X. Li and T. Yang, Well-posedness in Gevrey space for the Prandtl equations with nondegenerate points, J. Eur. Math. Soc. (JEMS), 22 (2020), 717-775.  doi: 10.4171/jems/931.  Google Scholar

[18]

X. Lin and T. Zhang, Almost global existence for 2D magnetohydrodynamics boundary layer system, Math. Methods Appl. Sci., 41 (2018), 7530-7553.  doi: 10.1002/mma.5217.  Google Scholar

[19]

X. Lin and T. Zhang, Almost global existence for the 3D Prandtl boundary layer equations, Acta Appl. Math., 169 (2020), 383-410.  doi: 10.1007/s10440-019-00303-y.  Google Scholar

[20]

C.-J. Liu, D. Wang, F. Xie and T. Yang, Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces, J. Funct. Anal., 279 (2020), 108637, 45 pp. doi: 10.1016/j.jfa.2020.108637.  Google Scholar

[21]

C.-J. LiuY.-G. Wang and T. Yang, Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 2011-2029.  doi: 10.3934/dcdss.2016082.  Google Scholar

[22]

C.-J. LiuY.-G. Wang and T. Yang, On the ill-posedness of the Prandtl equations in three space dimensions, Arch. Ration. Mech. Anal., 220 (2016), 83-108.  doi: 10.1007/s00205-015-0927-1.  Google Scholar

[23]

C.-J. LiuY.-G. Wang and T. Yang, A well-posedness theory for the Prandtl equations in three space variables, Adv. Math., 308 (2017), 1074-1126.  doi: 10.1016/j.aim.2016.12.025.  Google Scholar

[24]

C.-J. LiuF. Xie and T. Yang, A note on the ill-posedness of shear flow for the MHD boundary layer equations, Sci. China Math., 61 (2018), 2065-2078.  doi: 10.1007/s11425-017-9306-0.  Google Scholar

[25]

C.-J. LiuF. Xie and T. Yang, MHD boundary layers theory in Sobolev spaces without monotonicity Ⅰ: Well-posedness theory, Comm. Pure Appl. Math., 72 (2019), 63-121.  doi: 10.1002/cpa.21763.  Google Scholar

[26]

C.-J. LiuF. Xie and T. Yang, Justification of Prandtl ansatz for MHD boundary layer, SIAM J. Math. Anal., 51 (2019), 2748-2791.  doi: 10.1137/18M1219618.  Google Scholar

[27]

N. Masmoudi and 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.  doi: 10.1002/cpa.21595.  Google Scholar

[28]

O. A. Oleinik, On the mathematical theory of boundary layer for an unsteady flow of incompressible fluid, J. Appl. Math. Mech., 30 (1966), 951-974.  doi: 10.1016/0021-8928(66)90001-3.  Google Scholar

[29]

O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, Applied Mathematics and Mathematical Computation, 15., Chapman & Hall/CRC, Boca Raton, Fla., 1999.  Google Scholar

[30]

M. Paicu, P. Zhang and Z. Zhang, On the hydrostatic approximation of the Navier-Stokes equations in a thin strip, Adv. Math., 372 (2020), 107293, 42 pp. doi: 10.1016/j.aim.2020.107293.  Google Scholar

[31]

L. Prandtl, Über Flüssigkeitsbewegungen bei sehr Kleiner Reibung, In "Verh. Int. Math. Kongr., Heidelberg 1904, " Teubner, 1905. Google Scholar

[32]

X. QinT. YangZ. Yao and W. Zhou, A study on the boundary layer for the planar magnetohydrodynamics system, Acta Math. Sci. Ser. B (Engl. Ed.), 35 (2015), 787-806.  doi: 10.1016/S0252-9602(15)30022-9.  Google Scholar

[33]

X. QinT. YangZ. Yao and W. Zhou, Vanishing shear viscosity limit and boundary layer study for the planar MHD system, Math. Models Methods Appl. Sci., 29 (2019), 1139-1174.  doi: 10.1142/S0218202519500180.  Google Scholar

[34]

M. Sammartino and R. E. Caflisch, 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., 192 (1998), 433-461.  doi: 10.1007/s002200050304.  Google Scholar

[35]

M. Sammartino and R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half-space. Ⅱ. Construction of the Navier-Stokes solution, Comm. Math. Phys., 192 (1998), 463-491.  doi: 10.1007/s002200050305.  Google Scholar

[36]

Y.-G. Wang and 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.  doi: 10.5802/aif.2749.  Google Scholar

[37]

Y.-G. WangF. Xie and T. Yang, Local well-posedness of Prandtl equations for compressible flow in two space variables, SIAM J. Math. Anal., 47 (2015), 321-346.  doi: 10.1137/140978466.  Google Scholar

[38]

F. Xie and T. Yang, Global-in-time stability of 2D MHD boundary layer in the Prandtl-Hartmann regime, SIAM J. Math. Anal., 50 (2018), 5749-5760.  doi: 10.1137/18M1174969.  Google Scholar

[39]

F. Xie and T. Yang, Lifespan of solutions to MHD boundary layer equations with analytic perturbation of general shear flow, Acta Math. Appl. Sin. Engl. Ser., 35 (2019), 209-229.  doi: 10.1007/s10255-019-0805-y.  Google Scholar

[40]

Z. Xin and L. Zhang, On the global existence of solutions to the Prandtl's system, Adv. Math., 181 (2004), 88-133.  doi: 10.1016/S0001-8708(03)00046-X.  Google Scholar

[41]

C.-J. Xu and X. Zhang, Long time well-posedness of Prandtl equations in Sobolev space, J. Differential Equations, 263 (2017), 8749-8803.  doi: 10.1016/j.jde.2017.08.046.  Google Scholar

[42]

P. Zhang and Z. Zhang, Long time well-posednessof Prandtl system with small and analytic initial data, J. Funct. Anal., 270 (2016), 2591-2615.  doi: 10.1016/j.jfa.2016.01.004.  Google Scholar

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