March 2019, 18(2): 887-910. doi: 10.3934/cpaa.2019043

Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers

1. 

Department of Mathematics, Shanghai Normal University, ShangHai, 200234, China

2. 

Department of Mathematics, Shanghai University, ShangHai, 200444, China

* Corresponding author

Received  April 2018 Revised  June 2018 Published  October 2018

Fund Project: The first author is supported by NSFC grant 11771297; The second author is supported by NSFC grant 11771274

In this paper, we study the limiting behavior of solutions to a 1D two-point boundary value problem for viscous conservation laws with genuinely-nonlinear fluxes as $\varepsilon$ goes to zero. We here discuss different types of non-characteristic boundary layers occurring on both sides. We first construct formally the three-term approximate solutions by using the method of matched asymptotic expansions. Next, by energy method we prove that the boundary layers are nonlinearly stable and thus it is proved the boundary layer effects are just localized near both boundaries. Consequently, the viscous solutions converge to the smooth inviscid solution uniformly away from the boundaries. The rate of convergence in viscosity is optimal.

Citation: Jing Wang, Lining Tong. Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers. Communications on Pure & Applied Analysis, 2019, 18 (2) : 887-910. doi: 10.3934/cpaa.2019043
References:
[1]

B. Desjardins and E. Grenier, Linear instability implies nonlinear instability for various types of viscous boundary layers, Ann. Inst. H. Poincaré Anal, 20 (2003), 87-106. doi: 10.1016/S0294-1449(02)00009-4.

[2]

E. Grenier and O. Gues, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems, J. Differential Equations, 143 (1998), 110-146. doi: 10.1006/jdeq.1997.3364.

[3]

G. Jonathan and Z. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Rational Mech. Anal., 121 (1992), 235-265. doi: 10.1007/BF00410614.

[4]

G. Metivier, Small Viscosity and Boundary Layer Methods, Theory, stability analysis, and applications. Modeling and Simulation in Science, Engineering and Technology. Birkh user Boston, Inc., Boston, MA, 2004. doi: 10.1007/978-0-8176-8214-9.

[5]

G. Metivier and K. Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems, Mem. Amer. Math. Soc., 175 (2004), 243-252. doi: 10.1090/memo/0826.

[6]

T. Nguyen and K. Zumbrun, Long-time stability of multi-dimensional noncharacteristic viscous boundary layers, Comm. Math. Phys., 299 (2010), 1-44. doi: 10.1007/s00220-010-1095-7.

[7]

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

[8]

H. Schlichting, Boundary Layer Theory, 7th edition, McGraw-Hill, 1979.

[9]

D. Serre, Systems of Conservation Laws. 2. Geometric Structures, Oscillations, and Initial-boundary Value Problems, Translated from the 1996 French original by I. N. Sneddon. Cambridge University Press, Cambridge, 2000.

[10]

L. Tong and J. Wang, Stability of multiple boundary layers for 2D quasilinear parabolic equations, J. Math. Anal. Appl., 435 (2016), 349-368. doi: 10.1016/j.jmaa.2015.10.030.

[11]

J. Wang, Boundary layers for parabolic perturbations of quasi-linear hyperbolic problems, Math. Methods Appl. Sci., 32 (2009), 2416-2438. doi: 10.1002/mma.1144.

[12]

Z. Xin, Viscous boundary layers and their stability, J. Partial Differential Equations, 11 (1998), 97-124.

show all references

References:
[1]

B. Desjardins and E. Grenier, Linear instability implies nonlinear instability for various types of viscous boundary layers, Ann. Inst. H. Poincaré Anal, 20 (2003), 87-106. doi: 10.1016/S0294-1449(02)00009-4.

[2]

E. Grenier and O. Gues, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems, J. Differential Equations, 143 (1998), 110-146. doi: 10.1006/jdeq.1997.3364.

[3]

G. Jonathan and Z. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Rational Mech. Anal., 121 (1992), 235-265. doi: 10.1007/BF00410614.

[4]

G. Metivier, Small Viscosity and Boundary Layer Methods, Theory, stability analysis, and applications. Modeling and Simulation in Science, Engineering and Technology. Birkh user Boston, Inc., Boston, MA, 2004. doi: 10.1007/978-0-8176-8214-9.

[5]

G. Metivier and K. Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems, Mem. Amer. Math. Soc., 175 (2004), 243-252. doi: 10.1090/memo/0826.

[6]

T. Nguyen and K. Zumbrun, Long-time stability of multi-dimensional noncharacteristic viscous boundary layers, Comm. Math. Phys., 299 (2010), 1-44. doi: 10.1007/s00220-010-1095-7.

[7]

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

[8]

H. Schlichting, Boundary Layer Theory, 7th edition, McGraw-Hill, 1979.

[9]

D. Serre, Systems of Conservation Laws. 2. Geometric Structures, Oscillations, and Initial-boundary Value Problems, Translated from the 1996 French original by I. N. Sneddon. Cambridge University Press, Cambridge, 2000.

[10]

L. Tong and J. Wang, Stability of multiple boundary layers for 2D quasilinear parabolic equations, J. Math. Anal. Appl., 435 (2016), 349-368. doi: 10.1016/j.jmaa.2015.10.030.

[11]

J. Wang, Boundary layers for parabolic perturbations of quasi-linear hyperbolic problems, Math. Methods Appl. Sci., 32 (2009), 2416-2438. doi: 10.1002/mma.1144.

[12]

Z. Xin, Viscous boundary layers and their stability, J. Partial Differential Equations, 11 (1998), 97-124.

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