Scalar Green function bounds for instantaneous shock location and one-dimensional stability of viscous shock waves

Li, Yingwei

doi:10.1090/qam/1431

Author: Yingwei Li
Journal: Quart. Appl. Math. 74 (2016), 499-538
MSC (2010): Primary 35-XX, 35B35
DOI: https://doi.org/10.1090/qam/1431
Published electronically: June 16, 2016
MathSciNet review: 3518226
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Abstract: In this paper, we investigate and prove the nonlinear stability of viscous shock wave solutions of a scalar viscous conservation law, using the methods developed for general systems of conservation laws by Howard, Mascia, Zumbrun and others, based on instantaneous tracking of the location of the perturbed viscous shock wave. In some sense, this paper extends the treatment in a previous expository work of Zumbrun [“Instantaneous shock location…”] on Burgers equation to the general case, giving an exposition of these methods in the simplest setting of scalar equations. In particular we give, by a rescaling argument, a simple treatment of nonlinear stability in the small-amplitude case.

References

Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2005. MR 2169977, DOI 10.1007/3-540-29089-3
Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
Peter Howard, Pointwise Green’s function approach to stability for scalar conservation laws, Comm. Pure Appl. Math. 52 (1999), no. 10, 1295–1313. MR 1699970, DOI 10.1002/(SICI)1097-0312(199910)52:10<1295::AID-CPA6>3.3.CO;2-D
Tai-Ping Liu and Yanni Zeng, Time-asymptotic behavior of wave propagation around a viscous shock profile, Comm. Math. Phys. 290 (2009), no. 1, 23–82. MR 2520507, DOI 10.1007/s00220-009-0820-6
Andrew Majda and Robert L. Pego, Stable viscosity matrices for systems of conservation laws, J. Differential Equations 56 (1985), no. 2, 229–262. MR 774165, DOI 10.1016/0022-0396(85)90107-X
Corrado Mascia and Kevin Zumbrun, Pointwise Green function bounds for shock profiles of systems with real viscosity, Arch. Ration. Mech. Anal. 169 (2003), no. 3, 177–263. MR 2004135, DOI 10.1007/s00205-003-0258-5
C. Mascia and K. Zumbrun, Stability of small-amplitude shock profiles of symmetric hyperbolic-parabolic systems, Comm. Pure Appl. Math. 57 (2004), no. 7, 841–876. MR 2044067, DOI 10.1002/cpa.20023
Corrado Mascia and Kevin Zumbrun, Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems, Arch. Ration. Mech. Anal. 172 (2004), no. 1, 93–131. MR 2048568, DOI 10.1007/s00205-003-0293-2
Ramon Plaza and Kevin Zumbrun, An Evans function approach to spectral stability of small-amplitude shock profiles, Discrete Contin. Dyn. Syst. 10 (2004), no. 4, 885–924. MR 2073940, DOI 10.3934/dcds.2004.10.885
Kevin Zumbrun, Instantaneous shock location and one-dimensional nonlinear stability of viscous shock waves, Quart. Appl. Math. 69 (2011), no. 1, 177–202. MR 2807984, DOI 10.1090/S0033-569X-2011-01221-6
Kevin Zumbrun and Peter Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J. 47 (1998), no. 3, 741–871. MR 1665788, DOI 10.1512/iumj.1998.47.1604