Uniqueness and stability of entropy shocks to the isentropic Euler system in a class of inviscid limits from a large fam
- PDF / 862,482 Bytes
- 92 Pages / 439.37 x 666.142 pts Page_size
- 9 Downloads / 163 Views
Uniqueness and stability of entropy shocks to the isentropic Euler system in a class of inviscid limits from a large family of Navier–Stokes systems Moon-Jin Kang1 · Alexis F. Vasseur2
Received: 2 May 2019 / Accepted: 10 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We prove the uniqueness and stability of entropy shocks to the isentropic Euler systems among all vanishing viscosity limits of solutions to associated Navier–Stokes systems. To take into account the vanishing viscosity limit, we show a contraction property for any large perturbations of viscous shocks to the Navier–Stokes system. The contraction estimate does not depend on the strength of the viscosity. This provides a good control on the inviscid limit process. We prove that, for any initial value, there exist a vanishing viscosity limit to solutions of the Navier–Stokes system. The convergence holds in a weak topology. However, this limit satisfies some stability estimates measured by the relative entropy with respect to an entropy shock. In particular, our result provides the uniqueness of entropy shocks to the shallow water equation in a class of inviscid limits of solutions to the viscous shallow water equations. Mathematics Subject Classification 76N15 · 35B35 · 35Q30
B Moon-Jin Kang
[email protected] Alexis F. Vasseur [email protected]
1
Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea
2
Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USA
123
M.-J. Kang, A. F. Vasseur
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1.1 Main results . . . . . . . . . . . . . . . . . . . . . 2 Ideas of the proof . . . . . . . . . . . . . . . . . . . . . 3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 3.1 Transformation of the system (1.16) . . . . . . . . 3.2 Global and local estimates on the relative quantities 4 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . 4.1 Properties of small shock waves . . . . . . . . . . 4.2 Relative entropy method . . . . . . . . . . . . . . 4.3 Construction of the weight function . . . . . . . . . 4.4 Maximization in terms of h − h˜ ε . . . . . . . . . . 4.5 Main proposition . . . . . . . . . . . . . . . . . . 4.6 Proof of Theorem 3.1 from Proposition 4.1 . . . . . 4.7 Expansion in the size of the shock . . . . . . . . . 4.8 Truncation of the big values of | p(v) − p(v˜ε )| . . . 4.8.1 Proof of Proposition 4.3 . . . . . . . . . . . 4.9 Proof of Proposition 4.1 . . . . . . . . . . . . . . . 5 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . 5.1 Proof of (1.12) : Well-prepared initial data . . . . . 5.2 Proof for the main part of Theorem 1.1 . . . . . . . 5.2.1 Uniform estimates in ν . . . . . . . . . . . . 5.2.2 Proof of (1.13) . . . . . . . . . . . . . . . . 5.2.3 Convergence of {X ν }ν>0 . . . . . . . . . . . 5.2.4 Proof of (1.14) . . . . . . . . . . . . . . . . 5.2.5 Weak continuity of the limit v∞ . . . . . . . 5.2.6 Proof of
Data Loading...
