Related papers: Conditional stability of unstable viscous shocks
Solutions of constant-coefficient nonlinear hyperbolic PDEs generically develop shocks, even if the initial data is smooth. Solutions of hyperbolic PDEs with variable coefficients can behave very differently. We investigate formation and…
In this paper, the asymptotic-time behavior of solutions to an initial boundary value problem in the half space for 1-D isentropic Navier-Stokes system is investigated. It is shown that the viscous shock wave is stable for an impermeable…
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…
In this paper, we are concerned with the large time behavior of viscous shock wave for the convective porous-media equation with degenerate viscosity. We get the regularity of the solution for general initial data and prove the shock wave…
By reduction to a generalized Sturm Liouville problem, we establish spectral stability of hydraulic shock profiles of the Saint-Venant equations for inclined shallow-water flow, over the full parameter range of their existence, for both…
Extending previous results of Oh--Zumbrun and Johnson--Zumbrun, we show that spectral stability implies linearized and nonlinear stability of spatially periodic traveling-wave solutions of viscous systems of conservation laws for systems of…
This paper is concerned with the stability of planar viscous shock wave for the 3-dimensional compressible Navier-Stokes-Poisson (NSP) system in the domain $\Omega:=\mathbb{R}\times \mathbb{T}^2$ with…
We investigate $L^2$-contraction and time-asymptotic stability of large shock for scalar viscous conservation laws with polynomial flux. For the strictly convex flux $f(u)=u^p $ with $2\leq p \leq 4$, we can prove $L^2$-contraction and…
The recent theory of $a-$contraction with shifts provides $L^2$-stability for shock waves of $1-$D hyperbolic systems of conservation laws. The theory has been established at the inviscid level uniformly in the shock amplitude, and at the…
This paper concerns the stabilizing effect of viscosity on the vortex sheets. It is found that although a vortex sheet is not a time-asymptotic attractor for the compressible Navier-Stokes equations, a viscous wave that approximates the…
In this paper, a viscous shock wave under space-periodic perturbation of 1-D isentropic Navier-Stokes system in the half space is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small,…
We study extremal shocks of $1$-d hyperbolic systems of conservation laws which fail to be genuinely nonlinear. More specifically, we consider either $1$- or $n$-shocks in characteristic fields which are either concave-convex or…
We give the first proof of nonlinear stability for smooth shock profiles of second-order dissipative hyperbolic-hyperbolic systems under the assumption of spectral stability, showing stability of smooth small-amplitude profiles in…
We summarize recent progress on one- and multi-dimensional stability of viscous shock wave solutions of compressible Navier--Stokes equations and related symmetrizable hyperbolic--parabolic systems, with an emphasis on the large-amplitude…
Building on Evans function techniques developed to study the stability of viscous shocks, we examine the stability of viscous strong detonation wave solutions of the reacting Navier-Stokes equations. The primary result, following the work…
We prove the nonlinear stability of the planar viscous shock up to a time-dependent shift for the three-dimensional (3D) compressible Navier-Stokes equations under the generic perturbations, in particular, without zero mass conditions.…
We establish nonlinear $H^2\cap L^1 \to H^2$ stability with sharp rates of decay in $L^p$, $p\geq 2$, of general hydraulic shock profiles, with or without subshocks, of the inviscid Saint-Venant equations of shallow water flow, under the…
By using an equivalent form of the uniform Lopatinski condition for 1-shocks, we prove that the stability condition found by the energy method in [A. Morando, Y. Trakhinin, P. Trebeschi, Structural stability of shock waves in 2D…
In this paper we study small shocks of 1D scalar viscous conservation laws with uniformly convex flux and nonlinear dissipation. We show that such shocks are L2 stable independent of the strength of the dissipation, even with large…
Extending results of Humpherys-Lyng-Zumbrun in the one-dimensional case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the multidimensional stability of planar Navier--Stokes shocks…