Related papers: Conditional stability of unstable viscous shocks
We determine the modulational stability of standing waves with small group velocity in quasi-onedimensional systems slightly above the threshold of a supercritical Hopf bifurcation. The stability limits are given by two different…
We prove the linear orbital stability of spectrally stable stationary discrete shock profiles for conservative finite difference schemes applied to systems of conservation laws. The proof relies on an accurate description of the pointwise…
Coughlin et al. (2018) (Paper I) derived and analyzed a new regime of self-similarity that describes weak shocks (Mach number of order unity) in the gravitational field of a point mass. These solutions are relevant to low energy explosions,…
This paper studies the local stable and unstable manifolds of equilibria for quasilinear and fully nonlinear PDEs. These manifolds are fundamental objects in the analysis of local dynamics. While their existence is well understood for ODEs,…
We consider an ensemble of mass collisionless particles, which interact mutually either by an attraction of Newton's law of gravitation or by an electrostatic repulsion of Coulomb's law, under a background downward gravity in a…
This article studies a class of semilinear scalar field equations on the real line with variable coefficients in the linear terms. These coefficients are not necessarily small perturbations of a constant. We prove that under suitable…
We consider the nonlinear wave equation, with a large exponent, power-like non-linearity, outside a ball of the Euclidean 3-dimensional space. In a previous article, we have proved that any global solution converges, up to a radiation term,…
We study the behavior of perturbations of small nonlinear Dirac standing waves. We assume that the linear Dirac operator of reference $H=D_m+V$ has only two double eigenvalues and that degeneracies are due to a symmetry of $H$ (theorem of…
Linear stability of a plane shock waves in ultrarelativistic anisotropic hydrodynamics is investigated. The properties of the amplitudes of perturbations of physical quantities are studied depending on the components of the wave vector of a…
Following the original approach introduced by T. Cazenave and P.L. Lions in \cite{CaLi} we prove the existence and the orbital stability of standing waves for the following class of NLS: \label{intr1} i\partial_t u+ \Delta u - V(x) u + Q(x)…
In this paper, we consider scalar conservation laws with smoothly varying spatially heterogeneous flux that is convex in the conserved variable. We show that under certain assumptions, a shock wave connecting two constant states emerges in…
Refining previous work in \cite{Z.3, MaZ.3, Ra, HZ, HR}, we derive sharp pointwise bounds on behavior of perturbed viscous shock profiles for large-amplitude Lax or overcompressive type shocks and physical viscosity. These extend well-known…
We present a criterion for a shock wave existence in relativistic magnetic hydrodynamics with an arbitrary (possibly non-convex) equation of state. The criterion has the form of algebraic inequality that involves equation of state of the…
The evolutionary conditions for the dissipative continuous magnetohydrodynamic (MHD) shocks are studied. We modify Hada's approach in the stability analysis of the MHD shock waves. The matching conditions between perturbed shock structure…
We consider the incompressible Euler equations in the half cylinder $ \mathbb{R}_{>0}\times\mathbb{T}$. In this domain, any vorticity which is independent of $x_2$ defines a stationary solution. We prove that such a stationary solution is…
We study motion of a phase transition front at a constant temperature between stable and metastable states in fluids with the universal Van der Waals equation of state (which is valid sufficiently close to the fluid's critical point). We…
In this paper, we investigate the instability of the trivial steady states to the incompressible viscous fluid with Navier-slip boundary conditions. For the linear instability, the existence of infinitely many normal mode solutions to the…
Extending investigations of Yarahmadian and Zumbrun in the strictly parabolic case, we study time-asymptotic stability of arbitrary (possibly large) amplitude noncharacteristic boundary layers of a class of hyperbolic-parabolic systems…
Building on work of Barker, Humpherys, Lafitte, Rudd, and Zumbrun in the shock wave case, we study stability of compressive, or "shock-like", boundary layers of the isentropic compressible Navier-Stokes equations with gamma-law pressure by…
By a refinement of the technique used by Johnson and Zumbrun to show stability under localized perturbations, we show that spectral stability implies nonlinear modulational stability of periodic traveling-wave solutions of reaction…