Nonlinear stability of the composite wave of planar rarefaction waves and planar contact waves for viscous conservation laws with non-convex flux under multi-dimensional periodic perturbations

In this paper, we study the nonlinear stability of the composite wave consisting of planar rarefaction and planar contact waves for viscous conservation laws with degenerate flux under multi-dimensional periodic perturbations. To the level of our knowledge, it is the first stability result of the composite wave for conservation laws in several dimensions. Moreover, the perturbations studied in the present paper are periodic, which keep constantly oscillating at infinity. Suitable ansatz is constructed to overcome the difficulty caused by this kind of perturbation and delicate estimates are done on zero and non-zero modes of perturbations. We obtain satisfactory decay rates for zero modes and exponential decay rates for non-zero modes

Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws

We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion are known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is non-convex. This review compares the structure of solutions of Riemann problems for a conservation law with non-convex, cubic flux regularized by two different mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation; and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both classical (Lax) and non-classical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves.Comment: Revision from v2; 57 pages, 19 figure

Nonlinear asymptotic stability of planar viscous shocks for 3D compressible Navier-Stokes equations with periodic perturbations

This paper studies a Cauchy problem for the three-dimensional compressible isentropic Navier-Stokes equations, in which the initial data is a planar viscous shock with a periodic perturbation. It is shown that if the shock strength is weak and the perturbation is small and fulfills a zero-mass type condition, then the Cauchy problem admits a unique classical solution globally in time, which approaches the background planar viscous shock with a constant shift in the $W^{1,\infty}(\mathbb{R}^3)$ space as $t\to +\infty$. Moreover, an exponential decay rate of the non-zero mode of the solution is obtained in the $L^\infty(\mathbb{R}^3)$ space. The result reveals the nonlinear time-asymptotic stability of planar viscous shocks under the perturbations that oscillate in all spatial variables. The main ingredients of the proof consist of the construction of a suitable ansatz, a decomposition idea, an anti-derivative technique and a framework of $L^2$-theory.Comment: 47 page

Convergence rate toward shock wave under periodic perturbation for generalized Korteweg-de Vries-Burgers equation

In this paper, a viscous shock wave under space-periodic perturbation of generalized Korteweg-de Vries-Burgers equation is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover the exponential time decay rate toward the viscous shock wave is also obtained for some certain perturbations.Comment: 22 page

Multidimensional Conservation Laws: Overview, Problems, and Perspective

Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of multidimensional hyperbolic conservation laws are revealed, and some samples of multidimensional systems/models and related important problems are presented and analyzed with emphasis on the prototypes that have been solved or may be expected to be solved rigorously at least for some cases. In particular, multidimensional steady supersonic problems and transonic problems, shock reflection-diffraction problems, and related effective nonlinear approaches are analyzed. A theory of divergence-measure vector fields and related analytical frameworks for the analysis of entropy solutions are discussed.Comment: 43 pages, 3 figure