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Register a new userScalar Green Function Bounds for Instantaneous Shock Location and One-dimensional Stability of Viscous Shock Waves
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Yingwei Li
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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.
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In several space dimensions, scalar shock waves between two constant states u $pm$ are not necessarily planar. We describe them in detail. Then we prove their asymptotic stability, assuming that they are uniformly non-characteristic. Our result is conditional for a general flux, while unconditional for the multi-D Burgers equation.
We carry out the first rigorous numerical proof based on Evans function computations of stability of viscous shock profiles, for the system of isentropic gas dynamics with monatomic equation of state. We treat a selection of shock strengths ranging from the lower stability boundary of Mach number $approx 1.86 $, below which profiles are known by energy estimates to be stable, to the upper stability boundary of $approx 1669$, above which profiles are expected to be provable by rigorous asymptotic analysis to be stable. These results open the possibilities of: (i) automatic rigorous verification of stability or instability of individual shocks of general systems, and (ii) rigorous proof of stability of all shocks of particular systems.
Shock wave theory was first studied for gas dynamics, for which shocks appear as compression waves. A shock wave is characterized as a sharp transition, even discontinuity in the flow. In fact, shocks appear in many different physical situation and represent strong nonlinearity of the physical processes. Important progresses have been made on shock wave theory in recent years. We will survey the topics for which much more remain to be made. These include the effects of reactions, dissipations and relaxation, shock waves for interacting particles and Boltzmann equation, and multi-dimensional gas flows.
In this paper, the large time behavior of solutions of 1-D isentropic Navier-Stokes system is investigated. It is shown that a composite wave consisting of two viscous shock waves is stable for the Cauchy problem provided that the two waves are initially far away from each other. Moreover the strengths of two waves could be arbitrarily large.
The large time behavior of the solutions to a multi-dimensional viscous conservation law is considered in this paper. It is shown that the solution time-asymptotically tends to the planar rarefaction wave if the initial perturbations are multi-dimensional periodic. The time-decay rate is also obtained. Moreover, a Gagliardo-Nirenberg type inequality is established in the domain $ mathbb R times mathbb T^{n-1} (ngeq2) $, where $mathbb T^{n-1}$ is the $ n-1 $-dimensional torus.
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