Metastability for nonlinear parabolic equations with applications to viscous scalar conservation laws (original) (raw)
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Metastability for nonlinear parabolic equations with application to scalar viscous conservation laws
2013
The aim article is to contribute to the definition of a versatile language for metastability in the context of partial differential equations of evolutive type. A general framework suited for parabolic equations in one dimensional bounded domains is proposed, based on choosing a family of approximate steady states, and on the spectral properties of the linearized operators at such states. The slow motion for solutions belonging to a cylindrical neighborhood of the family of approximate steady states is analyzed by means of a system of an ODE for the parameter ξ that describes the family, coupled with a PDE describing the evolution of the perturbation v. We state and prove a general result concerning the reduced system for the couple (ξ,v), called quasi-linearized system, obtained by disregarding the nonlinear term in v, and we show how such approach suits to the prototypical example of scalar viscous conservation laws with Dirichlet boundary condition in a bounded one-dimensional in...
Metastability for parabolic equations with drift: part 1
Indiana University Mathematics Journal, 2015
We provide a self-contained analysis, based entirely on pde methods, of the exponentially long time behavior of solutions to linear uniformly parabolic equations which are small perturbations of a transport equation with vector field having a globally stable point. We show that the solutions converge to a constant, which is either the initial value at the stable point or the boundary value at the minimum of the associated quasi-potential. This work extends previous results of Freidlin and Wentzell and Freidlin and Koralov and applies also to semilinear elliptic pde.
arXiv (Cornell University), 2021
We study the inflow-outflow boundary value problem on an interval, the analog of the 1D shock tube problem for gas dynamics, for general systems of hyperbolic-parabolic conservation laws. In a first set of investigations, we study existence, uniqueness, and stability, showing in particular local existence, uniqueness, and stability of small amplitude solutions for general symmetrizable systems. In a second set of investigations, we investigate structure and behavior in the small-and large-viscosity limits. A phenomenon of particular interest is the generic appearance of characteristic boundary layers in the inviscid limit, arising from noncharacteristic data for the viscous problem, even of arbitrarily small amplitude. This induces an interesting new type of "transcharacteristic" hyperbolic boundary condition governing the formal inviscid limit.
Journal of Differential Equations, 2014
In this article we derive rigorously a nonlinear, steady, bifurcation through spectral bifurcation (i.e., eigenvalues of the linearized equation crossing the imaginary axis) for a class of hyperbolic-parabolic model in a strip. This is related to "cellular instabilities" occuring in detonation and MHD. Our results extend to multiple dimensions the results of [AS12] on 1D steady bifurcation of viscous shock profiles; en passant, changing to an appropriate moving coordinate frame, we recover and somewhat sharpen results of [TZ08a] on transverse Hopf bifurcation, showing that the bifurcating time-periodic solution is in fact a spatially periodic traveling wave. Our technique consists of a Lyapunov-Schmidt type of reduction, which prepares the equations for the application of other bifurcation techniques. For the reduction in transverse modes, a general Fredholm Alternative-type result is derived, allowing us to overcome the unboundedness of the domain and the lack of compact embeddings; this result apply to general closed operators. Contents 1. Introduction 1 2. A few technical lemmas 7 3. Some auxiliary spaces 12 4. Symmetry and Lyapunov-Schmidt method 20 5. A further step towards the reduced equations 22 6. The bifurcation analysis 26 Appendix A. The parametric contraction mapping theorem 27 Appendix B. Useful facts used in the proof 28 Appendix C. Differentiability with respect to parameters 30 Appendix D. Fredholm Alternative via Dunford integrals 32 References 34
Breakdown of smooth solutions in dissipative nonlinear hyperbolic equations
Quarterly of Applied Mathematics, 1982
Introduction. In this paper we study the nonexistence of global smooth solutions of one-dimensional motions for nonlinear viscoelastic fluids and solids by the method of Rozhdestvenskii [1], This method has been applied to prove the nonexistence of global smooth solutions for the shearing motions in an elastic circular tube in [2]. It is well known that the quasilinear hyperbolic equation V" = °(vx)x (0.1) exhibits the breakdown of smooth solutions in finite time for a certain class of initial data of arbitrary smoothness, no matter how small. This breakdown of smooth solutions is usually associated with the formation of a propagating singular surface often called a Shockwave. The absence of some dissipative or damping mechanism in the above equation causes this rather unrealistic result. Nishida [3] and Slemrod [4] have studied the equation v" = o<vx)x-ocv, (0.2) which includes the effect of first-order linear damping which is not present in (0.1). For (0.2) Nishida showed the existence of a global smooth solution for the small initial data. Slemrod showed the breakdown of smooth solutions for large initial data. His motivation for studying (0.2) was based on his model equation for shearing perturbations of steady shearing flows in a nonlinear, isotropic, incompressible, viscoelastic fluid, in the absence of an applied driving force. In experiments the analysis of the plane Poiseuille flow is more common. In Sec. 11 shall discuss the plane Poiseuille flow of the above fluid. MacCamy [6] considered the equation v" = a(0)ff(vx)x + a(t-z)a(vx)x dz+f (0.3) showed the existence of a global smooth solution for small initial data, and conjectured the breakdown of smooth solutions for large initial data. The effect of fading memory for elastic materials causing a dissipative mechanism is included in this model as the stress functional in the stress-strain relation. I shall show the breakdown of smooth solutions in this problem in Sec. 2.
Structure and regularity of solutions to nonlinear scalar conservation laws
2017
In this chapter we collect some preliminary and technical results that will be used in the main body of this thesis. More in details, Section 1.1 deals with several independent topics: first we recall the Kuratowski convergence of sets in a metric space, then we introduce a decomposition of piecewise monotone functions in “undulations”. Next we recall the notion of BV(R) space and we give estimates of the generalized variation of piecewise monotone functions in terms of their undulations. Finally we mention some elementary properties on smooth functions for future references. In Section 1.2 we review some result about scalar conservation laws: the general theory is only mentioned, with some emphasis on the more relevant point for the following chapters. After recalling the fundamental theorem by Kruzkov, we introduce the wave-front tracking algorithm and the notion of measure valued entropy solution. Then we consider the problem in bounded domains: the related notion of admissible b...
Scalar Conservation Laws on a Half-Line: A Parabolic Approach
Journal of Hyperbolic Differential Equations, 2010
The initial-boundary value problem for the (viscous) nonlinear scalar conservation law is considered, u ε t + f (u ε)x = εu ε xx , x ∈ R + = (0, ∞), 0 ≤ t ≤ T, ε > 0, u ε (x, 0) = u 0 (x), u ε (0, t) = g(t). The flux f (ξ) ∈ C 2 (R) is assumed to be convex (but not strictly convex, i.e. f (ξ) ≥ 0). It is shown that a unique limit u = lim ε→0 u ε exists. The classical duality method is used to prove uniqueness. To this end parabolic estimates for both the direct and dual solutions are obtained. In particular, no use is made of the Kružkov entropy considerations.
Quasilinear Parabolic Evolution Equations
Monographs in Mathematics, 2016
Our study of abstract quasi-linear parabolic problems in time-weighted Lp-spaces, begun in [17], is extended in this paper to include singular lower order terms, while keeping low initial regularity. The results are applied to reactiondiffusion problems, including Maxwell-Stefan diffusion, and to geometric evolution equations like the surface-diffusion flow or the Willmore flow. The method presented here will be applicable to other parabolic systems, including free boundary problems.