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On the speed rate of convergence of solutions to conservation laws with nonlinear diffusions

Raffaele Folino, Marta Strani

Abstract

In this paper we analyze the long-time behavior of solutions to conservation laws with nonlinear diffusion terms of different types: saturating dissipation (monotone and non monotone) and singular nonlinear diffusions are considered. In particular, the cases of mean curvature-type diffusions both in the Euclidean space and in Lorentz-Minkowski space enter in our framework. After dealing with existence and stability of monotone steady states in a bounded interval of the real line with Dirichlet boundary conditions, we discuss the speed rate of convergence to the asymptotic limit as $t\to+\infty$. Finally, in the particular case of a Burgers flux function, we show that the solutions exhibit the phenomenon of metastability.

On the speed rate of convergence of solutions to conservation laws with nonlinear diffusions

Abstract

In this paper we analyze the long-time behavior of solutions to conservation laws with nonlinear diffusion terms of different types: saturating dissipation (monotone and non monotone) and singular nonlinear diffusions are considered. In particular, the cases of mean curvature-type diffusions both in the Euclidean space and in Lorentz-Minkowski space enter in our framework. After dealing with existence and stability of monotone steady states in a bounded interval of the real line with Dirichlet boundary conditions, we discuss the speed rate of convergence to the asymptotic limit as . Finally, in the particular case of a Burgers flux function, we show that the solutions exhibit the phenomenon of metastability.

Paper Structure

This paper contains 19 sections, 13 theorems, 222 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Existence of monotone stationary solutions
  3. Stationary solutions on the whole line
  4. The case of a monotone and bounded dissipation flux function
  5. The case of a non monotone dissipation flux function
  6. The case of a monotone and unbounded dissipation flux function
  7. Stability of monotone stationary solutions
  8. Monotone and bounded dissipation flux function
  9. Non monotone and bounded dissipation flux function
  10. Monotone and unbounded dissipation flux function
  11. Metastability for the small dissipation limit
  12. Construction of the one-parameter family - nonmonotone case
  13. Construction of the one-parameter family - unbound case
  14. Linearization
  15. Asymptotics for the first eigenvalue
  16. ...and 4 more sections

Key Result

Proposition \oldthetheorem

Let $\varepsilon>0$ and $f,Q\in C^2(\mathbb{R})$, with $Q$ satisfying eq:Qmono, and consider the boundary value problem for some $u_-<u_+$. There exists a unique (smooth) increasing solution to eq:staz-mono-BV if and only if where

Figures (6)

  • Figure 1: The dynamics of the solution to the IBVP \ref{['eq:QnonmonoNUM']}-\ref{['BIC']} with initial datum $u_0(x)=0.8\left(\frac{1}{2}x^2-x-\frac{1}{2}\right)$ and $u_*=0.8$. In the left picture $\varepsilon=0.05$, in the right one $\varepsilon=0.025$.
  • Figure 2: The dynamics of the solution to the IBVP \ref{['eq:QnonmonoNUM']}-\ref{['BIC']}. In the left picture we consider a discontinuous initial datum with $\varepsilon=0.06$ and $u_*=0.7$, in the right picture the non-monotone one $u_0(x)=0.6\left(-\frac{25}{6}x^3+\frac{3}{4}x^2+\frac{19}{6}x-\frac{3}{4}\right)$ with $\varepsilon=0.04$ and $u_*=0.6$.
  • Figure 3: The dynamics of the solution to the IBVP \ref{['eq:QunboundNUM']}-\ref{['BIC']} with $u_* = 0.75$ and initial datum $u_0(x)=\frac{1}{9}x^2-\frac{1}{2}x-\frac{1}{4}$. In the left picture $\varepsilon=0.04$, in the right one $\varepsilon=0.02$.
  • Figure 4: The dynamics of the solution to the IBVP \ref{['eq:QunboundNUM']}-\ref{['BIC']} with $\varepsilon=0.07$, $u_*=1$ and initial datum $u_0(x)=-\frac{3}{2}x^3+\frac{3}{4}x^2+\frac{1}{2}x-\frac{3}{4}$.
  • Figure 5: The dynamics of the solution to the IBVP \ref{['eq:QnonmonoNUM']}-\ref{['BIC']} with $\varepsilon=0.004$ (left), and \ref{['eq:QunboundNUM']}-\ref{['BIC']} with $\varepsilon=0.001$ (right); in both case we choose $u_*=\varepsilon/2$ and the initial datum $u_0(x)=\frac{\varepsilon}{2}\left(\frac{1}{2}x^2-x-\frac{1}{2}\right)$.
  • ...and 1 more figures

Theorems & Definitions (33)

  • Proposition \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • Remark \oldthetheorem
  • ...and 23 more