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Nonlinear enhanced dissipation in viscous Burgers type equations

Tej-Eddine Ghoul, Nader Masmoudi, Eliot Pacherie

Abstract

We construct a class of infinite mass functions for which solutions of the viscous Burgers equation decay at a better rate than solution of the heat equation for initial data in this class. In other words, we show an enhanced dissipation coming from a nonlinear transport term. We compute the asymptotic profile in this class for both equations. For the viscous Burgers equation, the main novelty is the construction and description of a time dependent profile with a boundary layer, which enhanced the dissipation. This profile will be stable up to a computable nonlinear correction depending on the perturbation. We also extend our results to other convection-diffusion equations.

Nonlinear enhanced dissipation in viscous Burgers type equations

Abstract

We construct a class of infinite mass functions for which solutions of the viscous Burgers equation decay at a better rate than solution of the heat equation for initial data in this class. In other words, we show an enhanced dissipation coming from a nonlinear transport term. We compute the asymptotic profile in this class for both equations. For the viscous Burgers equation, the main novelty is the construction and description of a time dependent profile with a boundary layer, which enhanced the dissipation. This profile will be stable up to a computable nonlinear correction depending on the perturbation. We also extend our results to other convection-diffusion equations.
Paper Structure (41 sections, 20 theorems, 360 equations, 2 figures)

This paper contains 41 sections, 20 theorems, 360 equations, 2 figures.

Table of Contents

  1. Introduction and presentation of the results
  2. Profile for the viscous Burgers equation
  3. The rescaled problem and the underlying ODE
  4. The case $\varepsilon = 0$
  5. Viscosity solutions
  6. Construction of the profile for small $\varepsilon > 0$
  7. Stability of the profile
  8. Generalisation to equation (\ref{['gburgers']})
  9. Some related open problems
  10. Construction of the profile $\mathbbmss{h}_{\varepsilon}$
  11. Change of variable and viscosity conditions
  12. Computation of the underlying ODE problem
  13. Rankine-Hugoniot condition
  14. Some properties of solutions of $\frac{\alpha}{1 + \alpha} h + \left( \frac{\operatorname{Id}}{1 + \alpha} - h \right) h' = 0$
  15. The case $(z_0, b) \in \boldsymbol{A}$
  16. ...and 26 more sections

Key Result

Proposition 1.1

For $\kappa > 0, \alpha \in] 0, 1 [$, consider $f$ the solution of the heat equation $\partial_t f - \partial_x^2 f = 0$ for an initial condition $f_0 \in C^0 (\mathbb{R})$ that satisfies Then, uniformly in $z \in \mathbb{R}$, we have the convergence when $t \rightarrow + \infty$.

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (20)

  • Proposition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 10 more