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Nonlinear enhanced dissipation in viscous Burgers type equations II

Tej-Eddine Ghoul, Nader Masmoudi, Eliot Pacherie

Abstract

In this follow up paper, we focus on the viscous Burgers equation. There, using the Hopf-Cole transformation, we compute the long time behavior of solutions for some classes of infinite mass initial datas. We show that an enhanced dissipation effect occurs generically, that is the decay rate in time is better than if we considered instead the heat equations for the same inital value. We also show the existence of a kind of global attractor per class.

Nonlinear enhanced dissipation in viscous Burgers type equations II

Abstract

In this follow up paper, we focus on the viscous Burgers equation. There, using the Hopf-Cole transformation, we compute the long time behavior of solutions for some classes of infinite mass initial datas. We show that an enhanced dissipation effect occurs generically, that is the decay rate in time is better than if we considered instead the heat equations for the same inital value. We also show the existence of a kind of global attractor per class.
Paper Structure (26 sections, 10 theorems, 255 equations, 3 figures)

This paper contains 26 sections, 10 theorems, 255 equations, 3 figures.

Table of Contents

  1. Introduction and presentation of the results
  2. Nonlinear enhanced dissipation
  3. Computation of the asymptotic profile
  4. Other exemples of discontinuous asymptotic profiles
  5. Plan of the proofs
  6. Open problems
  7. Estimates on the viscous Burgers equation
  8. Proof of Proposition \ref{['prop211']}
  9. Estimate on $F$ in the case $\frac{| x |}{t^{\frac{1}{1 + \alpha}}} \leqslant \varepsilon$
  10. Estimate on $F$ in the case $\frac{x}{t^{\frac{1}{1 + \alpha}}} \geqslant \varepsilon$
  11. Estimate on $F$ in the case $- \frac{1}{\varepsilon} \leqslant \frac{x}{t^{\frac{1}{1 + \alpha}}} \leqslant - \varepsilon$
  12. Estimate on $F$ in the case $\frac{x}{t^{\frac{1}{1 + \alpha}}} \leqslant - \frac{1}{\varepsilon}$
  13. Proof of Theorem \ref{['mainth']} and Proposition \ref{['Prop14']}
  14. Computation of the first order
  15. Definition of the functions $y^{\ast}_{\pm}$ and $y_m^{\ast}$
  16. ...and 11 more sections

Key Result

Theorem 1.1

Consider the problem for an initial data $f_0 \in C^0 (\mathbb{R}, \mathbb{R})$, and suppose that there exists $\kappa_1, \kappa_2 > 0$ and $\alpha \in] 0, 1 [$ such that Then, there exists $K_1, K_2 > 0$ depending on $\kappa_1, \kappa_2, \alpha,$ such that, for all $t \geqslant 0$,

Figures (3)

  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (10)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Proposition 2.1