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Singularity formation for Burgers equation with transverse viscosity

Charles Collot, Tej-Eddine Ghoul, Nader Masmoudi

TL;DR

This work analyzes singularity formation for the Burgers equation with transverse viscosity by coupling a 1D self-similar shock along the x-direction with a parabolic, y-dependent modulation. The authors develop a novel modulation/energy framework for a mixed hyperbolic/parabolic blow-up, deriving a 2D blow-up profile Θ(X,Z) that captures the leading-order concentration of the shock and its y-dynamics, and performing a detailed spectral analysis of the linearized operator around the profile. They establish a stable blow-up corresponding to the first Burgers self-similar profile Ψ_1 and construct unstable flat blow-ups arising from higher Ψ_i and from the NLH dynamics, together with a coupled linear heat equation controlling the vertical axis. The results illuminate anisotropic blow-up mechanisms, provide a robust toolkit (self-similar variables, modulation equations, weighted energy estimates) for mixed-type blow-up problems, and open pathways to analyzing symmetry-breaking and continuation after singularity in anisotropic fluids.

Abstract

We consider Burgers equation with transverse viscosity $$\partial_tu+u\partial_xu-\partial_{yy}u=0, \ \ (x,y)\in \mathbb R^2, \ \ u:[0,T)\times \mathbb R^2\rightarrow \mathbb R.$$ We construct and describe precisely a family of solutions which become singular in finite time by having their gradient becoming unbounded. To leading order, the solution is given by a backward self-similar solution of Burgers equation along the $x$ variable, whose scaling parameters evolve according to parabolic equations along the $y$ variable, one of them being the quadratic semi-linear heat equation. We develop a new framework adapted to this mixed hyperbolic/parabolic blow-up problem, revisit the construction of flat blow-up profiles for the semi-linear heat equation, and the self-similarity in the shocks of Burgers equation.

Singularity formation for Burgers equation with transverse viscosity

TL;DR

This work analyzes singularity formation for the Burgers equation with transverse viscosity by coupling a 1D self-similar shock along the x-direction with a parabolic, y-dependent modulation. The authors develop a novel modulation/energy framework for a mixed hyperbolic/parabolic blow-up, deriving a 2D blow-up profile Θ(X,Z) that captures the leading-order concentration of the shock and its y-dynamics, and performing a detailed spectral analysis of the linearized operator around the profile. They establish a stable blow-up corresponding to the first Burgers self-similar profile Ψ_1 and construct unstable flat blow-ups arising from higher Ψ_i and from the NLH dynamics, together with a coupled linear heat equation controlling the vertical axis. The results illuminate anisotropic blow-up mechanisms, provide a robust toolkit (self-similar variables, modulation equations, weighted energy estimates) for mixed-type blow-up problems, and open pathways to analyzing symmetry-breaking and continuation after singularity in anisotropic fluids.

Abstract

We consider Burgers equation with transverse viscosity We construct and describe precisely a family of solutions which become singular in finite time by having their gradient becoming unbounded. To leading order, the solution is given by a backward self-similar solution of Burgers equation along the variable, whose scaling parameters evolve according to parabolic equations along the variable, one of them being the quadratic semi-linear heat equation. We develop a new framework adapted to this mixed hyperbolic/parabolic blow-up problem, revisit the construction of flat blow-up profiles for the semi-linear heat equation, and the self-similarity in the shocks of Burgers equation.

Paper Structure

This paper contains 22 sections, 42 theorems, 543 equations.

Table of Contents

  1. Blow-up for the reduced parabolic system
  2. Statement of the result
  3. Comments on the result and open problems
  4. Ideas of the proof and Organisation of the paper
  5. Notations
  6. Self similarity in shocks for Burgers equation
  7. Invariances
  8. Self-similar and discretely self-similar solutions
  9. Stability and convergence at blow-up to self-similar solutions
  10. Proof of the main Theorem \ref{['th:main']}
  11. Self-similar variables
  12. 2D Blow-up profile and spectral analysis
  13. Linear estimates
  14. Bootstrap analysis
  15. Analysis of the vertical axis
  16. ...and 7 more sections

Key Result

Theorem 1

Let $J\in \mathbb N$. There exists an open set for a suitable topology of even solutions to $(NLH)$ blowing up in finite time $T>0$ with where the remainder $\tilde{\xi}$ satisfies for $0\leq j \leq J$ for some constant $C>0$: For any $k\in \mathbb N$, $k\geq 2$, $a>0$, there exists $T^*>0$, such that for any $0<T<T^*$ there exists an even solution to (eq:NLH) blowing up at time $T$ with where

Theorems & Definitions (85)

  • Theorem 1
  • proof : Proof of Theorem \ref{['th:NLHinstable']}
  • Theorem 2
  • Theorem 3
  • proof
  • Proposition 4: Classification of self-similar solutions
  • Proposition 5: Self-similar solutions EF
  • proof
  • Proposition 6: Non-smooth discretely self-similar blow-up
  • Remark 7
  • ...and 75 more