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On the existence of solutions to some singular parabolic free boundary problems

Alessandro Audrito, Tomás Sanz-Perela

TL;DR

This work treats the singular parabolic free boundary problem $\partial_t u - \Delta u = - \frac{\mathrm{d}}{\mathrm{d}u} u_+^{\gamma}$ with $\gamma \in (0,1]$ by a robust regularization strategy that yields nonnegative weak solutions as $\varepsilon\to 0$. The authors develop sharp energy, regularity, and nondegeneracy estimates for the approximating problems, establish a Weiss-type monotonicity formula, and prove convergence to a parabolic FB problem where the FB is encoded by a weak formulation that recovers the sharp boundary condition $|\nabla(u^{1/\beta})|=\sqrt{2}/\beta$ on $\partial\{u>0\}$. They also show that the class of weak solutions is closed under blow-ups and extract detailed asymptotic behaviors of blow-up limits, including parabolic $\beta$-homogeneity, plus a rich collection of self-similar and traveling-wave solutions. The results advance the parabolic FB theory for singular nonlinearities and provide explicit examples illustrating FB geometry, stability, and possible singularities, with potential applications to combustion and phase-transition models.

Abstract

We construct nonnegative weak solutions to the singular parabolic free boundary problem \[ \partial_t u - Δu = - \frac{\mathrm{d}}{\mathrm{d} u} u_+^γ, \] where $γ\in (0,1]$, $u_+ := \max\{u,0\}$, and the term in the right-hand side denotes the formal derivative of the non-smooth function $u \mapsto u_+^γ$. Weak solutions are obtained as limits of a suitable approximation procedure. We show uniform optimal regularity, optimal growth and nondegeneracy estimates, and a Weiss-type monotonicity formula for solutions to the approximating problem. Such uniform estimates are then passed to limit: we prove the existence of a class of weak solutions to the free boundary problem which is closed under blow-up and whose weak formulation encodes the sharp free boundary condition. Finally, we construct several examples of weak solutions with self-similar and traveling wave form.

On the existence of solutions to some singular parabolic free boundary problems

TL;DR

This work treats the singular parabolic free boundary problem with by a robust regularization strategy that yields nonnegative weak solutions as . The authors develop sharp energy, regularity, and nondegeneracy estimates for the approximating problems, establish a Weiss-type monotonicity formula, and prove convergence to a parabolic FB problem where the FB is encoded by a weak formulation that recovers the sharp boundary condition on . They also show that the class of weak solutions is closed under blow-ups and extract detailed asymptotic behaviors of blow-up limits, including parabolic -homogeneity, plus a rich collection of self-similar and traveling-wave solutions. The results advance the parabolic FB theory for singular nonlinearities and provide explicit examples illustrating FB geometry, stability, and possible singularities, with potential applications to combustion and phase-transition models.

Abstract

We construct nonnegative weak solutions to the singular parabolic free boundary problem where , , and the term in the right-hand side denotes the formal derivative of the non-smooth function . Weak solutions are obtained as limits of a suitable approximation procedure. We show uniform optimal regularity, optimal growth and nondegeneracy estimates, and a Weiss-type monotonicity formula for solutions to the approximating problem. Such uniform estimates are then passed to limit: we prove the existence of a class of weak solutions to the free boundary problem which is closed under blow-up and whose weak formulation encodes the sharp free boundary condition. Finally, we construct several examples of weak solutions with self-similar and traveling wave form.

Paper Structure

This paper contains 18 sections, 31 theorems, 347 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Leading ideas and the notion of weak solutions
  3. Setting of the problem and main results
  4. Notation and terminology
  5. Energy estimates for the approximating problem
  6. Uniform regularity estimates and non-degeneracy
  7. Convergence to the free boundary problem
  8. Convergence of the solutions
  9. Convergence of the nonlinearity
  10. Limit equations
  11. Weiss monotonicity formula and blow-ups
  12. Weiss-type monotonicity formula
  13. Blow-ups and the proof of \ref{['thm:MAIN2Blowups']}
  14. Examples of homogeneous solutions
  15. Self-similar solutions
  16. ...and 3 more sections

Key Result

Theorem 1.3

Let $n \geqslant 1$, $\gamma \in (0,1]$, $\alpha \in (0,1)$, and let $u_\circ \in C_c^{2+\alpha}(\mathbb{R}^n)$ be nonnegative and nontrivial. Let $\{u_\varepsilon\}_{\varepsilon > 0}$ be a family of solutions to eq:ProbEps. Then there exist $\nu \in (0,1)$, $\varepsilon_j \downarrow 0$, and a nonne and as $j \uparrow \infty$. Furthermore:

Figures (3)

  • Figure 1: The qualitative graphs of $F_\varepsilon$ (left) and $f_\varepsilon$ (right).
  • Figure 2: Some trajectories in the phase-plane $(U,V)$ for $c=0$ (left), $c > 0$ (middle), and $c < 0$ (right). The admissible profiles are painted in red. In orange, examples of positive profiles, and in blue, examples of profiles reaching zero linearly ---and thus, not satisfying the FB condition \ref{['eq:DerFBTW']}.
  • Figure 3: The positivity sets of the "colliding TWs" (left) and the "TWs sliding on a line" (right).

Theorems & Definitions (78)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 3.1: Phillips1987:art, Lemma 2
  • ...and 68 more