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Existence and analyticity of solutions of nonlinear parabolic model equations with singular data

David Ambrose, Milton Lopes Filho, Helena Nussenzveig Lopes

TL;DR

The paper develops two complementary frameworks to establish existence and analyticity of nonlinear parabolic equations from rough data. The first uses an exponentially weighted Wiener algebra (Duchon-Robert style) to treat general nonlinearities and proves instant analyticity for small Wiener data in a 1D Kuramoto-Sivashinsky family, with global results when linearly growing modes are absent. The second adopts a two-norm fixed-point strategy (as in Bae, Cannone, and others) suitable for quadratic nonlinearities and is applied to dissipative and advection-diffusion Constantin-Lax-Majda models, yielding global solvability for data in negative-index Wiener and pseudomeasure spaces, and analyticity in many cases. Collectively, the results extend solvability to very rough data in negative-index spaces, clarify the role of dissipation in regularization, and provide analytic regularity at positive times with explicit radius growth, informing both numerical methods and the understanding of critical spaces. This work thus links abstract fixed-point theory, harmonic-analysis-based function spaces, and physically relevant nonlinear parabolic models to advance the theory of analytic smoothing from irregular initial data.

Abstract

We explore two approaches to proving existence and analyticity of solutions to nonlinear parabolic differential equations. One of these methods works well for more general nonlinearities, while the second method gives stronger results when the nonlinearity is simpler. The first approach uses the exponentially weighted Wiener algebra, and is related to prior work of Duchon and Robert for vortex sheets. The second approach uses two norms, one with a supremum in time and one with an integral in time, with the integral norm representing the parabolic gain of regularity. As an example of the first approach we prove analyticity of small solutions of a class of generalized one-dimensional Kuramoto-Sivashinsky equations, which model the motion of flame fronts and other phenomena. To illustrate the second approach, we prove existence and analyticity of solutions of the dissipative Constantin-Lax-Majda equation (which models vortex stretching), with and without added advection, with two classes of rough data. The classes of data treated include both data in the Wiener algebra with negative-power weights, as well as data in pseudomeasure spaces with negative-power weights.

Existence and analyticity of solutions of nonlinear parabolic model equations with singular data

TL;DR

The paper develops two complementary frameworks to establish existence and analyticity of nonlinear parabolic equations from rough data. The first uses an exponentially weighted Wiener algebra (Duchon-Robert style) to treat general nonlinearities and proves instant analyticity for small Wiener data in a 1D Kuramoto-Sivashinsky family, with global results when linearly growing modes are absent. The second adopts a two-norm fixed-point strategy (as in Bae, Cannone, and others) suitable for quadratic nonlinearities and is applied to dissipative and advection-diffusion Constantin-Lax-Majda models, yielding global solvability for data in negative-index Wiener and pseudomeasure spaces, and analyticity in many cases. Collectively, the results extend solvability to very rough data in negative-index spaces, clarify the role of dissipation in regularization, and provide analytic regularity at positive times with explicit radius growth, informing both numerical methods and the understanding of critical spaces. This work thus links abstract fixed-point theory, harmonic-analysis-based function spaces, and physically relevant nonlinear parabolic models to advance the theory of analytic smoothing from irregular initial data.

Abstract

We explore two approaches to proving existence and analyticity of solutions to nonlinear parabolic differential equations. One of these methods works well for more general nonlinearities, while the second method gives stronger results when the nonlinearity is simpler. The first approach uses the exponentially weighted Wiener algebra, and is related to prior work of Duchon and Robert for vortex sheets. The second approach uses two norms, one with a supremum in time and one with an integral in time, with the integral norm representing the parabolic gain of regularity. As an example of the first approach we prove analyticity of small solutions of a class of generalized one-dimensional Kuramoto-Sivashinsky equations, which model the motion of flame fronts and other phenomena. To illustrate the second approach, we prove existence and analyticity of solutions of the dissipative Constantin-Lax-Majda equation (which models vortex stretching), with and without added advection, with two classes of rough data. The classes of data treated include both data in the Wiener algebra with negative-power weights, as well as data in pseudomeasure spaces with negative-power weights.

Paper Structure

This paper contains 14 sections, 10 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Function spaces
  3. Abstract results
  4. First fixed point result
  5. Second fixed point result
  6. The 1D generalized Kuramoto-Sivashinsky equation
  7. Global theorem
  8. Local theorem
  9. The Constantin-Lax-Majda equation with dissipation
  10. The Constantin-Lax-Majda equation with advection and dissipation
  11. Existence of solutions with $Y^{-\sigma/2}$ data
  12. Existence of solutions with $PM^{r}$ data for $r\leq0$
  13. Analyticity of the solutions
  14. Discussion

Key Result

Lemma 1

Assume Assumption firstAssumption. If there exists $r_{1}>\|I^{+}\|_{op}\|N(0)\|_{\mathcal{Y}}$ and there exists $r_{0}>0$ such that $f(r_{0}+r_{1},0)(r_{0}+r_{1})\leq \frac{r_{1}}{\|I^{+}\|_{op}}-\|N(0)\|_{\mathcal{Y}},$ and if $\|I^{+}\|_{op}f(r_{0}+r_{1},r_{0}+r_{1})<1,$ then for any $u_{0}\in X$

Theorems & Definitions (20)

  • Lemma 1
  • proof
  • Corollary 2
  • proof
  • Lemma 3
  • Theorem 4
  • Theorem 5
  • Remark 6
  • Theorem 7
  • Theorem 8
  • ...and 10 more