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Existence, uniqueness, and numerical approximations for stochastic Burgers equations

Sara Mazzonetto, Diyora Salimova

TL;DR

The paper establishes an all-in-one framework for stochastic PDEs with non-globally monotone nonlinearities, proving existence, uniqueness, and spatial regularity of mild solutions while providing almost sure convergence of a fully explicit space-time discretization. The core technique combines a priori pathwise estimates with a Banach-space evolution approach and a nonlinearity-truncated accelerated exponential Euler scheme, yielding strong convergence for a broad class of SPDEs including stochastic Burgers equations with space-time white noise. The results extend prior work by simultaneously delivering existence, uniqueness, regularity, and convergence properties, and they apply directly to Burgers and related equations, ensuring robust numerical approximations with provable guarantees.

Abstract

In this paper we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. The proof of this result exploits the properties of an existent fully explicit space-time discrete approximation scheme and, in particular, the fact that it satisfies suitable a priori estimates. As a byproduct we obtain almost sure and strong convergence of the approximation scheme to the mild solutions of the considered SPDEs. We conclude by applying the main result of the paper to the stochastic Burgers equations with space-time white noise.

Existence, uniqueness, and numerical approximations for stochastic Burgers equations

TL;DR

The paper establishes an all-in-one framework for stochastic PDEs with non-globally monotone nonlinearities, proving existence, uniqueness, and spatial regularity of mild solutions while providing almost sure convergence of a fully explicit space-time discretization. The core technique combines a priori pathwise estimates with a Banach-space evolution approach and a nonlinearity-truncated accelerated exponential Euler scheme, yielding strong convergence for a broad class of SPDEs including stochastic Burgers equations with space-time white noise. The results extend prior work by simultaneously delivering existence, uniqueness, regularity, and convergence properties, and they apply directly to Burgers and related equations, ensuring robust numerical approximations with provable guarantees.

Abstract

In this paper we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. The proof of this result exploits the properties of an existent fully explicit space-time discrete approximation scheme and, in particular, the fact that it satisfies suitable a priori estimates. As a byproduct we obtain almost sure and strong convergence of the approximation scheme to the mild solutions of the considered SPDEs. We conclude by applying the main result of the paper to the stochastic Burgers equations with space-time white noise.

Paper Structure

This paper contains 10 sections, 7 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Notation
  4. Pathwise global solutions
  5. A priori bounds
  6. Pathwise existence, uniqueness, regularity, and approximation
  7. The main result: Existence, uniqueness, and strong convergence
  8. Example: Stochastic Burgers equations
  9. Properties of the nonlinearity
  10. Existence and uniqueness of the solution and strong convergence of the approximation scheme

Key Result

Lemma 2.2

Assume Setting sett:apriori, assume in addition that $\sup_{b\in \mathbb{H}} |\lambda_b|$$< \infty$, let $\beta \in (0, \infty)$, $h \in (0, \min\{1,T\}]$, and let $Y, O, \mathbb{O} \colon [0, T] \to H$, $F \in C(H, H)$, $\phi, \Phi \colon H \to [0,\infty)$ satisfy for all $v, w \in H$, $t \in [0,T and Then it holds that $\eta \mathbb{O} \in C([0,T],H)$ and for all $t \in [0, T]$ that

Theorems & Definitions (14)

  • Lemma 2.2: A priori bounds
  • proof : Proof of Lemma \ref{['lem:apriori1']}
  • Lemma 2.3: Pathwise convergence and non-explosion
  • proof : Proof of Lemma \ref{['lem:convergence1']}
  • Proposition 2.4: Global solutions
  • proof : Proof of Proposition \ref{['prop:euT:Xomega']}
  • Theorem 3.2: Existence, uniqueness, and almost sure convergence
  • proof : Proof of Theorem \ref{['thrm:EU']}
  • Corollary 3.3: Strong convergence
  • proof : Proof of Corollary \ref{['cor:strong']}
  • ...and 4 more