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Quantitative estimates for SPDEs on the full space with transport noise and $L^p$-initial data

Dejun Luo, Bin Xie, Guohuan Zhao

Abstract

For the stochastic linear transport equation with $L^p$-initial data ($1<p<2$) on the full space $\mathbb{R}^d$, we provide quantitative estimates, in negative Sobolev norms, between its solutions and that of the deterministic heat equation. Similar results are proved for the stochastic 2D Euler equations with transport noise.

Quantitative estimates for SPDEs on the full space with transport noise and $L^p$-initial data

Abstract

For the stochastic linear transport equation with -initial data () on the full space , we provide quantitative estimates, in negative Sobolev norms, between its solutions and that of the deterministic heat equation. Similar results are proved for the stochastic 2D Euler equations with transport noise.

Paper Structure

This paper contains 12 sections, 14 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Structure of the noise
  3. Stochastic linear transport equations
  4. Stochastic 2D Euler equations
  5. Preparations
  6. Quantitative estimates for stochastic transport equations
  7. Maximal estimates for stochastic convolutions
  8. Proof for Theorem \ref{['thm:STE']}
  9. Existence of weak solutions to stochastic $2$D Euler equations
  10. Quantitative estimates
  11. Well-posedness of deterministic 2D Navier-Stokes equation
  12. Proof for Theorem \ref{['thm:SEE']}

Key Result

Lemma 1.2

Let $\{\sigma_k\}_{k\in\mathbb N}$ be any complete orthonormal system of ${\mathcal{H}}$ made of smooth, divergence free functions. Then it holds where the series converges absolutely and uniformly on compact sets. Moreover, the noise admits the series representation where $\{B^k \}_k$ is a family of independent standard Brownian motions defined by

Theorems & Definitions (26)

  • Lemma 1.2
  • Proposition 1.3
  • proof
  • Example 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Proposition 2.4
  • ...and 16 more