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Local pathwise solutions and regularization by noises for the stochastic hyperbolic Keller-Segel equation

Tengyu Li, Lei Zhang

TL;DR

The paper analyzes the stochastic hyperbolic Keller-Segel equation on $\mathbb{T}^d$ with multiplicative noise, establishing local pathwise well-posedness in $H^s(\mathbb{T}^d)$ for $s>\frac{d}{2}+1$, under growth and Lipschitz assumptions on the noise. It then proves two noise-induced regularization phenomena: (i) nonlinear multiplicative noise with suitable intensity thresholds yields global pathwise solutions for large initial data (a.a.); (ii) linear multiplicative noise provides global existence with high probability for small initial data. The approach combines Galerkin approximations with cutoffs, rigorous a priori estimates, tightness arguments, Skorokhod embedding, and energy-type analyses (including logarithmic functionals) to pass to global pathwise solutions. These results demonstrate that stochastic perturbations can regularize a hyperbolic chemotaxis system, extending global well-posedness beyond the deterministic regime and offering insight into the interplay between noise structure and nonlinear nonlocalities in SHKS. The findings have significance for the mathematical understanding of stochastic chemotaxis models and their potential applications in modeling noisy biological environments.

Abstract

In this paper, we investigate the Cauchy problem associated with the stochastic hyperbolic Keller-Segel (SHKS) equation featuring multiplicative noises on the torus $\mathbb{T}^d$. First, we establish the local existence and uniqueness of pathwise solutions to the SHKS equation within Sobolev spaces $H^s(\mathbb{T}^d)$ for $s>\frac{d}{2}+1$, under appropriate regularity conditions imposed on the nonlinear multiplicative noises. Subsequently, we explore two global results pertaining to noise-induced regularization: (1) The first result demonstrates that for polynomial-type nonlinear noises, when the noise intensity parameters meet specific threshold conditions, the SHKS equation possesses a unique pathwise solution for large initial data with probability one. This finding provides a partial answer to a question that has remained unresolved in the deterministic setting; (2) The second result reveals that, for small initial data, or equivalently when dealing with linear multiplicative noises with sufficiently large intensity (allowed to be negative), the SHKS equation admits a unique pathwise solution with high probability.

Local pathwise solutions and regularization by noises for the stochastic hyperbolic Keller-Segel equation

TL;DR

The paper analyzes the stochastic hyperbolic Keller-Segel equation on with multiplicative noise, establishing local pathwise well-posedness in for , under growth and Lipschitz assumptions on the noise. It then proves two noise-induced regularization phenomena: (i) nonlinear multiplicative noise with suitable intensity thresholds yields global pathwise solutions for large initial data (a.a.); (ii) linear multiplicative noise provides global existence with high probability for small initial data. The approach combines Galerkin approximations with cutoffs, rigorous a priori estimates, tightness arguments, Skorokhod embedding, and energy-type analyses (including logarithmic functionals) to pass to global pathwise solutions. These results demonstrate that stochastic perturbations can regularize a hyperbolic chemotaxis system, extending global well-posedness beyond the deterministic regime and offering insight into the interplay between noise structure and nonlinear nonlocalities in SHKS. The findings have significance for the mathematical understanding of stochastic chemotaxis models and their potential applications in modeling noisy biological environments.

Abstract

In this paper, we investigate the Cauchy problem associated with the stochastic hyperbolic Keller-Segel (SHKS) equation featuring multiplicative noises on the torus . First, we establish the local existence and uniqueness of pathwise solutions to the SHKS equation within Sobolev spaces for , under appropriate regularity conditions imposed on the nonlinear multiplicative noises. Subsequently, we explore two global results pertaining to noise-induced regularization: (1) The first result demonstrates that for polynomial-type nonlinear noises, when the noise intensity parameters meet specific threshold conditions, the SHKS equation possesses a unique pathwise solution for large initial data with probability one. This finding provides a partial answer to a question that has remained unresolved in the deterministic setting; (2) The second result reveals that, for small initial data, or equivalently when dealing with linear multiplicative noises with sufficiently large intensity (allowed to be negative), the SHKS equation admits a unique pathwise solution with high probability.
Paper Structure (16 sections, 17 theorems, 181 equations)

This paper contains 16 sections, 17 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Formulation of problem
  3. Related works
  4. Notation and assumptions.
  5. Main results
  6. Pathwise solution in sufficiently regular spaces.
  7. Approximation scheme.
  8. A priori estimates
  9. Tightness of measures
  10. Global existence to the cut-off system \ref{['cutoff']}
  11. Local existence for SHKS \ref{['SHKS-trans']} with smooth initial profile
  12. Local well-posedness in $H^s(\mathbb{T}^d)$
  13. Global-in-time solutions
  14. Global result for nonlinear multiplicative noise
  15. Global Existence for linear multiplicative noise
  16. ...and 1 more sections

Key Result

Theorem 1.4

Let $s>\frac{d}{2}+1$ with $d\ge 1$. Assume that $u_0$ is an $H^s({\mathbb{T}}^d)$-valued $\mathcal{F}_0$-measurable random variable such that $\mathbb{E}\Vert u_0\Vert_{H^s}^2<\infty$, and the conditions (A1)-(A2) hold. Then the SHKS equation SHKS-trans has a unique local maximal pathwise solution

Theorems & Definitions (37)

  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4: Local-in-time pathwise solution
  • Remark 1.5
  • Theorem 1.6: Global solution with large initial data
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9: Global solution with small initial data
  • Remark 1.10
  • Remark 1.11
  • ...and 27 more