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Well-posedness of Generalized Fractional Singular Burgers equation driven by $|D|^{\frac{1}{2}}ξ$

Shuolin Zhang, Zhaonan Luo, Zhaoyang Yin

TL;DR

This work addresses the well-posedness of a generalized fractional Burgers equation driven by a singular noise with fractional dissipation. It introduces a Generalized Fractional Singular Burgers equation (GFSB) built on an enhanced data framework and para-controlled calculus to handle ill-defined nonlinearities, proving local well-posedness in the subcritical regime and showing convergence of approximation schemes to the generalized solution for $γ>\frac{3}{2}$. The analysis combines Bony paraproducts, a tailored contraction mapping principle, and an improved Gronwall lemma to control the evolution and stability of the remainder term. A detailed treatment of Gaussian trees and their convergence under the noise driving reveals the structure of the solution as a sum of stochastic objects plus a regular remainder, ensuring compatibility with the generalized equation. The work also discusses the critical case $γ=\frac{3}{2}$, outlining the need for more advanced renormalization techniques to extend results to this threshold. Overall, the paper provides a rigorous framework to interpret ill-defined nonlinearities in stochastic fractional Burgers equations and establishes a bridge between approximations and generalized solutions in the subcritical setting.

Abstract

In this paper, we study the generalized solution of Fractional Singular Burgers equation driving by $\vert D\vert^{\frac{1}{2}}ξ$. We establish a framework to describe the equations satisfied by generalized solutions, termed the Generalized Fractional Singular Burgers equation(GFSB), and prove its local well-posedness. Finally, we prove that the solution of GFSB can be the generalized solution of Fractional Singular Burgers equation for $γ>\frac{3}{2}$.

Well-posedness of Generalized Fractional Singular Burgers equation driven by $|D|^{\frac{1}{2}}ξ$

TL;DR

This work addresses the well-posedness of a generalized fractional Burgers equation driven by a singular noise with fractional dissipation. It introduces a Generalized Fractional Singular Burgers equation (GFSB) built on an enhanced data framework and para-controlled calculus to handle ill-defined nonlinearities, proving local well-posedness in the subcritical regime and showing convergence of approximation schemes to the generalized solution for . The analysis combines Bony paraproducts, a tailored contraction mapping principle, and an improved Gronwall lemma to control the evolution and stability of the remainder term. A detailed treatment of Gaussian trees and their convergence under the noise driving reveals the structure of the solution as a sum of stochastic objects plus a regular remainder, ensuring compatibility with the generalized equation. The work also discusses the critical case , outlining the need for more advanced renormalization techniques to extend results to this threshold. Overall, the paper provides a rigorous framework to interpret ill-defined nonlinearities in stochastic fractional Burgers equations and establishes a bridge between approximations and generalized solutions in the subcritical setting.

Abstract

In this paper, we study the generalized solution of Fractional Singular Burgers equation driving by . We establish a framework to describe the equations satisfied by generalized solutions, termed the Generalized Fractional Singular Burgers equation(GFSB), and prove its local well-posedness. Finally, we prove that the solution of GFSB can be the generalized solution of Fractional Singular Burgers equation for .
Paper Structure (15 sections, 32 theorems, 186 equations, 1 figure)

This paper contains 15 sections, 32 theorems, 186 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Generalized solution
  3. Generalized Fractional Singular Burgers equation
  4. Organized of this paper
  5. Basic tool
  6. Bony para-product
  7. Contraction mapping theorem
  8. Improving Gronwall lemma
  9. Well-posedness of GFSB
  10. Driving by $\vert D\vert^\frac{1}{2}\xi$
  11. Convergence of Gaussian trees
  12. Convergence of the approximation solution
  13. Discussion for $\gamma=\frac{3}{2}$
  14. Gaussian computation
  15. Other useful estimate

Key Result

Theorem 1.1

Let $\gamma\in(\frac{3}{2},2]$, given a approximation equation there exists $u\in C_TW^{-\frac{1}{4}+\delta}$ such that $u_\epsilon \to u$ in $C_TW^{-\frac{1}{4}+\delta}$ in probability sense for some small $\delta>0$. Moreover, $u$ has a structure

Figures (1)

  • Figure :

Theorems & Definitions (56)

  • Theorem 1.1
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.2
  • Remark 1.1
  • Definition 1.4
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • ...and 46 more