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

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 $γ>\frac{3}{2}$.

💡 Research Summary

The manuscript investigates a class of singular stochastic partial differential equations (SPDEs) that combine a fractional dissipation operator with a highly irregular driving noise. Specifically, the authors consider the fractional singular Burgers equation (FSB) driven by the spatial operator |D|^{1/2} applied to a space‑time white noise ξ. The deterministic part of the equation reads ∂_t u – Λ_γ u = ∂_x (u^2), where Λ_γ denotes the fractional Laplacian of order γ∈(3/2,2]. The stochastic forcing term |D|^{1/2} ξ is so singular that the nonlinear term ∂_x(u^2) cannot be interpreted directly when u lives in low‑regularity spaces such as C^{0−} or H^{0−}. To overcome this obstacle, the authors adopt the modern theory of paracontrolled distributions and the notion of “enhanced data” originally developed for the KPZ equation.

The paper proceeds in several logical stages. First, a regularized approximation (1.5) is introduced, where the noise is mollified (ξ_ε) and the solution is split into a linear stochastic component Y_ε solving ∂_t Y_ε – Λ_γ Y_ε = |D|^{1/2} ξ_ε and a remainder u^{(1)}_ε = u_ε – Y_ε. This decomposition yields a well‑defined equation for u^{(1)}_ε involving the product D((u^{(1)}_ε)^2 + 2 u^{(1)}_ε Y_ε + Y_ε^2). The term Y_ε^2 is still ill‑defined, prompting the authors to iterate the decomposition and introduce a hierarchy of random distributions {Y_τ} indexed by a combinatorial set T equipped with a commutative product.

A central conceptual contribution is the definition of an “enhanced data” X belonging to a Banach‑valued space 𝓧_{α,b}. The parameters α<0, b>0 satisfy the sub‑critical condition 2α+b>0, while b≤γ−1 ensures compatibility with the fractional dissipation. The enhanced data encodes all stochastic objects needed to give meaning to the nonlinearities, including the resonant products X_τ·X_{τ′} that appear in the generalized fractional singular Burgers (GFSB) equation (1.11). The authors then formulate the GFSB as a fixed‑point problem in a suitable paracontrolled space: the solution is written as u = Σ_{τ∈T′} c(τ) X_τ + u′≺≺Q + u♯, where Q is a convolution of X_{τ*} with the heat kernel and the symbols ≺≺ denote a modified paraproduct that captures the high‑frequency interaction.

The analytical backbone consists of three technical tools. First, Bony’s paraproduct decomposition together with a refined “double‑paraproduct” is used to split products into regular, singular, and resonant components, together with precise Schauder‑type estimates (Lemma 2.2, 2.3). Second, a contraction‑mapping theorem (Lemma 2.4) and its coupled version (Lemma 2.5) provide a robust framework for solving the nonlinear fixed‑point system, handling both the primary unknown and auxiliary variables such as the resonant terms. Third, an improved Grönwall inequality based on the Mittag‑Leffler function (Section 2.3) yields control over time‑dependent norms, crucial for establishing local existence.

The main results are encapsulated in four theorems. Theorem 1.2 establishes local well‑posedness for the sub‑critical GFSB in the space W^{s} with 0<s<α+b, together with a blow‑up criterion and Lipschitz dependence on initial data. Theorem 1.3 upgrades the solution to the full paracontrolled structure (u, u′, u♯) and proves existence, uniqueness, and continuity under the same regularity assumptions, while also providing boundedness of the linear operators I, J, K, M that appear in the expanded equation (1.14). Theorem 1.4 shows that for γ>3/2 the regularized solutions u_ε converge in probability to the generalized solution u in the space W^{−1/4+δ} for some small δ>0, thereby linking the abstract GFSB framework to the original stochastic Burgers equation. Finally, Theorem 1.5 discusses the borderline case γ=3/2, indicating that the same strategy yields convergence up to the maximal existence time but leaving global existence as an open problem.

The paper concludes with a discussion of the γ=3/2 case, a Gaussian computation appendix that details the covariance structure of the enhanced data, and additional estimates needed for the paracontrolled analysis. While the exposition is mathematically rigorous, some definitions (e.g., the precise form of Λ_γ and the operator Q) are introduced only implicitly, which may hinder reproducibility for readers not already familiar with the paracontrolled literature. Moreover, the work focuses exclusively on local well‑posedness; extending the results to global solutions, studying invariant measures, or developing numerical schemes would be natural next steps. Overall, the manuscript makes a significant contribution by extending paracontrolled techniques to fractional dissipation and half‑order noise, filling a gap in the theory of singular SPDEs and opening avenues for further research in both analysis and applications.