Non-linear parabolic PDEs with rough data and coefficients: existence, uniqueness and regularity of weak solutions in critical spaces

TL;DR

This work develops a rigorous framework for nonlinear parabolic PDEs with rough diffusion coefficients and rough initial data in critical Besov spaces. It introduces weighted Z-spaces and a novel theory of hypercontractive singular integral operators to achieve hypercontractivity and control nonlinearities via extrapolation, enabling existence, uniqueness, and regularity of both mild and weak solutions. A key contribution is the mild–weak transference principle together with self-improving reverse Hölder inequalities, which yield maximal weak solutions and precise lifespans for rough reaction–diffusion equations. The framework is designed to extend to Burgers-type, quasi-linear, and related nonlinear models, with potential applications to complex media and rough-coefficient diffusion processes.

Abstract

This article investigates the well-posedness of weak solutions to non-linear parabolic PDEs driven by rough coefficients with rough initial data in critical homogeneous Besov spaces. Well-posedness is understood in the sense of existence and uniqueness of maximal weak solutions in suitable weighted $Z$-spaces in the absence of smallness conditions. We showcase our theory with an application to rough reaction--diffusion equations. Subsequent articles will treat further classes of equations, including equations of Burgers-type and quasi-linear problems, using the same approach. Our toolkit includes a novel theory of hypercontractive singular integral operators (SIOs) on weighted $Z$-spaces and a self-improving property for super-linear reverse Hölder inequalities.

Figures (1)

• Figure 1: Set $n = 3$. The blue areas depict all pairs $(p,\alpha)$ with $p \in (1,\mathrm{RD}_{+}(n,\rho))$, $\alpha \in (-1,0)$ and $p \geq p(\alpha,A)$. The red line segment is the intersection of the critical line (that is, the points satisfying $2/\rho = n/p - \alpha$) with the blue area. The red line leaves the blue area on its left edge if and only if $\rho \geq 1$. If $\rho < 1$, it may happen that the red line leaves the blue area on its right-hand slope.

Theorems & Definitions (101)

• Theorem : Well-posedness of \ref{['eq:rd']}
• Theorem : Hypercontractive SIOs on weighted $\mathrm{Z}$-spaces
• Remark 1.1: Limitations of parabolic Sobolev embeddings
• Theorem : Self-improvement of super-linear RH-inequalities
• Definition 2.1: $\mathrm{Z}$-spaces
• Remark 2.2
• Definition 2.3: Weighted $\mathrm{L}^p$-spaces
• Proposition 2.4: "Hardy--Sobolev"-type embedding
• Lemma 2.5: Change of angle
• Lemma 2.6