Existence and uniqueness of weak solutions to quasilinear PDEs with critical data

Abstract: We establish existence and uniqueness of global, bounded weak solutions to quasilinear PDEs with bounded, uniformly continuous initial data and investigate their properties. Moreover, we establish existence of bounded weak solutions when the initial data is merely bounded.

• The paper proves well-posedness by establishing existence, uniqueness, stability, and regularity of bounded weak solutions under critical initial data conditions.
• It introduces weighted Z-spaces that capture parabolic scaling and local gradient estimates, enabling control over degenerate nonlinearities and rough coefficients.
• The methodology extends existence to merely bounded initial data via mollification and compactness, although uniqueness may not hold in that setting.

Existence and Uniqueness of Weak Solutions to Quasilinear Parabolic PDEs with Critical Data

Introduction and Problem Formulation

The paper “Existence and uniqueness of weak solutions to quasilinear PDEs with critical data” (2605.00439) addresses the well-posedness of quasilinear parabolic partial differential equations of the form: $$\partial_t u - \operatorname{div} (a(t,x,u) \nabla u) = 0, \quad u(0) = u_0,$$ on $(0,\infty)\times \mathbb{R}^n$, where $a$ is a possibly degenerate, non-linear, non-autonomous coefficient and $u_0$ is the initial data. Allowing general, possibly degenerate nonlinearities and only bounded initial data places this work at the threshold of the scaling-critical regime for well-posedness and regularity. In particular, classical theories such as those of Ladyzhenskaya-Solonnikov-Ural'tseva require stricter regularity on both $a$ and $u_0$.

The authors’ main contribution is a proof of well-posedness (existence, uniqueness, stability, and regularity) for bounded weak solutions under bounded and uniformly continuous initial data, and an extension of existence (without uniqueness) to the case when the initial data $u_0$ is merely bounded measurable. This is achieved under local ellipticity, local Lipschitz, and local continuity assumptions on the coefficient function $a$ with respect to $u$, but allowing $a$ to be degenerate or unbounded outside the data's essential range.

Main Theoretical Innovations

Minimal Regularity and Criticality

The arguments hinge on the analysis in critical function spaces, leveraging the fact that boundedness is scale-invariant for parabolic equations. The framework circumvents traditional compactness approaches, which fail in $(0,\infty)\times \mathbb{R}^n$0 due to lack of integrability at infinity and the absence of a reverse Poincaré inequality. Instead, the key is an analytic approach in weighted $(0,\infty)\times \mathbb{R}^n$1-spaces—function spaces capturing the appropriate parabolic scaling and averaging behavior for the gradient $(0,\infty)\times \mathbb{R}^n$2. Notably, the authors establish that weak solutions enjoy a priori bounds in these $(0,\infty)\times \mathbb{R}^n$3-spaces, even at the minimal (critical) regularity allowed by scaling.

Formulation of Weak Solutions & A Priori Properties

A bounded weak solution is defined via the integral formulation with test functions, coupled with the natural distributional trace at $(0,\infty)\times \mathbb{R}^n$4. The paper proves:

• Uniform continuity of the solution for all positive time and, when $(0,\infty)\times \mathbb{R}^n$5, up to $(0,\infty)\times \mathbb{R}^n$6;
• Conservation of essential range: $(0,\infty)\times \mathbb{R}^n$7;
• Quantitative regularity: for all $(0,\infty)\times \mathbb{R}^n$8 and $(0,\infty)\times \mathbb{R}^n$9,

$a$0

Furthermore, the solution’s spatial gradient vanishes in small time in $a$1-space norm, corresponding to instantaneous parabolic smoothing.

Existence and Uniqueness via Fixed Point Argument

• Local existence is established via a contraction mapping argument on a ball in the aforementioned $a$2-spaces, using as base flow the autonomous linear problem with frozen coefficients at $a$3. The construction exploits the short-time smallness of the nonlinearity, which is justified by the local Lipschitz structure in $a$4—even though $a$5 may be unbounded in $a$6 outside the data range.
• Global existence is obtained by a standard continuation/maximal time argument, leveraging the uniform a priori bounds and propagation of the essential range.
• Uniqueness for bounded, uniformly continuous initial data follows from local uniqueness in the function class provided by the $a$7-space estimates, globalized via a branching time analysis. Importantly, the authors show that bounded weak solutions in this setting instantaneously enjoy enough regularity to admit the fixed point argument at arbitrary positive times.

Extension to Merely Bounded Data

When $a$8, the authors employ a mollification and compactness approach: approximate $a$9 by smooth, uniformly continuous data, construct the associated solutions, and extract a limit exploiting Aubin–Lions-type compactness. The constructed weak solution is likewise bounded and achieves the prescribed initial value in the distributional sense. However, uniqueness and short-time regularity may fail at this level, in line with limitations from low-regularity theory for parabolic problems.

Technical Contributions

Analytic Function Spaces and Non-Autonomous Regularity

A central technical device is the introduction and use of weighted $u_0$0-spaces. These spaces encode local space-time averages of $u_0$1 uniformly over all space-time cylinders matching parabolic scaling, allowing crucial control in non-autonomous settings even when pointwise (or global) $u_0$2 control is unavailable. The approach interfaces with advanced results on maximal parabolic regularity for both the heat equation and equations with non-autonomous, rough coefficients. The authors generalize earlier approaches for linear problems and show how these analytic techniques transfer to control the nonlinearity through the fixed-point map.

They also discuss the propagation of range and the maximum principle using Aronson’s Gaussian kernel bounds for the fundamental solution, in a context where vector-field compactness is unavailable. Regularity theory (including a local Nash–Moser–De Giorgi theory) is invoked for temporal-spatial Hölder regularity away from $u_0$3. The methods avoid any reliance on integrability at infinity, Poincaré inequalities, or finite energy solutions, thus also encompassing cases with infinite mass or unbounded spatial domain.

Discussion, Limitations, and Open Problems

The paper explicitly identifies several directions for further investigation:

• Extension to Systems: The approach potentially adapts to parabolic systems under analogous structure conditions on the nonlinearity, suggesting directions for analysis of vector-valued equations with critical data.
• Uniqueness for Mere $u_0$4 Data: While existence is gained for bounded data, suitable regularity mechanisms ensuring uniqueness are unclear absent some form of trace regularity or improved continuity near $u_0$5.
• Further Irregular Data: Potential extension to data in VMO$u_0$6 or other function spaces that only marginally exceed $u_0$7 in local regularity is raised.

Implications

From a theoretical perspective, this work closes several gaps in the analysis of quasilinear parabolic equations at critical regularity. The analytic well-posedness in infinite domains for degenerate nonlinearities sharply delineates the precise threshold data regularity required for uniqueness, and identifies the key analytic tools—weighted $u_0$8-spaces and maximal regularity—which control critical phenomena in parabolic PDEs with rough coefficients.

Practically, these results lay foundations for further investigation of nonlinear parabolic models modeling, e.g., porous medium or reaction–diffusion phenomena in unbounded or non-integrable settings, where classical energy or $u_0$9-based techniques are unavailable.

The analytic framework and function space technology in this work are expected to impact future research in the treatment of non-autonomous and non-linear PDEs not only in parabolic but potentially in nonlocal or nondivergence structures where scale-invariant properties are central. Similarly, the framework may have implications for data assimilation and inverse problems where initial data is only indirectly accessible or lacks regularity.

Conclusion

This work establishes existence, uniqueness, and regularity properties for bounded weak solutions of general quasilinear parabolic equations with critical, minimal regularity initial data. The analytical tools and function spaces developed allow for sharp propagation of a priori properties and enable a robust fixed-point approach, filling significant theoretical gaps at the interface of non-autonomous, nonlinear, and critical data PDE theory. The results provide a rigorous foundation for further study of critical phenomena and degenerate parabolic models in broad analytic and applied settings.