Existence of weak solutions for nonlinear drift-diffusion equations with measure data

TL;DR

This work proves the existence of nonnegative weak solutions for nonlinear drift-diffusion equations of porous medium and fast diffusion types with measure-valued forcing and divergence-form drift A, providing gradient energy estimates that relate $\nabla u^{\frac{m}{2}}$ to the data $\mu$ and $V$. A key novelty is the identification of scaling-based drift classes $\mathcal{S}_{m}^{(q_1,q_2)}$, $\mathfrak{S}_{m}^{(q_1,q_2)}$ (and their sigma variants in the divergence-free case) that ensure existence under subcritical and supercritical regimes, respectively, and the derivation of energy inequalities that accommodate measure data. In the divergence-free setting, the drift constraints are notably relaxed, enabling broader applicability and sharper gradient bounds for $\nabla u^{m/2}$ with $\alpha\in(0,2)$. As an application, the authors construct weak solutions for a Keller-Segel-fluid type system with a measure forcing, yielding global energy bounds and highlighting the framework’s relevance to coupled nonlinear diffusion and incompressible flow problems. Overall, the paper advances the theory of nonlinear diffusion with measure data by integrating drift effects through scaling-informed classes and robust energy methods.

Abstract

We consider nonlinear drift-diffusion equations (both porous medium equations and fast diffusion equations) with a measure-valued external force. We establish existence of nonnegative weak solutions satisfying gradient estimates, provided that the drift term belongs to a sub-scaling class relevant to $L^1$-space. If the drift is divergence-free, such a class is, however, relaxed so that drift suffices to be included in a certain supercritical scaling class, and the nonlinear diffusion can be less restrictive as well. By handling both the measure data and the drift, we obtain a new type of energy estimates. As an application, we construct weak solutions for a specific type of nonlinear diffusion equation with measure data coupled to the incompressible Navier-Stokes equations.

Figures (3)

• Figure 1: The condition on $V$ in Theorem \ref{['T:PME']} (i) when $1\leq m \leq 2$.
• Figure 2: The condition on V in Theorem \ref{['T:PME']} (ii) when $1-\frac{1}{d} < m < 1$.
• Figure 3: The condition of $V$ in Theorem \ref{['T:divfree']} in case $\nabla \cdot V = 0$ when $m> (1-\frac{2}{d})_{+}$.

Theorems & Definitions (33)

• Definition 1.1
• Definition 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Theorem 1.7
• Remark 1.8
• Remark 1.9
• Remark 1.10