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Existence of weak solutions for fast diffusion equation with a divergence type of drift term

Sukjung Hwang, Kyungkeun Kang, Hwa Kil Kim

TL;DR

This work establishes the existence of nonnegative $L^q$-weak solutions to the fast diffusion equation with a divergence-form drift in a bounded domain, under various structural assumptions on the drift term and initial data. The authors develop an energy–speed framework in Wasserstein spaces, introducing new classes of drift fields and a $\delta$-distance to capture narrow-convergence behavior, and prove existence for three drift structures: $V\in\mathsf{S}_{m,q}^{(q_1,q_2)}$, $\nabla V\in\tilde{\mathsf{S}}_{m,q}^{(\tilde{q}_1,\tilde{q}_2)}$, and $\nabla\cdot V=0$, with corresponding AC weak solutions and energy estimates. In the divergence-free case, the work extends existence to broader, sometimes supercritical regimes via compactness arguments and speed estimates, and also treats the porous medium equation in parallel, yielding improved results. As an application, the paper proves the existence of weak solutions for a viscous Boussinesq system of fast-diffusion type, illustrating the broad applicability of the developed methods to coupled drift–diffusion-fluid models and underscoring the potential for further PDE and gradient-flow analyses in nonuniform drift fields.

Abstract

We construct non-negative weak solutions of fast diffusion equations with a divergence type of drift term satisfying the $L^q$-energy inequality and speed estimate in Wasserstein spaces under some integrability conditions on the drift term. Furthermore, in the case that the drift term has a divergence-free structure, it turns out that its integrability conditions can be relaxed, which is also applicable to porous medium equations, thereby improving previous results. As an application, the existence of weak solutions is also discussed for a viscous Boussinesq system of the fast diffusion type.

Existence of weak solutions for fast diffusion equation with a divergence type of drift term

TL;DR

This work establishes the existence of nonnegative -weak solutions to the fast diffusion equation with a divergence-form drift in a bounded domain, under various structural assumptions on the drift term and initial data. The authors develop an energy–speed framework in Wasserstein spaces, introducing new classes of drift fields and a -distance to capture narrow-convergence behavior, and prove existence for three drift structures: , , and , with corresponding AC weak solutions and energy estimates. In the divergence-free case, the work extends existence to broader, sometimes supercritical regimes via compactness arguments and speed estimates, and also treats the porous medium equation in parallel, yielding improved results. As an application, the paper proves the existence of weak solutions for a viscous Boussinesq system of fast-diffusion type, illustrating the broad applicability of the developed methods to coupled drift–diffusion-fluid models and underscoring the potential for further PDE and gradient-flow analyses in nonuniform drift fields.

Abstract

We construct non-negative weak solutions of fast diffusion equations with a divergence type of drift term satisfying the -energy inequality and speed estimate in Wasserstein spaces under some integrability conditions on the drift term. Furthermore, in the case that the drift term has a divergence-free structure, it turns out that its integrability conditions can be relaxed, which is also applicable to porous medium equations, thereby improving previous results. As an application, the existence of weak solutions is also discussed for a viscous Boussinesq system of the fast diffusion type.
Paper Structure (25 sections, 27 theorems, 229 equations, 5 figures, 1 table)

This paper contains 25 sections, 27 theorems, 229 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Existence results
  4. Existence for case: $V \in \mathsf{S}_{m,q}^{(q_1,q_2)}$
  5. Existence for case: $V \in \tilde{\mathsf{S}}_{m,q}^{(q_1,q_2)}$
  6. Existence for case: $\nabla \cdot V =0$
  7. Application
  8. Preliminaries
  9. Technical lemmas
  10. Wasserstein space
  11. Flows on $\mathcal{P}_p(\Omega)$ generated by vector fields
  12. A priori estimates
  13. Energy estimates
  14. Estimates of speed
  15. Estimates of temporal and spatial derivatives
  16. ...and 10 more sections

Key Result

Theorem 2.6

Let $d\geq 2$ and $1-\frac{1}{d} < m <1$. Let $q\geq 1$ and suppose that

Figures (5)

  • Figure 1: The class $\mathcal{D}_{m,q}^{(q_1,q_2)}$.
  • Figure 2: AC $L^q$-weak solutions of FDE in case $V \in \mathsf{S}_{m,q}^{(q_1,q_2)}$
  • Figure 3: AC $L^q$-weak solutions of FDE in case $V\in \tilde{\mathsf{S}}_{m,q}^{(\tilde{q}_1, \tilde{q}_2)}$
  • Figure 4: (AC) $L^q$-weak solutions of FDE in case $\nabla \cdot V = 0$
  • Figure 5: $L^q$-weak and AC $L^q$-weak solutions of PME in case $\nabla \cdot V = 0$ and $q>m$

Theorems & Definitions (52)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • Remark 2.9
  • Theorem 2.10
  • ...and 42 more