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

This paper establishes the existence of non-negative weak solutions for fast diffusion equations with divergence-type drift terms under specific integrability conditions, demonstrating that these conditions can be relaxed for divergence-free drifts to improve prior results and applying the findings to viscous Boussinesq systems.

Original authors: Sukjung Hwang, Kyungkeun Kang, Hwa Kil Kim

Published 2026-06-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Sukjung Hwang, Kyungkeun Kang, Hwa Kil Kim

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Crowd in a Windy Room

Imagine a large, enclosed room (the domain $\Omega$ ) filled with a crowd of people. These people represent a substance, like heat or a gas, spreading out over time.

In a normal situation, people naturally spread out to fill the empty space. This is diffusion. However, this paper studies a specific, tricky kind of diffusion called Fast Diffusion.

Normal Diffusion: Like a slow, gentle spreading of perfume.
Fast Diffusion: Imagine the crowd is so eager to spread out that they move incredibly fast when the room is crowded, but they slow down drastically when the room is empty. It's a chaotic, "super-fast" spreading that can get messy near the edges where the crowd thins out to nothing (the "vacuum").

Now, imagine there is a wind blowing through the room. This wind is the Drift Term ( $V$ ). It pushes the people around, either helping them spread or pushing them into corners.

The authors of this paper are mathematicians asking a very specific question: "If we have this super-fast spreading crowd and a wind blowing them around, can we mathematically prove that a solution exists? In other words, does the crowd's movement make sense and stay under control, or does it explode into chaos?"

The Challenge: The Wind is Too Strong or Too Weird

In math, proving a solution exists is like proving a bridge won't collapse.

If the wind is gentle, it's easy to prove the bridge holds.
If the wind is too strong, or if it changes direction in a very jagged, unpredictable way (mathematically, if the wind doesn't have enough "smoothness" or "integrability"), the bridge might collapse. The equations might break, and the solution might vanish or become infinite.

The authors found that for Fast Diffusion, the wind has to be much more "well-behaved" than for normal diffusion. They had to draw a strict map of exactly how strong and how smooth the wind could be before the math breaks.

The Three Types of Wind Conditions

The paper categorizes the wind into three scenarios, each with different rules:

1. The General Wind (The "Sub-Critical" Case)

Imagine the wind is blowing, but it's not perfectly smooth. It has some bumps.

The Rule: The authors found that if the wind isn't too wild, the crowd will still spread out in a controlled way. They proved that as long as the wind fits into a specific "box" of mathematical limits (called the class $S$ ), a solution exists.
The Catch: For Fast Diffusion, this "box" is smaller than it is for normal diffusion. The wind has to be weaker or smoother than we might expect. If the wind gets too strong (crossing into a "supercritical" zone), the standard math tools fail to prove the solution exists.

2. The "Gradient" Wind (The "Smoothness" Case)

Sometimes, we don't know exactly how the wind blows, but we know how fast the wind changes direction (its gradient).

The Rule: If the changes in the wind are smooth enough, the crowd still behaves. The authors showed that even if the wind itself is a bit rough, as long as its shifts are controlled, the solution holds. This is like saying, "Even if the wind gusts are jerky, as long as the jerks aren't too violent, the crowd won't panic."

3. The "No-Spin" Wind (The Divergence-Free Case)

This is the most special and powerful part of the paper.

The Analogy: Imagine a wind that swirls around but never creates a vacuum or a pile-up. It's like water flowing in a pipe: it moves, but it doesn't compress or expand the air. In math, this is called divergence-free ( $\nabla \cdot V = 0$ ).
The Breakthrough: When the wind doesn't create or destroy mass (it just moves it around), the math becomes much more forgiving.
The Result: The authors proved that in this specific "swirling" case, the wind can be much wilder and much stronger than in the other cases. They expanded the "box" of allowed winds significantly. They showed that even if the wind is in a "supercritical" zone (usually considered too dangerous for math), the solution still exists because the wind isn't "squeezing" the crowd, just spinning it.

The Tools They Used: The "Wasserstein" Map

To prove these things, the authors didn't just look at the crowd as a blob of gas. They treated the crowd as a map of probability.

They used a concept called Wasserstein Space. Think of this as a special map where the distance between two crowd configurations isn't just about how far apart they are, but how much "work" it takes to move one crowd shape into another.
They proved that the crowd moves along a "smooth path" on this map (an "absolutely continuous curve"). This ensures the crowd doesn't teleport or vanish; it flows continuously from one moment to the next.

The Real-World Application: The Boussinesq System

The paper doesn't just stay in theory. They applied their findings to a real-world physics problem: The Viscous Boussinesq System.

The Scenario: Imagine a pot of water being heated from the bottom. The hot water rises (buoyancy), and the cold water sinks. This creates a circulation pattern.
The Connection: The temperature in the pot acts like our "Fast Diffusion" crowd. The water moving around acts like the "Wind" (Drift).
The Twist: In this system, the water is incompressible (it doesn't get squished). This means the "wind" (water flow) is divergence-free.
The Payoff: Because the authors proved that Fast Diffusion works even with very wild, swirling winds (as long as they don't compress), they could prove that this heating system has a valid mathematical solution. They showed that the temperature and the water flow will behave predictably, even in complex 3D environments, provided the heat diffuses fast enough.

Summary of the Achievement

The Problem: Fast Diffusion is unstable and hard to control mathematically, especially when a wind (drift) is blowing.
The Discovery: The authors mapped out exactly how strong and rough that wind can be before the math breaks.
The Surprise: If the wind is "swirling" (divergence-free) and doesn't compress the crowd, the math is much more robust. The wind can be much stronger than previously thought possible.
The Application: This new understanding allows mathematicians to prove that complex fluid dynamics problems (like heated fluids rising) have valid solutions, even when the heat spreads in a "fast" and singular way.

In short, they built a stronger bridge for a very tricky type of crowd movement, showing that even in a chaotic, swirling wind, order can still be mathematically guaranteed.

Technical Summary: Existence of Weak Solutions for Fast Diffusion Equation with a Drift Term and Its Application

Problem Statement
This paper investigates the existence of non-negative weak solutions for the fast diffusion equation (FDE) with a divergence-type drift term, formulated as:
$\partial_t \rho = \nabla \cdot (\nabla \rho^m - V \rho), \quad 0 < m < 1,$
in a bounded domain $\Omega \subset \mathbb{R}^d$ ( $d \geq 2$ ) over a time interval $(0, T)$ . The system is subject to no-flux boundary conditions and non-negative initial data $\rho_0 \in L^q(\Omega)$ . The primary challenge addressed is the construction of global weak solutions under various integrability conditions on the drift vector field $V(x,t)$ , particularly when $V$ belongs to scaling-invariant or super-critical classes. The authors distinguish between three structural cases for the drift: general drifts satisfying specific integrability conditions, drifts where the gradient $\nabla V$ satisfies integrability conditions, and divergence-free drifts ( $\nabla \cdot V = 0$ ).

Methodology
The authors employ a constructive approach based on the splitting method and compactness arguments within the framework of Wasserstein spaces. The methodology proceeds in three main stages:

Existence of Regular Solutions: Using a splitting method in the Wasserstein space $P_2(\Omega)$ , the authors first establish the existence of regular solutions for smooth initial data and smooth drift fields. This involves solving the homogeneous FDE and the transport equation separately in alternating time steps.
A Priori Estimates: The core of the analysis relies on deriving uniform a priori estimates for the sequence of regular solutions. These include:
- Energy Estimates: $L^q$ -energy bounds and entropy bounds (for $q=1$ ) are derived. Crucially, the authors demonstrate that for divergence-free drifts, the energy estimates are independent of the drift term $V$ , allowing for weaker conditions on $V$ .
- Speed Estimates: Estimates on the "speed" of the solution curve in the Wasserstein space are obtained. This involves bounding the term $\int |w|^2 \rho \, dx dt$ , where $w = -\nabla \rho^m / \rho + V$ , ensuring the solution curves are absolutely continuous (AC) in the Wasserstein metric.
- Compactness: A refined compactness argument is developed. Unlike previous works that established compactness at the level of $\rho^q$ , this paper establishes compactness directly for $\rho$ by estimating temporal and spatial derivatives ( $\partial_t \rho$ and $\nabla \rho$ ) in suitable negative Sobolev spaces. This allows for the handling of drift terms in super-critical regimes relative to the scaling-invariant class.
Passage to the Limit: A sequence of regular solutions is shown to converge to a weak solution via the Aubin-Lions lemma and the properties of narrow convergence in Wasserstein spaces. The convergence of the drift term $\rho_n V_n$ is carefully handled using interpolation inequalities and the specific integrability classes defined for $V$ .

Key Contributions and Results

New Drift Classes: The paper introduces and analyzes specific classes of drift fields, denoted as $S^{(q_1, q_2)}_{m,q}$ $S_{m, q}^{(q_{1}, q_{2})}$ , $\tilde{S}^{(\tilde{q}_1, \tilde{q}_2)}_{m,q}$ $\tilde{S}_{m, q}^{(\tilde{q}_{1}, \tilde{q}_{2})}$ , and $D^{(q_1, q_2)}_{m,q}$ $D_{m, q}^{(q_{1}, q_{2})}$ (and its variants).
- For general drifts, the class $S^{(q_1, q_2)}_{m,q}$ imposes sub-critical integrability conditions relative to the scaling-invariant class to ensure energy estimates hold.
- For divergence-free drifts, the authors introduce the classes $D^{(q_1, q_2)}_{m,q}$ and $D^{(q_1, q_2)}_{m,q,s}$ . These classes extend into super-critical regimes relative to the scaling-invariant class, a significant improvement over previous results for the porous medium equation (PME) and FDE.
Existence Theorems:
- Theorem 2.6 & 2.8: Establish the existence of absolutely continuous (AC) $L^q$ -weak solutions for general drifts where $V \in S^{(q_1, q_2)}_{m,q}$ or $\nabla V \in \tilde{S}^{(\tilde{q}_1, \tilde{q}_2)}_{m,q}$ . These results cover the range $1 - 1/d < m < 1$ .
- Theorem 2.10 & 2.11: Establish the existence of $L^q$ -weak solutions (and AC solutions) for divergence-free drifts. Notably, these theorems allow for a wider range of $m$ , specifically $(1 - 2q/d)_+ \leq m < 1$ , and admit drifts in the super-critical classes $D^{(q_1, q_2)}_{m,q}$ and $D^{(q_1, q_2)}_{m,q,s}$ .
Improved Compactness: The paper claims that the direct compactness argument for $\rho$ (Proposition 4.6) is a key technical advancement. It allows the authors to bypass the limitations of previous methods that required drifts to be in the scaling-invariant class, thereby extending the existence theory to super-critical drift regimes for divergence-free fields.
Application to Boussinesq System: As an application, the authors apply their existence theory to a viscous Boussinesq system of the fast-diffusive type. They prove the existence of weak solutions for the coupled system where the temperature equation is an FDE with a divergence-free drift (the fluid velocity). The divergence-free nature of the velocity field allows the temperature equation to satisfy the energy estimates required by the main theorems, even though the drift is coupled and not prescribed.

Significance and Claims
The paper positions its work as an extension of the existence theory for porous medium equations with drift (developed in [18, 19]) to the fast diffusion range ( $0 < m < 1$ ). The authors highlight that the transition from $m > 1$ to $m < 1$ is non-trivial due to weaker coercive control in the fast diffusion case.

The primary significance claimed is the relaxation of integrability conditions on the drift term $V$ in the divergence-free case. By utilizing improved compactness arguments and speed estimates in Wasserstein spaces, the authors demonstrate that weak solutions exist even when the drift belongs to super-critical classes relative to the scaling-invariant threshold. This improves upon previous results for both FDE and PME. Furthermore, the application to the Boussinesq system illustrates the physical relevance of these theoretical improvements, showing that the theory can handle coupled systems where the drift is determined by the fluid dynamics itself.

The authors explicitly note that for $q > 1$ , the admissible drift classes for FDE are more restrictive than the scaling-invariant class unless the divergence-free condition is met, in which case the super-critical extension is possible. The work also clarifies the distinction between $L^1$ -based and $L^q$ -based ( $q>1$ ) initial data, showing that the latter allows for different integrability regimes for the drift.