Local boundedness for weak solutions to fractional porous medium equation

1. Problem Statement

The paper investigates the local boundedness of weak solutions to the fractional porous medium equation in the fast diffusion regime. The equation is given by:
$$\partial_t u^q - \mathcal{L}^s_K u = 0 \quad \text{in } \Omega_T$$
where:

• $\Omega_T = \Omega \times (0, T]$ is a space-time cylinder with $\Omega \subset \mathbb{R}^N$ ($N \ge 2$).
• $s \in (0, 1)$ is the fractional order.
• $q > 1$ is the exponent governing the nonlinearity (fast diffusion implies $q > 1$, contrasting with the slow diffusion $0 < q < 1$).
• $\mathcal{L}^s_K$ is a nonlocal operator defined via a kernel $K(x, y, t)$ satisfying ellipticity conditions comparable to the fractional Laplacian $(-\Delta)^s$.

The Core Challenge:
In the study of doubly nonlinear parabolic equations, the boundedness of solutions is a prerequisite for higher regularity (e.g., Harnack inequalities, Hölder continuity). For local equations, there exists a critical exponent $q_c$ separating regimes where solutions are automatically bounded from regimes where they may blow up. In the nonlocal setting, the interplay between the fractional order $s$, the dimension $N$, and the exponent $q$ is delicate. The paper aims to establish boundedness under optimal tail assumptions (conditions on the behavior of the solution at infinity) without requiring unnecessary a priori integrability in the subcritical and critical regimes.

2. Methodology

The author employs a combination of De Giorgi's iteration technique and Caccioppoli-type energy estimates, adapted specifically for the nonlocal and time-dependent nature of the problem.

A. Time-Dependent Truncation (The "Tail" Method)

A central innovation is the handling of the nonlocal "tail" contribution. In nonlocal problems, the value of the solution at a point depends on values far away.

• Standard Approach: Usually involves a fixed truncation level $k$.
• De Filippis' Approach: Introduces a time-dependent truncation level $k(t) = k + \ell(t)$.
  • The term $\ell(t)$ is explicitly constructed to depend on the tail of the solution outside the domain of interest:
    $$\ell(t) \approx \left( \int_{t_0-\theta}^t \int_{\mathbb{R}^N \setminus B_R} \frac{u(y, \tau)}{|y-x_0|^{N+2s}} dy d\tau \right)^{1/q}$$
  • Purpose: The time derivative of this truncation level generates a term in the energy estimate that exactly cancels the nonlocal tail contribution arising from the fractional operator. This allows the iteration to proceed without the tail term destabilizing the estimates.

B. Iteration Schemes

The proof relies on a recursive sequence of cylinders $Q_n$ and levels $k_n$ shrinking towards the center.

• Caccioppoli Estimate: A refined energy inequality (Proposition 2.6) is derived for the truncated function $(u - k(t))_+$.
• Parabolic Sobolev Embedding: Lemma 2.5 is used to upgrade integrability from $L^2$ (energy space) to $L^m$, where $m$ is a critical exponent derived from the embedding:
  $$m = \frac{2}{N}(N + (q+1)s)$$
• Geometric Convergence: The sequence of integrals $Y_n = \iint_{Q_n} (u - k_n)_+^r dx dt$ is shown to converge to zero using Lemma 2.3 (fast geometric convergence), provided the initial data satisfies specific bounds.

3. Key Contributions and Results

The paper establishes two main theorems distinguishing between subcritical/critical regimes and supercritical regimes.

The Critical Threshold

The paper identifies the critical exponent for boundedness in the fractional setting:
$$q_c = \frac{N + 2s}{N - 2s}$$
This threshold separates the behavior of the solutions.

Theorem 1.1: Qualitative Boundedness (Subcritical & Critical)

• Condition: $q \le q_c$ (i.e., $q \le \frac{N+2s}{N-2s}$).
• Result: Any local, nonnegative weak sub-solution is locally bounded.
• Significance: This result holds without any additional a priori integrability assumptions (e.g., $u \in L^A_{loc}$ for large $A$). This confirms that the critical exponent $q_c$ is the exact borderline for automatic boundedness, mirroring results known for local doubly nonlinear equations (specifically the work of Bögelein et al. for $p=2$).

Theorem 1.2: Quantitative $L^\infty$ Estimates

This theorem provides explicit bounds depending on the integrability of the solution. It distinguishes two cases based on an auxiliary exponent $r$:

1. Case 1: Subcritical ($r \le m$ and $\Lambda_r > 0$)

   • Condition: $q < q_c$.
   • Result: The solution satisfies a quantitative bound involving the $L^r$ norm of the solution and the Tail term:
     $$\text{ess sup } u \le C \left( \text{Tail term} + \left( \iint u^r \right)^{\dots} + \dots \right)$$
   • Implication: Boundedness is guaranteed solely by the natural integrability of the weak solution ($u \in L^{q+1}_{loc}$) and the tail condition.

2. Case 2: Supercritical/Critical ($r > m$ and $\Lambda_r \le 0$)

   • Condition: $q \ge q_c$.
   • Requirement: An additional higher integrability assumption is necessary: $u \in L^A_{loc}$ for some $A > \frac{N}{2s}(q-1)$.
   • Result: Under this extra assumption, a quantitative bound is derived involving the $L^A$ norm.
   • Significance: This aligns with the local theory where, beyond the critical exponent, boundedness is not automatic and requires the solution to be "better" than the minimal weak solution definition.

4. Significance and Impact

1. Optimality of Assumptions: The paper proves that the tail condition used is optimal. It removes the need for extra integrability assumptions in the subcritical and critical regimes ($q \le q_c$), which were previously required in some nonlocal analyses.
2. Resolution of the Critical Case: It settles the behavior at the critical exponent $q = q_c$, showing that solutions remain bounded without extra hypotheses, a result that was previously open or required stronger assumptions in the nonlocal context.
3. Methodological Advancement: The introduction of the time-dependent truncation to neutralize the nonlocal tail is a significant technical tool. It simplifies the De Giorgi iteration for nonlocal parabolic equations, making the analysis of the "tail" contribution more robust and applicable to other nonlocal doubly nonlinear problems.
4. Bridge to Local Theory: The results successfully extend the known regularity theory for local porous medium equations (specifically the work of Bögelein, Duzaar, et al.) to the fractional setting, identifying the precise role of the fractional order $s$ in shifting the critical exponent.

Summary

Filomena De Filippis establishes that weak solutions to the fractional porous medium equation in the fast diffusion regime are locally bounded if the exponent $q$ is below or equal to the critical threshold $q_c = \frac{N+2s}{N-2s}$. This is achieved under optimal tail assumptions and without requiring higher integrability. For supercritical exponents, boundedness is recovered only if the solution possesses additional integrability. The proof relies on a novel time-dependent truncation technique that effectively manages nonlocal interactions, providing a rigorous foundation for further regularity studies (such as Hölder continuity) in nonlocal parabolic equations.