Existence and regularity of solutions to parabolic-elliptic nonlinear systems

1. Problem Statement

The paper investigates the existence, regularity, and summability of solutions to a parabolic-elliptic system of partial differential equations (PDEs) with discontinuous coefficients and low-regularity data. The system is defined on a bounded domain $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) over a time interval $(0, T)$, denoted as $\Omega_T = \Omega \times (0, T)$.

The system is given by:
$$\begin{cases} u_t - \text{div}(A(x, t)\nabla u) = -\text{div}(u M(x)\nabla \psi) + f(x, t) & \text{in } \Omega_T, \\ -\text{div}(M(x)\nabla \psi) = |u|^\theta & \text{in } \Omega_T, \\ u = \psi = 0 & \text{on } \partial\Omega \times (0, T), \\ u(x, 0) = 0 & \text{in } \Omega. \end{cases}$$

Key Characteristics:

• Coefficients: $A(x,t)$ and $M(x)$ are measurable, bounded, and uniformly elliptic matrices.
• Source Term: $f \in L^1(\Omega_T)$ with $f \geq 0$.
• Nonlinearity: The coupling term $u M(x) \nabla \psi$ represents a drift (convection) term. The elliptic equation depends on $u$ via a sublinear power $|u|^\theta$ where $\theta \in (0, 2/N)$.
• Context: This system is a discontinuous-coefficient variant of the Keller-Segel chemotaxis model. Unlike the fully parabolic Keller-Segel system, the second equation here is elliptic. The sublinear dependence ($\theta < 1$) distinguishes it from the linear dependence in classical models.

2. Methodology

The author employs a regularization and approximation strategy combined with a priori estimates and fixed-point arguments.

1. Approximating System:
   The problem is approached by truncating the nonlinearity and the source term using the truncation operator $T_n(s) = \max(-n, \min(s, n))$. The approximating system (3.1) replaces $u$ with $T_n(u)$ and $f$ with $T_n(f)$ in the drift and source terms, and $|u|^\theta$ with $|T_n(u)|^\theta$.

   • Existence of Approximate Solutions: The existence of solutions $(u_n, \psi_n)$ to the truncated system is proven using Schauder's Fixed Point Theorem. A mapping $S$ is constructed where $w \mapsto v$ involves solving the elliptic equation for $\psi$ (given $w$) and then the parabolic equation for $v$.

2. A Priori Estimates:
   The core of the proof relies on deriving uniform bounds for the sequence $(u_n, \psi_n)$ in various Lebesgue and Sobolev spaces, depending on the summability of $f$ (specifically $f \in L^m(\Omega_T)$).

   • $L^1$ Estimates: Using $T_h(u_n)$ as a test function to establish $L^\infty(0, T; L^1(\Omega))$ bounds.
   • $L^\infty$ Estimates: Using Stampacchia's iteration technique (Lemma 3.7) to prove boundedness of $u_n$ when $f$ has high summability.
   • Energy Estimates: Using powers of $u_n$ (e.g., $u_n^{2\gamma-1}$) as test functions to derive bounds in $L^2(0, T; H^1_0(\Omega))$ and $L^p(\Omega_T)$.
   • Gagliardo-Nirenberg Inequality: The parabolic version is used to interpolate between time and space norms to establish higher integrability.

3. Passage to the Limit:

   • Compactness: The Aubin-Lions-Simon compactness lemma (referenced as Corollary 4 of [14]) is used to extract strongly convergent subsequences for $u_n$ and $\psi_n$.
   • Convergence of Drift Term: A critical difficulty is passing to the limit in the term $u_n M(x) \nabla \psi_n$. The author uses the strong convergence of $u_n$ (in $L^1$ or $L^2$) and the strong convergence of $\nabla \psi_n$ (derived from Stampacchia's results on the elliptic equation) to handle the product.
   • Entropy Solutions: For highly singular data ($f \in L^1$ but $m$ is too low for distributional solutions), the concept of entropy solutions is introduced. This involves testing with truncated functions $T_k(u - \phi)$ and using the inequality formulation (2.4) to handle the lack of integrability in the drift term.

3. Key Contributions and Results

The paper establishes existence and regularity results based on the integrability exponent $m$ of the source term $f \in L^m(\Omega_T)$. The results are categorized into three regimes:

A. High Regularity Data ($m > \frac{N+2}{2}$)

• Result (Theorem 2.1): If $f \in L^m$ with $m > \frac{N+2}{2}$, there exists a bounded solution.
• Regularity: $u \in L^\infty(\Omega_T) \cap C^0([0, T]; L^2(\Omega)) \cap L^2(0, T; H^1_0(\Omega))$.
• Significance: The interaction between the parabolic and elliptic equations improves the regularity of $u$ significantly, yielding a bounded solution even with discontinuous coefficients.

B. Intermediate Regularity Data ($\frac{2N+4}{N+4} \leq m < \frac{N+2}{2}$)

• Result (Theorem 2.2 & 2.4): If $f$ has intermediate summability, the solution $u$ is not necessarily bounded but possesses specific summability.
• Regularity:
  • $u \in L^{m^{\star\star}}(\Omega_T)$ where $m^{\star\star} = \frac{(N+2)m}{N+2-2m}$.
  • $u \in L^2(0, T; H^1_0(\Omega))$ (if $m$ is high enough in this range) or $L^{m^\star}(0, T; W^{1, m^\star}_0(\Omega))$ (if $m$ is lower).
• Significance: The solution obeys the $L^p$-regularity theorems typical of parabolic equations, adapted for the specific dimension $N+2$. The paper clarifies the threshold where the drift term $u M \nabla \psi$ remains in $L^1$, allowing for distributional solutions.

C. Singular Data ($1 \leq m < \frac{2N+4}{N+6}$)

• Result (Theorem 2.6): For very singular data (including $f \in L^1$), distributional solutions may not exist because the drift term is not integrable.
• Regularity: The paper proves the existence of Entropy Solutions.
  • $u \in L^s(\Omega_T)$ for $s < 1^{\star\star}$ and $u \in L^q(0, T; W^{1,q}_0(\Omega))$ for $q < 1^\star$.
  • The solution satisfies the entropy inequality (2.4) involving truncations $T_k$.
• Significance: This extends the theory to the most general case of $L^1$ data, utilizing the specific structure of the sublinear elliptic equation to control the singularity.

4. Significance and Novelty

• Discontinuous Coefficients: The results hold for measurable, discontinuous coefficients $A$ and $M$, which is a significant generalization over models requiring smooth coefficients.
• Sublinear Coupling: The paper analyzes the specific case where the elliptic source is sublinear ($\theta < 1$). This contrasts with the linear case ($\theta=1$) found in standard Keller-Segel models, where blow-up is more common. The sublinear nature here aids in proving existence and regularity.
• Regularity Improvement: The work demonstrates that the coupling between the parabolic and elliptic equations leads to a "regularizing effect." Even with $L^1$ data, the solution $u$ gains higher integrability ($L^s$ with $s>1$) and gradient integrability compared to standard parabolic equations with drift.
• Unified Framework: The paper provides a comprehensive classification of solution types (bounded, finite energy, entropy) based strictly on the summability of the source term $f$, bridging the gap between elliptic and parabolic regularity theories in the context of chemotaxis models.

5. Conclusion

Marco Picerni successfully establishes the well-posedness of a parabolic-elliptic system with discontinuous coefficients and $L^1$ data. By combining fixed-point techniques, Stampacchia's iteration, and entropy methods, the author proves that the specific structure of the system (sublinear elliptic source) ensures the existence of solutions with improved regularity properties, ranging from bounded solutions for high-regularity data to entropy solutions for singular data.