Counterexamples to maximal regularity for operators in divergence form

Technical Summary: Counterexamples to Maximal Regularity for Operators in Divergence Form

Problem Statement
This paper addresses two fundamental questions regarding the regularity of solutions to parabolic partial differential equations (PDEs) in divergence form with time-dependent coefficients. The authors consider the abstract Cauchy problem (CP):
$$\begin{cases} u' - Au = f, & \text{on } (0, 1), \\ u(0) = 0, \end{cases}$$
where $A(t)$ is a strongly measurable family of operators mapping a Hilbert space $V$ to its dual $V^*$, satisfying coercivity and boundedness conditions. Specifically, the authors focus on the concrete setting where $H = L^2(\mathbb{R}^d)$, $V = H^1(\mathbb{R}^d)$, and $A(t)v = -\text{div}(B(t)\nabla v)$, with $B$ being a uniformly elliptic matrix field.

The study investigates two specific regularity problems:

1. Problem 1 (Maximal $L^p$-regularity on $V$): For $p \in (1, \infty) \setminus \{2\}$, does the condition $f \in L^p(0, 1; V^*)$ imply that the unique weak solution $u$ belongs to $L^p(0, 1; V)$?
2. Problem 2 (Maximal $L^2$-regularity on $H$): Does the condition $f \in L^2(0, 1; H)$ imply that the time derivative of the solution satisfies $u' \in L^2(0, 1; H)$?

While Lions' theory guarantees a unique weak solution in $L^2(0, 1; V) \cap H^1(0, 1; V^*)$ under coercivity alone, and positive results exist for autonomous operators or operators with sufficient time-regularity (e.g., continuity or Hölder continuity), the behavior for general time-dependent coefficients without such regularity assumptions remains open. Previous counterexamples by Dier and Fackler were abstract and not differential operators. This paper aims to construct counterexamples that are genuine differential operators.

Methodology
The authors' approach relies on a specific construction of solutions to linear, uniformly parabolic PDEs with complex coefficients, originally developed by the second-named author in [19]. The methodology proceeds as follows:

1. Scaling Symmetry Construction: The authors utilize a self-similar ansatz for the solution $\zeta(t, x)$ and the coefficient matrix $B(t, x)$:
   $$\zeta(t, x) = (1-t)^{-\mu/2} e^{-i \log(1-t)/2} w\left(\frac{x}{(1-t)^{1/2}}\right), \quad B(t, x) = a\left(\frac{x}{(1-t)^{1/2}}\right).$$
   This reduces the parabolic PDE to an elliptic PDE for the profile function $w$ and the coefficient field $a$ on $\mathbb{R}^d$.

2. Existence of Singular Profiles: Citing [19], the authors note that for any parameter $\mu \in [0, d/2)$, there exist smooth (away from the origin) uniformly elliptic coefficients $a$ and solutions $w$ such that $w$ is locally Lipschitz but exhibits specific decay rates at infinity ($|x|^{-\mu}$). Crucially, the imaginary part of $B$ is chosen to be symmetric, allowing strong coupling between the real and imaginary parts of the equation without violating the ellipticity condition.

3. Localization and Counterexample Generation: To adapt these global solutions to the bounded time interval $(0, 1)$ and the specific problem setting, the authors introduce a smooth cutoff function $\eta$. They define a new solution candidate $u(t, x) = t \zeta(t, x) \eta(x)$ and derive the corresponding forcing term $f$.

4. Regularity Analysis: The core of the proof involves estimating the integrability of $u$ and $f$. By carefully selecting the parameters $\mu$, $r$, and $s$, the authors demonstrate that while $f$ possesses high regularity (e.g., $f \in L^\infty(0, 1; L^2)$), the solution $u$ fails to belong to specific mixed-norm spaces $L^r(0, 1; L^s(\mathbb{R}^d))$.

5. Duality and Interpolation:

   • For Problem 1 with $p > 2$, the failure of $L^r(L^s)$ integrability directly implies the failure of $L^p(0, 1; H^1)$ regularity via Sobolev embedding.
   • For Problem 1 with $p < 2$, a duality argument is employed. Assuming maximal regularity holds for $p$ leads to a contradiction with the known failure for the dual exponent $p'$.
   • For Problem 2, complex interpolation and Sobolev embedding theorems are used to show that if $u$ possessed the required time-regularity ($H^\nu$), it would necessarily belong to the $L^r(L^s)$ space, which has already been proven false.

Key Contributions and Results

• Theorem 2.4 (Failure of Higher Integrability): For dimensions $d \ge 2$ and exponents $r, s \in (1, \infty)$ satisfying $\frac{2}{r} + \frac{d}{s} < \frac{d}{2}$, there exists an elliptic matrix $B$ and a forcing term $f \in L^\infty(0, 1; L^2)$ such that the unique solution $u$ to the divergence form equation is not in $L^r(0, 1; L^s)$.
• Theorem 2.5 (Failure of Maximal $L^p$-Regularity for Problem 1): For every $p \in (1, \infty) \setminus \{2\}$, there exists an elliptic operator in divergence form and a forcing term $f \in L^p(0, 1; H^{-1})$ such that the solution $u$ does not belong to $L^p(0, 1; H^1)$. This demonstrates that the positive result of Theorem 2.2 (which guarantees regularity for $p$ in a small neighborhood of 2) is sharp and cannot be extended to all $p$.
• Theorem 2.7 (Failure of Maximal $L^2$-Regularity for Problem 2): For every $\nu \in (1/2, 1]$, there exists an elliptic operator and $f \in L^\infty(0, 1; L^2)$ such that the solution $u$ is not in the fractional Sobolev space $H^\nu(0, 1; L^2)$. This provides a negative answer to Lions' problem for general elliptic operators in divergence form, showing that the solution can fail to have even $1/2$-order time regularity in $L^2$.

Significance and Claims
The paper claims to resolve the question of whether maximal regularity holds for general time-dependent divergence form operators by providing explicit counterexamples that are differential operators, rather than abstract operators.

• Sharpness of Existing Theory: The results confirm that the small interval of $p$ for which maximal $L^p$-regularity holds (Theorem 2.2) is optimal and cannot be extended to the full range $(1, \infty)$ without additional assumptions on the time-regularity of the coefficients.
• Limitations of Divergence Form Structure: Unlike the Kato square root problem, where the differential structure provided positive results, this work shows that the divergence form structure alone is insufficient to guarantee maximal regularity when coefficients lack time-continuity.
• Regularity of Coefficients: The authors note that their counterexamples rely on coefficients $B$ that are not continuous in time (specifically, not continuous as maps into $L^\infty$). They explicitly state that finding a counterexample for Problem 2 with Hölder continuous coefficients (specifically for $\nu=1$) remains an open and interesting problem, as positive results exist for coefficients with slightly more regularity (e.g., $C^{1/2+\epsilon}$).

The paper concludes by acknowledging open questions, including the case of symmetric/hermitian operators and the one-dimensional setting ($d=1$), where the behavior may differ.