Global Calderon-Zygmund estimates for irregular double-phase evolution problem with non-divergence data

Technical Summary: Global Calderón-Zygmund Estimates for Irregular Double-Phase Evolution Problems with Non-Divergence Data

Problem Formulation
This work investigates the existence and global regularity of strong solutions to a class of irregular double-phase parabolic equations with non-divergence data. The problem is defined on a bounded domain $\Omega \subset \mathbb{R}^N$ ($N \ge 2$) over a time interval $(0, T)$, denoted as $Q_T = \Omega \times (0, T)$:

$$\begin{cases} u_t - \text{div} \left( F(z, \nabla u) \nabla u \right) = f(z), & z = (x, t) \in Q_T, \\ u = 0, & \text{on } \partial\Omega \times (0, T), \\ u(x, 0) = u_0(x), & \text{in } \Omega, \end{cases}$$

where the flux function is given by $F(z, \xi) = a(z)|\xi|^{p(z)-2} + b(z)|\xi|^{q(z)-2}$. The exponents $p(z)$ and $q(z)$ are variable and Lipschitz continuous, while the non-negative modulating coefficients $a(z)$ and $b(z)$ satisfy the condition $a(z) + b(z) > 0$. The problem is characterized by a heterogeneous growth structure: in regions where both coefficients are positive, the flux exhibits mixed growth governed by both exponents, while in regions where one vanishes, it reduces to a single-phase model. The data $f$ is assumed to be in $L^\sigma(Q_T)$ with $\sigma > 2$, representing non-divergence form forcing.

Methodology
The authors employ a regularization approach combined with global higher integrability estimates and second-order regularity analysis. The methodology proceeds in several stages:

1. Regularization: The original problem is approximated by a family of regularized problems with smooth data and a regularized flux $F_\epsilon(z, \nabla u)$, where the singularity at $\nabla u = 0$ is handled by introducing a parameter $\epsilon \in (0, 1)$. Classical solutions to these regularized problems are established.
2. A Priori Estimates: The core of the analysis involves deriving uniform estimates independent of the regularization parameter $\epsilon$. This is achieved by:
   • Multiplying the regularized equation by specific test functions involving the time derivative and second-order spatial derivatives.
   • Utilizing the Green formula to handle boundary integrals, which are estimated using the geometry of the domain and the regularity of coefficients.
   • Establishing interpolation inequalities and pointwise estimates for the Hessian matrix of the solution.
   • Applying Hölder and Young inequalities to manage the variable exponents and the non-divergence term $f$.
3. Handling Non-Divergence Data: A significant technical challenge addressed is the presence of non-divergence data ($f$ not in divergence form). Unlike the elliptic case, parabolic equations do not allow reduction to divergence form via Bogovskiĭ-type regularity. The authors develop specific estimates to transfer the integrability of $f$ directly to the gradient and flux terms.
4. Passage to the Limit: Using the uniform estimates, compactness arguments (including the Vitali convergence theorem and properties of Musielak-Orlicz-Sobolev spaces), and pointwise convergence of gradients, the authors pass to the limit as $\epsilon \to 0$ and as the approximating data sequences converge to the original data.

Key Contributions and Results
The paper extends previous results on double-phase problems (specifically those in [1]) to the full range of integrability parameters $r \ge 0$ and strengthens the regularity conclusions. The main results are summarized as follows:

• Existence and Uniqueness: Under suitable gap conditions on the variable exponents $p(z)$ and $q(z)$ and regularity assumptions on the coefficients $a(z)$ and $b(z)$, the problem admits a unique strong solution $u$.
• Global Transfer of Integrability (Calderón-Zygmund Theory): The solution inherits the integrability of the initial data and the forcing term. Specifically, if the initial data satisfies a weighted integrability condition involving $r$, and $f \in L^\sigma(Q_T)$, then the double-phase flux maintains this integrability globally in time.
• Higher Integrability of the Gradient: The gradient of the solution exhibits self-improving properties. For any $s \in (0, \frac{4}{N+2})$, the gradient satisfies:
  $$|\nabla u|^{2(\min\{p(z), q(z)\}-1)+r+s} \in L^1(Q_T).$$
  This indicates that the gradient possesses higher integrability than the natural energy space suggests.
• Second-Order Regularity: The solution possesses enhanced second-order spatial regularity. The authors prove that:
  $$F((\cdot, t), \nabla u(\cdot, t))|\nabla u(\cdot, t)|^{r+2} \in L^1(\Omega) \quad \text{for a.e. } t \in (0, T),$$
  and
  $$F(z, \nabla u)|\nabla u|^{\frac{r+2}{2}} \in L^2(0, T; W^{1,2}(\Omega)).$$
  This implies that the weighted flux belongs to a Sobolev space with respect to spatial variables.

Significance and Scope
The paper claims to address a gap in the literature regarding double-phase parabolic equations with non-divergence data. While Calderón-Zygmund type estimates for such equations with divergence-form data or in the elliptic setting are established (citing works by Kim, Acerbi, Mingione, etc.), analogous results for the non-divergence case with variable exponents were previously unavailable.

The significance of the work lies in:

1. Generalization: Extending the range of the integrability parameter $r$ from $r \ge \max\{2, p_+, q_+\}$ (as in previous works) to the full range $r \ge 0$.
2. Non-Divergence Data: Successfully handling the non-divergence form of the source term $f$, which requires distinct techniques compared to the divergence case.
3. Irregular Coefficients: Accommodating modulating coefficients $a(z)$ and $b(z)$ that are not necessarily smooth but satisfy specific $L^d$ integrability conditions on their derivatives, depending on the gap parameter $\beta$ between exponents.

The results are presented as improvements and complements to existing theory, providing a robust framework for the regularity of solutions to heterogeneous, variable-exponent evolution problems arising in nonlinear elasticity and related fields. The authors do not propose new experimental applications but rather provide the theoretical foundation (existence and regularity estimates) necessary for the analysis of such models.