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

Abstract: We study irregular double-phase parabolic equations with variable exponents and non-divergence data, [ u_t-\operatorname{div} \left(\mathcal{F}(z,\nabla u)\nabla u \right)=f(z),\quad z=(x,t)\in Q_T:=Ω\times (0,T), ] under the homogeneous Dirichlet boundary conditions. Here, $Ω\subset \mathbb{R}^N$, $N \geq 2$, is a bounded domain, $T>0$, [ \mathcal{F}(z,\nabla u)=a(z)|\nabla u|^{p(z)-2} + b(z) |\nabla u |^{q(z)-2} ] with given Lipschitz-continuous exponents $p,q$ that satisfy a suitable balance condition. The nonnegative coefficients $a(z), b(z)$ satisfy the inequality $a(z)+b(z)>0$ in $Q_T$, the space and time derivatives of $a$ and $b$ belong to $L^d(Q_T)$ with some $d$ depending on the data. If [ f\in L^σ(Q_T) \quad \text{for} \ σ\in (2, N+2] \quad \text{and} \quad \mathcal{F}((\cdot,0),\nabla u_0)\,|\nabla u_0|^{r+2}\in L^1(Ω), ] where (0\le r\le K(N,σ,p,q)) if (σ<N+2), while (r\ge0) is arbitrary if (σ=N+2), then the problem has a unique strong solution, for which we prove the global transfer of integrability from the initial data and the forcing term to the double-phase flux in the spirit of Calderón-Zygmund theory, higher integrability of the gradient, and the second-order space regularity: [ \begin{split} & \text{$\mathcal{F}((\cdot,t),\nabla u(\cdot,t))|\nabla u(\cdot,t)|^{r+2}\in L^1(Ω)$ for a.e. $t\in (0,T)$}, \ & \text{$|\nabla u|^{2(\min{p(z),q(z)}-1)+r+s}\in L^1(Q_T)$ for every $s\in\left(0,\frac{4}{N+2}\right)$}, \ & \mathcal{F}(z,\nabla u)|\nabla u|^{\frac{r+2}{2}} \in L^2(0,T;W^{1,2}(Ω)). \end{split} ] The results improve and complement the results in \cite{Arora-Shmarev-JGA-2026} and extend them to the full range $r \geq 0$.

• The paper demonstrates that under precise balance conditions, unique global strong solutions achieve transferred integrability from the data to the flux.
• It establishes higher gradient integrability and second-order regularity through advanced parabolic interpolation, regularization, and compactness techniques.
• The work broadens CZ theory to irregular, non-divergence, double-phase models, impacting areas like nonlinear elasticity and non-Newtonian flows.

Global Calderón-Zygmund Theory for Irregular Double-Phase Evolution Problems with Non-Divergence Data

Problem Formulation and Analytical Setting

The paper investigates parabolic PDEs of the form

$$u_t-\operatorname{div}\left(\mathcal{F}(z,\nabla u)\nabla u\right) = f(z)$$

over a bounded $C^{2+\gamma}$ domain $\Omega \subset \mathbb{R}^N$ for $z=(x,t)\in Q_T := \Omega \times (0,T)$, with $$\mathcal{F}(z,\nabla u) = a(z) |\nabla u|^{p(z)-2} + b(z)|\nabla u|^{q(z)-2}$$. The exponents $p(z), q(z)$ and the modulating coefficients $a(z), b(z)$ are space-time dependent, with minimal regularity and positivity constraints: $a(z) + b(z) > 0$ and $p, q \in C^{0,1}(\overline{Q}_T)$. Non-divergence right-hand sides distinguish the model from more classical double-phase evolution problems.

The flux $\mathcal{F}$ models transitions between single-phase and double-phase regimes according to the supports of $C^{2+\gamma}$0, $C^{2+\gamma}$1, generalizing the classical nonstandard growth setting. Musielak-Orlicz and variable exponent spaces serve as the analytic framework, capturing the nonuniform and multi-regime character of the PDE.

Main Results: Existence, Uniqueness, and Regularity

Well-Posedness and Calderón-Zygmund Integrability Transfer

Under natural balance (gap) conditions on $C^{2+\gamma}$2, $C^{2+\gamma}$3 and integrability for initial data and the forcing, the study establishes unique global strong solutions that propagate integrability in the spirit of the Calderón-Zygmund (CZ) theory to the double-phase parabolic context. Specifically, for $C^{2+\gamma}$4 with $C^{2+\gamma}$5 and suitably integrable initial data, the solution satisfies: $C^{2+\gamma}$6 together with higher integrability for $C^{2+\gamma}$7 for any $C^{2+\gamma}$8.

A crucial element is the global transfer of integrability from the data to the nonstandard flux, generalizing classical CZ estimates beyond constant exponent or divergence-form settings. This is achieved for the full range $C^{2+\gamma}$9 when the source satisfies $\Omega \subset \mathbb{R}^N$0 and, with explicit bounds, for $\Omega \subset \mathbb{R}^N$1 when $\Omega \subset \mathbb{R}^N$2. The technical apparatus hinges on careful balance conditions on the exponents, Lipschitz-type regularity of the modulating coefficients, refined interpolation, and trace inequalities adapted to the parabolic and double-phase context.

Higher Integrability and Second-Order Regularity

The propagation of integrability in initial data and forcing yields an $\Omega \subset \mathbb{R}^N$3 bound with $\Omega \subset \mathbb{R}^N$4-regularity of the time derivative. The study rigorously shows that the solution gradient achieves higher global integrability, i.e.

$\Omega \subset \mathbb{R}^N$5

for $\Omega \subset \mathbb{R}^N$6 and all $\Omega \subset \mathbb{R}^N$7 in a small parameter range depending on the space dimension. This self-improving property is crucial for CZ regularity and consequently the compactness framework for passing to the limit in regularized (smooth) problems.

Global second-order regularity is established in the sense that

$\Omega \subset \mathbb{R}^N$8

and

$\Omega \subset \mathbb{R}^N$9

with bounds in terms only of the data. These results generalize and reinforce elliptic and homogeneous estimates to the evolutionary, non-divergence, and irregular double-phase setting, allowing for variable exponents and low regularity data.

Analytical Techniques

The analysis deploys sophisticated regularization, interpolation, and compactness arguments. A notable technical contribution is the global control of second-order terms, exploiting delicate versions of parabolic interpolation inequalities adjusted for variable exponents and nonuniform ellipticity. Importantly, the parabolic nature precludes reduction to divergence-form analogues (e.g., via Bogovskiĭ operators), necessitating a fundamentally parabolic, direct approach.

Functional analysis is conducted predominantly in Musielak-Orlicz-Sobolev spaces, ensuring the framework flexibly accommodates transitions in growth regime, and handles the regularity-breakdown induced by variable exponents and degeneracy.

Comparison to and Improvement over Previous Work

The paper simultaneously extends and sharpens previous existence and regularity results for double-phase (and $z=(x,t)\in Q_T := \Omega \times (0,T)$0-type) parabolic equations [Arora-Shmarev-JGA-2026, Kim-JMAA-2025, RACSAM-2023]. Prior works, while addressing double-phase evolution in divergence-form, with constant exponents or in stationary regimes, did not establish global CZ-type regularity for non-divergence data, nor for the general variable exponent and multi-coefficient structure considered here.

This study covers the full admissible range $z=(x,t)\in Q_T := \Omega \times (0,T)$1 for the inherited gradient integrability, significantly broadening applicability. The explicit gap and balance conditions distilled here are optimal in light of recent counterexamples and sharp regularity constraints in both elliptic and parabolic scenarios.

Implications and Prospects

Theoretical Impact

The rigorous extension of Calderón-Zygmund theory to variable exponent, non-divergence form, and evolutionary double-phase models fills a critical theoretical gap, illustrating that integrability and regularity transfer can persist in multi-phased, irregular, and time-dependent regimes. The results clarify the functional analytic and PDE constraints required—especially concerning the balance of exponents, coefficient regularity, and minimal integrability for initial and forcing data.

These findings provide a template for future extensions of CZ theory in related settings—such as parabolic systems (not scalar equations), interface problems, or problems with even more intricate modulatory structures.

Practical Relevance

The models addressed naturally arise in nonlinear elasticity, composite materials, and non-Newtonian flows, where energy density may shift between distinct growth regimes and the coefficients exhibit heterogeneity in space and time. The present results lay solid groundwork for numerical analysis and simulation techniques that can exploit these regularity properties for error estimates and convergence proofs in adaptive FEM and related schemes.

Future Directions

Potential developments include:

• Extension to systems of equations and to equations/systems with lower order terms or boundary reactions.
• Weakening regularity assumptions on the coefficients (e.g., Dini or $z=(x,t)\in Q_T := \Omega \times (0,T)$2-type continuity).
• Analysis under anisotropic, non-convex, or dynamically evolving domains.
• Applications to homogenization, $z=(x,t)\in Q_T := \Omega \times (0,T)$3-convergence, and multi-scale phenomena in composite and complex media.

Conclusion

This work establishes a global Calderón-Zygmund theory for irregular double-phase parabolic equations with non-divergence data, variable exponents, and minimal coefficient regularity. The analysis guarantees the transfer of integrability from initial and forcing data to the solution flux, higher gradient integrability, and global second-order regularity, all under explicit structural and gap conditions. In doing so, the results generalize and strengthen the theoretical foundation for nonstandard growth problems in both PDE theory and applications in mathematical models of heterogeneous materials, setting the stage for further advances in the field.

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Reference: "Global Calderon-Zygmund estimates for irregular double-phase evolution problem with non-divergence data" (2607.04492)