Fractional higher differentiability of solutions to strongly nonlinear Stokes systems

1. Problem Statement

The paper investigates the fractional regularity (higher differentiability) of local weak solutions to stationary Stokes-type systems governed by strongly nonlinear differential operators. The system is given by:
$$\begin{cases} -\text{div } a(x, Eu) + \nabla \pi = f \\ \text{div } u = 0 \end{cases}$$
in an open set $\Omega \subset \mathbb{R}^n$ ($n \ge 2$), where:

• $u$ is the velocity field and $\pi$ is the pressure.
• $Eu = \frac{1}{2}(\nabla u + (\nabla u)^T)$ is the symmetric gradient (deformation tensor).
• $a(x, P)$ is a nonlinear stress tensor depending on the spatial variable $x$ and the symmetric matrix $P$.
• $f$ is an external force.

Key Challenges:

1. Non-Power Nonlinearities: The operator $a$ does not necessarily follow a power-law growth (e.g., $p$-Laplacian type). Instead, it is governed by general Young functions $\phi$ and $\psi$, requiring an Orlicz-Sobolev functional framework.
2. Lack of Smoothness in $x$: The dependence of $a$ on the spatial variable $x$ is not assumed to be smooth (e.g., $C^1$ or Lipschitz). Instead, it satisfies a fractional Hölder-type condition involving a function $k \in L^m_{loc}$.
3. Symmetric Gradient vs. Full Gradient: Unlike standard elliptic systems involving the full gradient $\nabla u$, Stokes systems involve the symmetric gradient $Eu$. This complicates the analysis because standard tools for $\nabla u$ do not directly apply without handling the divergence-free constraint.

2. Methodology

The authors employ a sophisticated combination of techniques from nonlinear potential theory, Orlicz space theory, and harmonic analysis:

• Functional Framework: The analysis is conducted in Orlicz spaces $L^\phi$ and Orlicz-Sobolev spaces $W^{1,\phi}$. The growth and ellipticity of the operator are defined via Young functions $\phi$ and $\psi$.
• Nonlinear Transformation: To handle the degeneracy of the operator, the authors analyze the regularity of the transformed quantity:
  $$V(Eu) = \sqrt{\frac{\phi'(|Eu|)}{|Eu|}} Eu$$
  This transformation linearizes the behavior of the operator near zero and infinity, a standard technique in $p$-Laplacian regularity theory extended here to Orlicz settings.
• Difference Quotient Method: The core of the proof involves estimating the difference quotients $\tau_h V(Eu)$ to establish membership in Besov spaces $B^{\sigma, 2, \infty}_{loc}$.
• Bogovskiĭ Operator: To handle the divergence-free constraint ($\text{div } u = 0$) in the Stokes system, the authors utilize the Bogovskiĭ operator to construct divergence-free test functions. This allows them to eliminate the pressure term $\pi$ in the weak formulation when estimating the velocity gradient.
• Modular Poincaré-Sobolev Inequality: A significant technical innovation is the derivation of a modular Poincaré-Sobolev inequality in Orlicz spaces on domains of finite measure. This inequality avoids restrictive assumptions on the Young function near zero (specifically condition (2.36) in the text) by introducing a modified conjugate function $\phi_{0,n}$.
• Interpolation and Approximation: The proof proceeds in two steps:
  1. A Priori Estimates: Assuming higher regularity for the data and coefficients, the authors derive estimates for $V(Eu)$ in Besov spaces.
  2. Approximation: They use a mollification argument to remove the regularity assumptions on $a$ and $f$, passing to the limit using compactness and convergence properties in Orlicz spaces.

3. Key Contributions

A. Generalization to Orlicz Spaces

The paper extends known results for $p$-Laplacian systems (where $\phi(t) = t^p$) to the general Orlicz setting. This covers a vast class of non-standard growth conditions, including logarithmic perturbations and other non-polynomial behaviors, which are crucial for modeling complex non-Newtonian fluids.

B. Optimal Balance Condition

The authors establish a precise balance condition between the nonlinearity of the operator and the integrability of the datum $f$.

• Let $\phi$ govern the growth of the operator and $\psi$ govern the integrability of $f$.
• The condition (4.11) requires that $\phi^{-1}(t)^\theta \phi_n^{-1}(t)^{1-\theta} \psi^{-1}(t) \le ct$, where $\phi_n$ is the Sobolev conjugate of $\phi$ and $\theta \in (0, 1]$.
• This condition ensures that $f$ lies in an Orlicz space slightly "smaller" than the one required merely for the existence of weak solutions, which is necessary to gain fractional differentiability.

C. Fractional Regularity of Pressure

While much literature focuses on the velocity gradient, this paper explicitly establishes the fractional regularity of the pressure term $\pi$. The pressure is shown to belong to an Orlicz-Besov space $B^{\varrho, \tilde{\phi}, \infty}_{loc}$, where the smoothness index $\varrho$ depends on the smoothness of the velocity and the indices of the Young function.

D. New Inequality

The paper introduces a new modular Poincaré-Sobolev inequality (Theorem 5.2) tailored for Orlicz spaces on sets of finite measure. This inequality is critical for the proof and removes technical restrictions on the behavior of Young functions near zero that were previously necessary for such embeddings.

4. Main Results

Theorem 4.1 (Main Result):
Let $(u, \pi)$ be a local weak solution to the Stokes system (1.1) under the stated assumptions on $a$ and $f$.

1. Regularity of Velocity: The nonlinear expression $V(Eu)$ belongs to the local Besov space:
   $$V(Eu) \in B^{\sigma, 2, \infty}_{loc}(\Omega, \mathbb{R}^{n \times n})$$
   where the smoothness exponent is $\sigma = \min\{\alpha, \theta, \theta s'_\phi / 2\}$. Here, $\alpha$ is the fractional smoothness of $a$ in $x$, $\theta$ relates to the integrability of $f$, and $s'_\phi$ is related to the upper Simonenko index of $\phi$.
   Consequently, $V(Eu)$ also belongs to fractional Sobolev spaces $W^{\rho, 2}_{loc}$ for any $\rho < \sigma$.

2. Regularity of Pressure: The pressure $\pi$ belongs to the local Orlicz-Besov space:
   $$\pi \in B^{\varrho, \tilde{\phi}, \infty}_{loc}(\Omega)$$
   where $\varrho = \sigma / \max\{1, i'_\phi/2\}$ and $\tilde{\phi}$ is the Young conjugate of $\phi$.

Theorem 4.2 (Elliptic System):
Parallel results are established for the associated plain elliptic system $-\text{div } a(x, Eu) = f$, where the divergence-free constraint is absent. The proof simplifies as divergence-free test functions are not required.

5. Significance and Impact

• Modeling Non-Newtonian Fluids: The results provide a rigorous mathematical foundation for the regularity of solutions to models describing non-Newtonian fluids with complex, non-power-law rheology (e.g., shear-thinning or shear-thickening fluids with logarithmic corrections).
• Optimality: The assumptions on the integrability of the source term $f$ are shown to be nearly optimal, matching known sharp results for the standard $p$-Laplace equation.
• Methodological Advancement: The development of the modular Poincaré-Sobolev inequality and the successful application of Besov space techniques to Stokes systems with non-smooth coefficients in Orlicz spaces represent a significant methodological breakthrough.
• Pressure Regularity: Explicitly characterizing the regularity of the pressure in non-standard growth settings fills a gap in the literature, as pressure regularity is often more difficult to establish than velocity regularity in Stokes systems.

In summary, this paper significantly advances the regularity theory for nonlinear Stokes systems by moving beyond power-type growth to general Orlicz growth, handling non-smooth coefficients, and providing a unified framework for the regularity of both velocity and pressure.