Interior Second Order Hölder Regularity for Stokes Systems with Variable Coefficients

Core Concepts

This paper establishes interior second order Hölder regularity for the velocity and first order Hölder regularity for the pressure in spatial variables for Stokes systems with variable coefficients in non-divergence form. Notably, the gradient of the curl of velocity exhibits Hölder continuity in both space and time, despite the absence of temporal continuity assumptions for the coefficients and right-hand side terms.

Abstract

• Bibliographic Information: Dong, R., Li, D., & Wang, L. (2024). Interior Second Order Hölder Regularity for Stokes systems. arXiv preprint arXiv:2401.09841v2.
• Research Objective: To investigate the interior second order Hölder regularity for Stokes systems with variable coefficients in non-divergence form.
• Methodology: The authors employ a pointwise estimation technique inspired by Caffarelli's work, utilizing Campanato's characterization of Hölder continuity. They approximate the velocity and its curl's gradient with polynomials in spatial variables through an iterative process.
• Key Findings:
  • The paper establishes interior C^2,α regularity for velocity and C^1,α regularity for pressure in spatial variables.
  • The gradient of the curl of velocity (D∇×u) exhibits C^{α, α/2} regularity, indicating Hölder continuity in both space and time, even without assuming temporal continuity for coefficients and right-hand side terms.
  • The sharpness of the results is demonstrated through a counterexample.
• Main Conclusions: The study successfully derives interior second order Hölder regularity results for Stokes systems with variable coefficients, contributing novel insights to the field. The findings are particularly significant as they highlight the regularity of the curl of velocity in both space and time, a result not previously established.
• Significance: This research enhances the understanding of the regularity properties of solutions to Stokes systems, a fundamental topic in fluid dynamics and partial differential equations. The results have implications for both theoretical analysis and numerical simulations of fluid flow problems.
• Limitations and Future Research: The study focuses on interior regularity, leaving room for investigating regularity up to the boundary. Further research could explore extending these results to Stokes systems with more general boundary conditions or involving more complex fluid models.

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Interior Second Order H\"{o}lder Regularity for Stokes systems

by Rong Dong, D... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2401.09841.pdf

Deeper Inquiries

How can these interior regularity results be extended to address regularity up to the boundary of the domain?

Extending the interior regularity results for Stokes systems to encompass regularity up to the boundary presents significant challenges and necessitates careful consideration of the boundary conditions and the geometry of the domain. Here's a breakdown of the key aspects and potential approaches:
1. Boundary Conditions:

Dirichlet (No-Slip): The most common boundary condition for Stokes systems, enforcing zero velocity at the boundary, introduces complexities due to the interaction between the fluid and the solid boundary.
Neumann (Stress-Free): These conditions prescribe the stress at the boundary and can lead to different regularity properties compared to Dirichlet conditions.
Mixed Conditions: Combinations of Dirichlet and Neumann conditions on different parts of the boundary further complicate the analysis.
2. Domain Geometry:

Smooth Boundaries: For domains with sufficiently smooth boundaries (e.g., $C^{2,\alpha}$), regularity theory is more tractable. Techniques from elliptic and parabolic regularity theory, combined with appropriate extensions and reflections, can be employed.
Corners and Edges: Domains with corners or edges introduce singularities in the solutions, making regularity analysis considerably more delicate. Weighted Sobolev spaces or other specialized function spaces are often required to capture the behavior near these singularities.
Approaches for Boundary Regularity:

Flattening the Boundary: For domains with smooth boundaries, one approach is to locally flatten the boundary using a smooth diffeomorphism. This transforms the problem into one posed on a half-space, where reflection techniques can be applied.
Difference Quotient Techniques: These methods involve studying the difference quotients of the solution in directions tangential to the boundary. By carefully estimating these difference quotients, one can derive regularity estimates up to the boundary.
Layer Potential Techniques: For constant or smooth coefficient Stokes systems, layer potential representations of solutions can be utilized. These representations allow for a detailed analysis of the boundary behavior of solutions and their derivatives.
Challenges and Considerations:

Compatibility Conditions: For higher-order regularity up to the boundary, compatibility conditions between the initial data, boundary data, and the Stokes system itself become crucial. These conditions ensure the well-posedness of the problem and the existence of sufficiently regular solutions.
Corner Singularities: As mentioned earlier, corners and edges in the boundary can lead to singularities in the solution. Analyzing these singularities and their impact on regularity requires sophisticated techniques from singular integral operators and weighted function spaces.

Could the regularity results differ if the coefficients of the Stokes system exhibit some form of temporal continuity?

Yes, the regularity results for Stokes systems can indeed differ if the coefficients exhibit temporal continuity.
Here's why:

Regularizing Effect of Time: Temporal continuity in the coefficients introduces a regularizing effect on the solution. This is analogous to the regularizing effect observed in parabolic equations, where even if the initial data is rough, the solution becomes smoother over time due to the smoothing properties of the parabolic operator.
Potential Differences in Regularity:

Higher-Order Spatial Regularity: Temporal continuity can potentially lead to higher-order spatial regularity in the solution. For instance, if the coefficients are Hölder continuous in time, one might expect to gain an additional half-derivative of regularity in space compared to the case with no temporal continuity.
Improved Temporal Regularity: More importantly, temporal continuity in the coefficients can directly improve the temporal regularity of the solution. This is in stark contrast to the counterexample mentioned in the context, where the lack of temporal continuity in the coefficients allowed for solutions with poor temporal regularity.
Techniques and Considerations:

Exploiting Parabolic Structure: To leverage temporal continuity, techniques that exploit the parabolic structure of the Stokes system become essential. This might involve using anisotropic function spaces that treat spatial and temporal variables differently or employing methods like the method of continuity, where one starts with a problem with smooth coefficients and gradually perturbs it to the desired problem.
Bootstrapping Arguments: If the coefficients possess higher-order temporal regularity, bootstrapping arguments can be employed. These arguments involve repeatedly using the regularity of the coefficients and the equation to improve the regularity of the solution iteratively.
Implications:

Enhanced Regularity Theory: The presence of temporal continuity in the coefficients can significantly enrich the regularity theory for Stokes systems, leading to stronger results and a more complete understanding of the solution's behavior.

What are the implications of these findings for the numerical analysis of Stokes systems, particularly in developing and analyzing numerical schemes?

The interior regularity findings for Stokes systems have substantial implications for numerical analysis, particularly in the design and analysis of effective numerical schemes. Here's an elaboration:
1. Convergence Rates and Accuracy:

A Priori Error Estimates: Regularity results are crucial for deriving a priori error estimates, which provide bounds on the difference between the exact solution and the numerical approximation. Higher regularity allows for the use of higher-order numerical schemes, leading to improved convergence rates and increased accuracy.
Choice of Discretization: The regularity of the solution informs the choice of appropriate discretization methods. For instance, if the solution is known to be smooth, higher-order finite element methods or spectral methods can be employed to achieve optimal convergence.
2. Stability and Robustness:

Control of Numerical Errors: Regularity estimates provide a means to control the growth of numerical errors. Knowing that the solution and its derivatives are bounded in certain norms helps ensure the stability of the numerical scheme.
Robustness to Data Perturbations: Regularity results contribute to the robustness of numerical methods to perturbations in the data. Small changes in the data (e.g., due to measurement errors) will have a controlled effect on the numerical solution if the solution is known to be sufficiently regular.
3. Adaptive Mesh Refinement:

Efficient Use of Computational Resources: In regions where the solution is less regular (e.g., near corners in the domain), adaptive mesh refinement strategies can be employed. Regularity estimates guide the refinement process, concentrating computational effort where it is most needed and improving overall efficiency.
4. Development of Specialized Schemes:

Handling Irregularities: The understanding of regularity and potential singularities motivates the development of specialized numerical schemes. For instance, graded meshes or locally refined meshes can be used to effectively capture the behavior of solutions near corners or edges.
Specific Examples:

Finite Element Methods: For Stokes systems, the regularity of the solution directly influences the convergence rates of finite element methods. Higher regularity allows for the use of higher-order polynomial spaces, leading to faster convergence.
Finite Difference Methods: Regularity results are essential for analyzing the consistency and stability of finite difference schemes. The order of accuracy of the scheme is limited by the regularity of the solution.
In summary: The interior regularity findings provide a theoretical foundation for developing and analyzing robust, accurate, and efficient numerical methods for Stokes systems. They guide the choice of discretization, the analysis of convergence, and the development of strategies to handle potential irregularities in the solution.