insight - Scientific Computing - # Fluid Dynamics Regularity

An Elementary Proof of Double Hölder Regularity of the Hydrodynamic Pressure for Weak Solutions of Euler Equations in Bounded C2-Domains

Q: Could the regularity results be different if the domain is not C2 but has corners or edges, and how would the proof need to be adapted?

Yes, the regularity results could be different if the domain is not C2 but has corners or edges. The presence of corners or edges introduces singularities in the solutions of elliptic equations, and the pressure, being governed by an elliptic equation, is not immune to these singularities. Here's how the regularity might be affected and how the proof might need to be adapted: Reduced Regularity: The double Hölder regularity (C2γ) obtained in the context relies heavily on the C2 regularity of the domain. In domains with corners or edges, the pressure regularity is expected to be lower near these geometric irregularities. The exact regularity would depend on the angle of the corners or edges. Weighted Spaces: To account for the reduced regularity near corners, one might need to work with weighted Hölder spaces, where the weight function is chosen to vanish or decay appropriately near the corners. These weighted spaces allow for singularities in the solution while still providing control over its behavior. Singular Solutions: The Green function estimates used in the proof, which rely on the C2 regularity of the domain, would need to be modified. Near corners, the Green function for the Laplacian operator exhibits singular behavior, and these singularities would need to be carefully accounted for in the estimates. Decomposition Techniques: One possible approach to handle domains with corners is to decompose the solution into a regular part and a singular part. The regular part would possess the desired regularity away from the corners, while the singular part would capture the behavior near the corners. In summary: Analyzing pressure regularity in non-smooth domains requires sophisticated tools from the theory of elliptic equations in non-smooth domains. The proof would need substantial modifications to account for the singular behavior of solutions near corners or edges.

Q: What are the implications of this pressure regularity result for the development of numerical methods for simulating fluid flows in bounded domains?

This pressure regularity result has important implications for developing and analyzing numerical methods for simulating fluid flows in bounded domains: Convergence Rates: The double Hölder regularity of pressure provides theoretical justification for expecting higher convergence rates for numerical methods that approximate the pressure. Knowing the regularity of the solution is crucial for establishing error estimates and proving convergence of numerical schemes. Choice of Discretization: The regularity result can guide the choice of appropriate spatial discretization schemes. Higher-order methods, such as finite element methods with higher-order polynomials or spectral methods, might be more suitable for approximating the pressure accurately, given its C2γ regularity. Pressure-Velocity Coupling: Many numerical methods for incompressible flows require handling the pressure-velocity coupling, often through a projection method or a coupled system. Understanding the pressure regularity can inform the design of stable and accurate coupling strategies. Boundary Conditions: The proof highlights the importance of carefully handling boundary conditions for the pressure, especially near the boundary. Numerical methods need to accurately impose boundary conditions to avoid spurious oscillations or inaccuracies in the pressure field. Adaptive Mesh Refinement: The regularity result suggests that regions near the boundary might require finer mesh resolution compared to the interior, as the pressure gradients might be steeper near the boundary. This information can be leveraged for developing adaptive mesh refinement strategies to optimize computational resources. Overall: This theoretical result provides valuable insights for developing, analyzing, and improving numerical methods for simulating fluid flows in bounded domains. By understanding the regularity of the pressure, we can design more accurate, stable, and efficient numerical schemes.

Core Concepts

This note presents a simplified proof for the double Hölder regularity of the hydrodynamic pressure in Euler equations for incompressible inviscid fluids in bounded domains, utilizing elementary elliptic PDE techniques and a novel approach with two cutoff functions.

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arxiv.org

Li, S., & Wang, Y. (2024). A note for double Hölder regularity of the hydrodynamic pressure for weak solutions of Euler equations [Preprint]. arXiv:2409.09433v2.

This note aims to provide an alternative, elementary proof for the double Hölder regularity (p ∈ C0,2γ(Ω) for 0 < γ < 1/2) of the hydrodynamic pressure (p) in the Euler equations for incompressible inviscid fluids in a bounded C2-domain Ω ⊂ Rd (d ≥ 3), given that the velocity (u) is Hölder continuous (u ∈ C0,γ(Ω; Rd)).

Key Insights Distilled From

A note for double H\"{o}lder regularity of the hydrodynamic pressure for weak solutions of Euler equations

by Siran Li, Ya... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2409.09433.pdf

$A note for double H\"{o}lder regularity of the hydrodynamic pressure for weak solutions of Euler equations$

Deeper Inquiries

How might this simplified proof for the double Hölder regularity of pressure in bounded domains be applied to other fluid dynamics problems or more complex fluid models?

This simplified proof, relying on elementary elliptic PDE techniques and a clever choice of cutoff functions, has the potential to be extended to other fluid dynamics problems involving pressure regularity in bounded domains. Here are some potential applications:

Navier-Stokes Equations: A natural extension would be to investigate if similar techniques can be applied to the Navier-Stokes equations, which are the fundamental equations governing viscous, incompressible fluid flow. The pressure in Navier-Stokes equations also exhibits a close relationship with the nonlinear convective term, and adapting this proof could provide insights into pressure regularity for weak solutions of Navier-Stokes.
Magnetohydrodynamics (MHD): MHD studies electrically conducting fluids where the magnetic field influences the fluid flow. The MHD equations also involve a pressure term coupled with the fluid's velocity and magnetic field. This simplified approach could potentially be adapted to study pressure regularity in MHD, particularly in bounded domains where boundary effects are significant.
Non-Newtonian Fluids: Many fluids encountered in industrial applications and biological systems exhibit non-Newtonian behavior, meaning their viscosity is not constant.  The constitutive laws for such fluids are more complex, leading to different pressure-velocity couplings.  Investigating if this proof technique can be modified to handle such complex constitutive laws could be valuable.
Multi-phase Flows:  Flows involving multiple fluids or phases (e.g., gas-liquid mixtures) present additional challenges due to interfacial dynamics and potential pressure jumps across interfaces. Adapting this proof to handle interfacial conditions and pressure discontinuities could be a fruitful direction.
Key Challenges and Considerations:

Nonlinearity: Extending this proof to more complex models will require carefully addressing the specific nonlinear terms and their interactions with the pressure.
Boundary Conditions: Different fluid dynamics problems involve various boundary conditions. Adapting the cutoff function approach to different boundary conditions will be crucial.
Regularity of the Domain: As mentioned in the context, the regularity of the domain (C2 in this case) plays a significant role. Adapting the proof to less regular domains will require careful consideration.

Could the regularity results be different if the domain is not C2 but has corners or edges, and how would the proof need to be adapted?

Yes, the regularity results could be different if the domain is not C2 but has corners or edges. The presence of corners or edges introduces singularities in the solutions of elliptic equations, and the pressure, being governed by an elliptic equation, is not immune to these singularities.
Here's how the regularity might be affected and how the proof might need to be adapted:

Reduced Regularity: The double Hölder regularity (C2γ) obtained in the context relies heavily on the C2 regularity of the domain. In domains with corners or edges, the pressure regularity is expected to be lower near these geometric irregularities. The exact regularity would depend on the angle of the corners or edges.
Weighted Spaces: To account for the reduced regularity near corners, one might need to work with weighted Hölder spaces, where the weight function is chosen to vanish or decay appropriately near the corners. These weighted spaces allow for singularities in the solution while still providing control over its behavior.
Singular Solutions: The Green function estimates used in the proof, which rely on the C2 regularity of the domain, would need to be modified.  Near corners, the Green function for the Laplacian operator exhibits singular behavior, and these singularities would need to be carefully accounted for in the estimates.
Decomposition Techniques: One possible approach to handle domains with corners is to decompose the solution into a regular part and a singular part. The regular part would possess the desired regularity away from the corners, while the singular part would capture the behavior near the corners.
In summary: Analyzing pressure regularity in non-smooth domains requires sophisticated tools from the theory of elliptic equations in non-smooth domains. The proof would need substantial modifications to account for the singular behavior of solutions near corners or edges.

What are the implications of this pressure regularity result for the development of numerical methods for simulating fluid flows in bounded domains?

This pressure regularity result has important implications for developing and analyzing numerical methods for simulating fluid flows in bounded domains:

Convergence Rates: The double Hölder regularity of pressure provides theoretical justification for expecting higher convergence rates for numerical methods that approximate the pressure.  Knowing the regularity of the solution is crucial for establishing error estimates and proving convergence of numerical schemes.
Choice of Discretization:  The regularity result can guide the choice of appropriate spatial discretization schemes. Higher-order methods, such as finite element methods with higher-order polynomials or spectral methods, might be more suitable for approximating the pressure accurately, given its C2γ regularity.
Pressure-Velocity Coupling:  Many numerical methods for incompressible flows require handling the pressure-velocity coupling, often through a projection method or a coupled system. Understanding the pressure regularity can inform the design of stable and accurate coupling strategies.
Boundary Conditions: The proof highlights the importance of carefully handling boundary conditions for the pressure, especially near the boundary.  Numerical methods need to accurately impose boundary conditions to avoid spurious oscillations or inaccuracies in the pressure field.
Adaptive Mesh Refinement:  The regularity result suggests that regions near the boundary might require finer mesh resolution compared to the interior, as the pressure gradients might be steeper near the boundary. This information can be leveraged for developing adaptive mesh refinement strategies to optimize computational resources.
Overall: This theoretical result provides valuable insights for developing, analyzing, and improving numerical methods for simulating fluid flows in bounded domains. By understanding the regularity of the pressure, we can design more accurate, stable, and efficient numerical schemes.