[Paper Review] On the reduction of PDE's problems in the half-space, under the slip boundary condition, to the corresponding problems in the whole space

H. Beirão da Veiga, Francesca Crispo+1 more|arXiv (Cornell University)|Aug 19, 2010

Navier-Stokes equation solutions24 references3 citations

TL;DR

This paper presents a reflection-based method to reduce boundary value problems for partial differential equations (PDEs) in the half-space under slip boundary conditions to equivalent problems in the whole space. By exploiting flat boundaries and symmetry, it establishes that strong norm results—such as existence, uniqueness, and vanishing viscosity limits—for the Navier-Stokes and non-Newtonian fluid equations in the whole space directly imply corresponding results in the half-space, enabling sharp convergence and regularity estimates under slip conditions.

ABSTRACT

The resolution of a very large class of linear and non-linear, stationary and evolutive partial differential problems in the half-space (or similar) under the slip boundary condition is reduced here to that of the corresponding results for the same problem in the whole space. The approach is particularly suitable for proving new results in strong norms. To determine whether this extension is available, turns out to be a simple exercise. The verification depends on a few general features of the functional space X related to the space variables. Hence, we present an approach as much as possible independent of the particular space X. We appeal to a reflection technique. Hence a crucial assumption is to be in the presence of flat boundaries (see below). Instead of stating "general theorems" we rather prefer to illustrate how to apply our results by considering a couple of interesting problems. As a main example, we show that the resolution of a class of problems for the evolution Navier-Stokes equations under a slip boundary condition can be reduced to that of the corresponding results for the Cauchy problem. In particular, we show that sharp vanishing viscosity limit results that hold for the evolution Navier-Stokes equations in the whole space can be extended to the boundary value problem in the half-space. We also show some applications to non-Newtonian fluid problems.

Motivation & Objective

• To establish a general framework for transferring strong norm results from the whole space to the half-space under slip boundary conditions.
• To extend sharp vanishing viscosity limit results from the Cauchy problem to the half-space with slip conditions.
• To provide a systematic method applicable to both evolutionary and stationary PDEs, including non-Newtonian fluid models.
• To demonstrate that regularity and convergence properties in the whole space imply equivalent properties in the half-space without requiring new proofs for each problem.

Proposed method

• Utilizes a reflection technique to extend functions defined on the half-space ℝ³₊ to the whole space ℝ³, preserving boundary conditions.
• Applies a symmetric extension across the boundary x₃ = 0, modifying components of the vector field to satisfy slip conditions (no normal flow, zero tangential stress).
• Employs a mirror-reflection strategy that preserves divergence-free structure and regularity, ensuring the extended function belongs to the same functional space X.
• Relies on the flatness of the boundary to ensure that the stress tensor condition reduces to a simple form involving the vorticity and normal vector.
• Reduces the analysis of boundary value problems to the corresponding whole-space problems by verifying that the extended solution satisfies the same PDE and functional framework.
• Applies the method to evolution problems (e.g., Navier-Stokes, non-Newtonian fluids) and stationary problems, with results transferred via extension and symmetry.

Experimental results

Research questions

• RQ1 Can strong norm results for PDEs in the whole space be extended to the half-space under slip boundary conditions?
• RQ2 Under what conditions does the reflection method preserve the solution structure and regularity for evolution and stationary PDEs?
• RQ3 Can sharp vanishing viscosity limits for the Navier-Stokes equations in the whole space be transferred to the half-space with slip conditions?
• RQ4 How does the method apply to non-Newtonian fluid models, particularly shear-thinning fluids?
• RQ5 What functional space properties are necessary for the reflection method to preserve solution regularity and convergence?

Key findings

• The resolution of the evolution Navier-Stokes equations under slip boundary conditions in the half-space ℝ³₊ can be reduced to the corresponding Cauchy problem in ℝ³, preserving strong convergence and regularity.
• Sharp vanishing viscosity limits in the whole space imply the same limits in the half-space, with convergence in C([0,T]; X) for initial data in Hˡ,²₋σ(ℝ³₊).
• For shear-thinning fluids with p ∈ (7/5, 2), existence and uniqueness of solutions in L∞(0,T; Ṽₚ) ∩ L²(0,T; W²,²(ṽΩ)) is established via extension and symmetry.
• The method avoids compatibility conditions in the periodic cube case, as the reflection preserves the required symmetry and boundary behavior.
• The approach is general and applies to any functional space X with sufficient regularity and divergence-free structure, provided the boundary is flat.
• The reflection technique ensures that if a solution exists and satisfies regularity in the whole space, then its restriction to the half-space solves the original boundary value problem with slip conditions.