[Paper Review] Local energy weak solutions for the Navier-Stokes equations in the half-space

Yasunori Maekawa, Hideyuki Miura+1 more|arXiv (Cornell University)|Nov 13, 2017

Navier-Stokes equation solutions20 references3 citations

TL;DR

This paper establishes the existence of global-in-time local energy weak solutions to the 3D Navier-Stokes equations in the half-space ℝ³₊ with no-slip boundary conditions. By introducing an explicit pressure decomposition into Helmholtz-Leray and harmonic parts due to the boundary, the authors prove existence and further show that such solutions recover the blow-up of the scale-critical L³(ℝ³₊) norm at potential finite-time singularities, extending results of Seregin and Barker.

ABSTRACT

The purpose of this paper is to prove the existence of global in time local energy weak solutions to the Navier-Stokes equations in the half-space $\mathbb R^3_+$. Such solutions are sometimes called Lemarié-Rieusset solutions in the whole space $\mathbb R^3$. The main tool in our work is an explicit representation formula for the pressure, which is decomposed into a Helmholtz-Leray part and a harmonic part due to the boundary. We also explain how our result enables to reprove the blow-up of the scale-critical $L^3(\mathbb R^3_+)$ norm obtained by Barker and Seregin for solutions developing a singularity in finite time.

Motivation & Objective

• To extend the theory of Lemarié-Rieusset-type local energy weak solutions from the whole space ℝ³ to the half-space ℝ³₊.
• To resolve an open problem regarding the existence of such solutions in the half-space, as noted by Barker and Seregin.
• To provide a framework for analyzing singularities via the blow-up of scale-critical norms, particularly L³.
• To establish a representation formula for the pressure that separates the Helmholtz-Leray and harmonic components due to the boundary.
• To reprove the blow-up of the L³(ℝ³₊) norm for solutions developing a finite-time singularity, using the constructed local energy solutions.

Proposed method

• Derive an explicit representation formula for the pressure in the half-space, decomposing it into a Helmholtz-Leray part and a harmonic part arising from the boundary.
• Use the boundary integral representation to control the harmonic part of the pressure via the Dirichlet problem and singular integral estimates.
• Construct local energy weak solutions via a Galerkin-type approximation with cutoff functions and mollification, ensuring the solutions satisfy the local energy inequality.
• Employ the localized energy method and the ε-regularity theory adapted to the half-space, relying on the boundary ε-regularity results from [32, 35, 34].
• Apply the theory of local energy solutions to rederive the blow-up of the L³(ℝ³₊) norm at a potential singularity time T, following the scheme of Seregin [31].
• Use the characterization of L²_uloc,σ(ℝ³₊) via cut-off functions and divergence problems to ensure the solenoidal and boundary conditions are satisfied in the limit.

Experimental results

Research questions

• RQ1 Can local energy weak solutions to the Navier-Stokes equations be constructed globally in time in the half-space ℝ³₊ with no-slip boundary conditions?
• RQ2 Does the pressure in the half-space admit a decomposition into a Helmholtz-Leray part and a harmonic part due to the boundary, and can this be used to control the solution?
• RQ3 Can the blow-up of the scale-critical L³(ℝ³₊) norm at a finite-time singularity be recovered using the constructed local energy weak solutions?
• RQ4 Is the notion of local energy weak solutions in the half-space consistent with existing notions in the literature, such as those of Maremonti and Shimizu?
• RQ5 How does the boundary affect the structure of the pressure and the regularity theory for local energy solutions?

Key findings

• The paper proves the existence of global-in-time local energy weak solutions to the Navier-Stokes equations in the half-space ℝ³₊ with no-slip boundary conditions.
• An explicit pressure representation is derived, decomposing the pressure into a Helmholtz-Leray part and a harmonic part due to the boundary, enabling precise control of the solution.
• The harmonic part of the pressure is shown to be controlled via singular integral estimates, leveraging the boundary geometry.
• The constructed solutions satisfy the local energy inequality and are suitable in the sense of Caffarelli, Kohn, and Nirenberg, enabling ε-regularity estimates in the half-space.
• The theory is applied to reprove the blow-up of the L³(ℝ³₊) norm at a potential singularity time T, confirming the result of Barker and Seregin for q=3.
• The solution class is shown to be consistent with the local limit of rescaled finite-energy solutions near a potential singularity, validating its role in singularity analysis.