[Paper Review] Free Boundary Problems via Da Prato-Grisvard Theory

Raphaël Danchin, Matthias Hieber+2 more|arXiv (Cornell University)|Nov 16, 2020

Navier-Stokes equation solutions4 citations

TL;DR

This paper develops an L¹-maximal regularity theory for parabolic evolution equations using Da Prato-Grisvard's framework, enabling global-in-time well-posedness for free boundary problems in fluid mechanics. It establishes strong solutions for incompressible viscous fluids and compressible pressureless gases in the half-space with small initial data, leveraging endpoint L¹-regularity for the Stokes and Lamé operators under Neumann-type boundary conditions.

ABSTRACT

An $\mathrm{L}_1$-maximal regularity theory for parabolic evolution equations inspired by the pioneering work of Da Prato and Grisvard is developed. Besides of its own interest, the approach yields a framework allowing global-in-time control of the change of Eulerian to Lagrangian coordinates in various problems related to fluid mechanics. This property is of course decisive for free boundary problems. This concept is illustrated by the analysis of the free boundary value problem describing the motion of viscous, incompressible Newtonian fluids without surface tension and, secondly, the motion of compressible pressureless gases. For this purpose, an endpoint maximal $\mathrm{L}_1$-regularity approach to the Stokes and Lamé systems is developed. It is applied then to establish global, strong well-posedness results for the free boundary problems described above in the case where the initial domain coincides with the half-space, and the initial velocity is small with respect to a suitable scaling invariant norm.

Motivation & Objective

• Address the long-standing open problem of global-in-time existence for free boundary problems in fluid dynamics with infinite depth.
• Develop a functional analytic framework based on L¹-maximal regularity to control the transformation between Eulerian and Lagrangian coordinates.
• Establish strong well-posedness for the motion of viscous incompressible fluids and compressible pressureless gases in the half-space.
• Provide a systematic approach to maximal regularity for the Stokes and Lamé systems in homogeneous Besov spaces with Neumann-type boundary conditions.
• Achieve global existence results under scaling-invariant smallness assumptions on initial data in critical function spaces.

Proposed method

• Adapt Da Prato-Grisvard's theorem on maximal regularity to homogeneous Besov spaces, focusing on L¹-regularity for parabolic systems.
• Develop endpoint L¹-regularity estimates for the Stokes and Lamé operators in the upper half-space with Neumann-type boundary conditions.
• Use real interpolation theory and homogeneous function spaces to characterize the domain of the Stokes operator and its maximal regularity properties.
• Apply product and interpolation estimates for diffeomorphisms to handle nonlinear terms arising from coordinate transformations.
• Construct a fixed-point argument in a critical function space framework to prove global existence for the nonlinear free boundary problems.
• Derive resolvent estimates for the Neumann Laplacian and use them to control the full system via parameter-ellipticity and sectoriality arguments.

Experimental results

Research questions

• RQ1 Can L¹-maximal regularity techniques be extended to handle free boundary problems with infinite depth in fluid mechanics?
• RQ2 What is the role of homogeneous Besov spaces and Neumann-type boundary conditions in achieving global well-posedness for the Stokes system?
• RQ3 How can the transformation between Eulerian and Lagrangian coordinates be globally controlled in free boundary problems?
• RQ4 Under what smallness conditions on initial data does the system admit a global strong solution for incompressible viscous fluids?
• RQ5 Can the same framework be applied to compressible pressureless gases, and what are the structural differences in the analysis?

Key findings

• The paper establishes L¹-maximal regularity for the Stokes operator in homogeneous Besov spaces on the half-space, with estimates uniformly bounded in the spectral parameter.
• Global-in-time strong solutions exist for the free boundary problem of incompressible viscous fluids in the half-space when the initial velocity is small in a scaling-invariant norm.
• The Lamé system in the half-space admits endpoint L¹-regularity, enabling the analysis of compressible pressureless gas flows with global well-posedness under small initial data.
• The transformation from Eulerian to Lagrangian coordinates is globally controlled via the maximal regularity framework, which is essential for handling free boundary evolution.
• Resolvent estimates for the Neumann Laplacian are derived and used to bound the full system, including normal derivatives and higher-order derivatives.
• The fixed-point argument succeeds in the critical function space framework due to the stability of the L¹-maximal regularity estimates under nonlinear perturbations.