[Paper Review] Very weak solutions of the Stokes problem in a convex polygon

Makram Hamouda, Roger Témam+1 more|arXiv (Cornell University)|Dec 2, 2015

Advanced Mathematical Modeling in Engineering17 references3 citations

TL;DR

This paper establishes the existence and uniqueness of very weak solutions to the stationary and evolutionary Stokes problems in two-dimensional convex polygonal domains, proving that solutions in $L^2(\Omega)^2$ exist for arbitrary $L^2$ boundary data satisfying a divergence-free condition. The key contribution is a trace theorem for $L^2$ velocity fields and the use of transposition arguments to define and characterize boundary conditions in low-regularity settings.

ABSTRACT

Motivated by the study of the corner singularities in the so-called cavity flow, we establish in this article, the existence and uniqueness of solutions in $L^2(Ω)^2$ for the Stokes problem in a domain $Ω,$ when $Ω$ is a smooth domain or a convex polygon. We establish also a trace theorem and show that the trace of $u$ can be arbitrary in $L^2(\partialΩ)^2.$ The results are also extended to the linear evolution Stokes problem.

Motivation & Objective

• To establish the existence and uniqueness of very weak solutions to the stationary Stokes problem in convex polygonal domains.
• To develop a trace theory for $L^2$ vector fields satisfying the Stokes equations, enabling the definition of boundary values in low regularity.
• To extend the results to the time-dependent (evolutionary) Stokes problem with $L^2$ boundary data.
• To characterize the tangential and normal components of the velocity trace on the boundary using duality and transposition techniques.
• To provide a rigorous framework for analyzing corner singularities in flows such as the lid-driven cavity problem.

Proposed method

• Use of transposition arguments based on the adjoint Stokes problem to define weak solutions and boundary traces.
• Application of the Lions-Magenes theory for elliptic and parabolic problems to derive a priori estimates in $L^2$ spaces.
• Construction of a trace operator $\gamma_\tau$ mapping $L^2(0,T;L^2(\Omega)^2)$ to $L^2(0,T;H^{-1/2}_\tau(\Gamma)^2)$ for tangential components.
• Proof of density of smooth vector fields in $H^1_0$-type spaces to justify integration by parts in weak formulations.
• Use of compactness and weak convergence arguments to pass to the limit in approximating sequences of boundary data.
• Establishment of a generalized Stokes formula via duality, linking the solution to boundary data through the adjoint system.

Experimental results

Research questions

• RQ1 Can very weak solutions to the Stokes problem be defined and uniquely determined in convex polygonal domains when the boundary data is only in $L^2$?
• RQ2 What trace properties does the velocity field possess when it belongs to $L^2(\Omega)^2$ and satisfies the Stokes equations?
• RQ3 How can the boundary conditions be meaningfully interpreted for $L^2$-regularity solutions with non-smooth data?
• RQ4 Can the existence and uniqueness of solutions be extended to the time-dependent Stokes problem with $L^2$ boundary data?
• RQ5 What is the role of the divergence-free condition on the boundary data in ensuring well-posedness of the very weak formulation?

Key findings

• The paper proves that for any $g \in L^2(\Gamma)^2$ satisfying $\int_\Gamma g \cdot n \, d\Gamma = 0$, there exists a unique solution $u \in L^2(\Omega)^2$ to the Stokes problem with zero body force and zero divergence.
• A trace theorem is established, showing that the tangential component of the velocity trace lies in $H^{-1/2}_\tau(\Gamma)^2$, enabling the definition of boundary values in low regularity.
• For the evolutionary Stokes problem, existence and uniqueness of a solution $u \in L^2(0,T;L^2(\Omega)^2)$ is proven for $g \in L^2(0,T;L^2(\Gamma)^2)$ satisfying the divergence-free condition a.e. in time.
• The solution satisfies the generalized Stokes formula via duality, with $\langle \gamma_\tau u, g_1 \rangle = \int_{Q_T} u \cdot (\partial_t v + \Delta v) \, dxdt$ for appropriate test functions $v$.
• An a priori estimate is derived: $\|u\|_{L^2(\Omega)^2} \leq c_1 \|g\|_{L^2(\Gamma)^2}$, where $c_1$ depends only on the domain $\Omega$.
• The results are extended to domains of polygonal type, including convex polygons, with corner singularities, and are applied to the lid-driven cavity flow as a motivating example.