On $L^p$-semigroup to Stokes equation with dynamic slip boundary condition in the half-space

1. Problem Statement

The paper investigates the evolutionary Stokes system in a $d$-dimensional half-space $\Omega = \{x \in \mathbb{R}^d : x_d > 0\}$, coupled with a dynamic slip boundary condition on the boundary $\Gamma = \{x_d = 0\}$. The system is given by:
$$\begin{aligned} \partial_t u - \text{div}(2\nu Du) + \nabla \pi &= f, \\ \text{div } u &= 0, \end{aligned}$$
in $(0, T) \times \Omega$, subject to the boundary conditions:
$$\begin{aligned} \beta \partial_t u + \alpha u + 2\nu [(Du)n]_\tau &= \beta h, \\ u \cdot n &= 0, \end{aligned}$$
on $(0, T) \times \Gamma$.

Key Features:

• Dynamic Boundary Condition: Unlike standard slip or no-slip conditions, the boundary equation contains a time derivative ($\partial_t u$), coupling the evolution of the fluid velocity with the boundary dynamics.
• Parameters: $\alpha, \beta, \nu$ are positive constants. The authors set $\beta = \nu = 1$ for simplicity without loss of generality.
• Goal: To establish the $L^p$-theory (for $p \in (1, \infty)$) for this system, specifically proving the existence of an analytic semigroup and deriving optimal regularity estimates for both weak ($W^{1,p}$) and strong ($W^{2,p}$) solutions.

2. Methodology

The authors employ an elementary approach based on Fourier analysis and classical multiplier theorems, avoiding heavy functional analytic machinery like $R$-boundedness or the full $H^\infty$-calculus framework (though the results imply $H^\infty$-calculus properties).

Step-by-Step Methodology:

1. Resolvent Problem: The authors analyze the stationary resolvent problem associated with the system:
   $$(\lambda - \Delta)u + \nabla \pi = f, \quad \text{div } u = 0,$$
   with the dynamic boundary condition $(\lambda + \alpha)u + 2[(Du)n]_\tau = h$.
2. Partial Fourier Transform: They apply the Fourier transform with respect to the tangential variables $x' \in \mathbb{R}^{d-1}$, converting the PDE system into an ODE system in the normal variable $x_d$.
3. Explicit Fundamental Solution:
   • They decouple the system to solve for the normal component $u_d$ and the pressure $\pi$.
   • They derive an explicit formula for the resolvent in the Fourier domain. The solution is expressed in terms of specific multiplier functions $m_i(\lambda, |\xi|, x_d)$ involving the roots of the characteristic equation ($\sqrt{\lambda + |\xi|^2}$ and $|\xi|$).
   • A correction term is added to satisfy the boundary conditions exactly.
4. Multiplier Estimates:
   • The core of the proof lies in estimating the multiplier functions $m_i$.
   • They utilize the classical Mikhlin Multiplier Theorem to establish $L^p$ bounds.
   • They prove that these multipliers belong to the class of bounded analytic functions in a sector $\Sigma_\theta$, effectively establishing $H^\infty$-calculus for the operator $(-\Delta)^{1/2}$ implicitly.
5. Decomposition Strategy: To handle general data, the solution $u$ is decomposed into $u = u_1 + u_2$:
   • $u_1$: Solves the problem with zero boundary data but non-zero source $f$ (standard Dirichlet Stokes problem).
   • $u_2$: Solves the problem with zero source $f$ but non-zero boundary data $h$ (the focus of the new estimates).

3. Key Contributions and Results

A. Optimal Regularity Estimates

The paper establishes sharp estimates for the resolvent problem in terms of data belonging to negative Sobolev or fractional Besov spaces.

• Weak Solutions ($W^{1,p}$):
  For data $f \in W^{-1,p}(\Omega)$ and $h \in B^{-1/p, p}_p(\Gamma)$, there exists a unique weak solution satisfying:
  $$\|u\|_{W^{1,p}(\Omega)} \leq C \left( \|f\|_{W^{-1,p}(\Omega)} + \|h\|_{B^{-1/p, p}_p(\Gamma)} \right).$$
• Strong Solutions ($W^{2,p}$):
  For data $f \in L^p_\sigma(\Omega)$ and $h \in B^{1-1/p, p}_p(\Gamma)$, the solution is strong, and the pressure gradient is in $L^p$:
  $$\|u\|_{W^{2,p}(\Omega)} + \|\nabla \pi\|_{L^p(\Omega)} \leq C \left( \|f\|_{L^p(\Omega)} + \|h\|_{B^{1-1/p, p}_p(\Gamma)} \right).$$
• Non-zero Divergence: The results are extended to cases where $\text{div } u = g \neq 0$, which is crucial for extending these results to general domains via localization.

B. Analytic Semigroup Generation

Corollary 2.1 proves that the system generates an analytic semigroup in the product space $X = L^p_\sigma(\Omega) \times L^p_n(\Gamma)$.

• The domain of the generator $A$ is identified as a subspace of the Besov space $B^{1+1/p-\epsilon, p}_{p; \sigma, n}(\Omega)$.
• This result generalizes previous $L^2$ results to the full $L^p$ scale.

C. Explicit Resolvent Formula

The paper provides an explicit representation of the resolvent operator in the Fourier domain (Equation 19), which is a significant technical achievement for dynamic boundary conditions where such formulas are often intractable.

4. Significance and Impact

1. Filling a Gap in Literature: Prior to this work, the $L^p$-theory for the Stokes system with dynamic slip boundary conditions (containing $\partial_t u$ on the boundary) was largely unexplored, especially in unbounded domains like the half-space. Most existing literature focused on bounded domains or static boundary conditions.
2. Elementary yet Powerful: The authors demonstrate that deep results (analytic semigroups, optimal regularity) can be obtained using classical tools (Fourier transform, Mikhlin theorem) rather than relying on the complex machinery of $R$-boundedness. This makes the theory more accessible and potentially applicable to other coupled systems.
3. Foundation for Nonlinear Problems: The optimal regularity estimates for the stationary problem are a prerequisite for analyzing nonlinear evolutionary problems (e.g., Navier-Stokes with dynamic boundary conditions) via bootstrap arguments. This allows for the study of global attractors and higher regularity of solutions.
4. Extension to General Domains: By explicitly handling non-zero divergence and working in the half-space, the authors lay the groundwork for extending these results to general bounded or unbounded domains using standard localization and perturbation techniques.

Conclusion

Pražák and Zelina successfully establish a comprehensive $L^p$-theory for the Stokes system with dynamic slip boundary conditions. By deriving explicit resolvent formulas and proving optimal regularity estimates using elementary Fourier methods, they confirm the existence of an analytic semigroup and provide the necessary analytical tools to tackle more complex nonlinear fluid dynamics problems involving dynamic boundaries.