The Hydrostatic Approximation for the Primitive Equations by the Scaled Navier-Stokes Equations under the No-Slip Boundary Condition

In this paper we justify the hydrostatic approximation of the primitive equations in the maximal $L^p$-$L^q$-setting in the three-dimensional layer domain \Omega = \Torus^2 \times (-1, 1) under the no-slip (Dirichlet) boundary condition in any time interval $(0, T)$ for $T>0$. We show that the solution to the scaled Navier-Stokes equations with Besov initial data $u_0 \in B^{s}_{q,p}(\Omega)$ for $s > 2 - 2/p + 1/ q$ converges to the solution to the primitive equations with the same initial data in $\mathbb{E}_1 (T) = W^{1, p}(0, T ; L^q (\Omega)) \cap L^p(0, T ; W^{2, q} (\Omega))$ with order $O(\epsilon)$ where $(p,q) \in (1,\infty)^2$ satisfies \frac{1}{p} \leq \min \bracket{ 1 - 1/q, 3/2 - 2/q }. The global well-posedness of the scaled Navier-Stokes equations in $\mathbb{E}_1 (T)$ is also proved for sufficiently small $\epsilon>0$. Note that $T = \infty$ is included.Comment: 24page

Boussinesq Equations with Partial or Fractional Dissipation

The two-dimensional (2D) incompressible Boussinesq system is not only an important model in geophysics, but also retains some key features of the 3D Euler and Navier-Stokes equations such as the vortex stretching mechanism. Especially, the inviscid 2D Boussinesq equations are identical to the Euler equations for the 3D axisymmetric swirling flows. Even though the global regularity of full dissipative Boussinesq equations is well known, the global regularity problem of inviscid case is still left open. First, we prove the global existence and uniqueness of 2D Boussinesq equations with partial dissipation in bounded main with Navier type boundary conditions. Secondly, we investigate Boussinesq equations with fractional dissipation on a d-dimensional periodic domain, and apply a re-developed tool of LittlewoodPaley decomposition to achieve global existence and uniqueness of weak solutions. Lastly, we focus on several variants of the 2D incompressible Euler equations. It is not known whether global well-posedness result would hold if there is only partially damping term for 2D Euler equation. Besides, in the vorticity equations, the partially damping term becomes a non-local operator \mathcal R_2^2 \omega. Our numerical simulations show that by replacing \mathcal R_2^2 \omega with different operators (e.g. \mathcal R_1\mathcal R_2 \omega), the solutions will behave quite differently

Rotation-Based Mixed Formulations for an Elasticity-Poroelasticity Interface Problem

In this paper we introduce a new formulation for the stationary poroelasticity equations written using the rotation vector and the total fluid-solid pressure as additional unknowns, and we also write an extension to the elasticity-poroelasticity problem. The transmission conditions are imposed naturally in the weak formulation, and the analysis of the effective governing equations is conducted by an application of Fredholm's alternative. We also propose a monolithically coupled mixed finite element method for the numerical solution of the problem. Its convergence properties are rigorously derived and subsequently confirmed by a set of computational tests that include applications to subsurface flow in reservoirs as well as to dentistry-oriented problems.Fondo Nacional de Desarrollo Científico y Tecnológico/[11160706]/FONDECYT/ChilePrograma Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia/[AFB170001]/PIA/ChileUCR::Sedes Regionales::Sede de OccidenteUCR::Vicerrectoría de Docencia::Ciencias Básicas::Facultad de Ciencias::Escuela de Matemátic

Optimal control of the two-dimensional evolutionary Navier-Stokes equations with measure valued controls

In this paper, we consider an optimal control problem for the two-dimensional evolutionary Navier-Stokes system. Looking for sparsity, we take controls as functions of time taking values in a space of Borel measures. The cost functional does not involve directly the control but we assume some constraints on them. We prove the well-posedness of the control problem and derive necessary and sufficient conditions for local optimality of the controls.The first author was supported by Spanish Ministerio de Economía, Industria y Competitividad under research project MTM2017-83185-P. The second author was supported by the ERC advanced grant 668998 (OCLOC) under the EU's H2020 research program