Uniform stabilization of 3D Navier-Stokes equations in critical Besov spaces with finite dimensional, tangential-like boundary, localized feedback controllers

Uniform stabilization of Boussinesq systems in critical $\mathbf{L}^q$-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls

We consider the d-dimensional Boussinesq system defined on a sufficiently smooth bounded domain, with homogeneous boundary conditions, and subject to external sources, assumed to cause instability. The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of explicitly constructed, feedback controls, which are localized on an arbitrarily small interior subdomain. In addition, they will …

Finite dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory

We consider the d-dimensional Boussinesq system defined on a sufficiently smooth bounded domain, and subject to a pair $\{ v, \boldsymbol{u} \}$ of controls localized on $\{ \widetildeΓ, ω\}$. Here, $v$ is a scalar Dirichlet boundary control for the thermal equation, acting on an arbitrary small connected portion $\widetildeΓ$ of the boundary $Γ= \partial Ω$. Instead, $\boldsymbol{u}$ is a $d$-dimensional internal control for the fluid equation acting on an arbitrary small collar $ω$ supported by $\widetildeΓ$ (Fig 1). The initial conditions for both fluid and heat equations are taken of low r…

On the mixed problem for the semilinear Darcy-Forchheimer-Brinkman PDE system in Besov spaces on creased Lipschitz domains

The purpose of this paper is to study the mixed Dirichlet-Neumann boundary value problem for the semilinear Darcy-Forchheimer-Brinkman system in $L_p$-based Besov spaces on a bounded Lipschitz domain in ${\mathbb R}^3$, with $p$ in a neighborhood of $2$. This system is obtained by adding the semilinear term $|{\bf u}|{\bf u}$ to the linear Brinkman equation. {First, we provide some results about} equivalence between the Gagliardo and non-tangential traces, as well as between the weak canonical conormal derivatives and the non-tangential conormal derivatives. Various mapping and invertibility p…

On the energy of inviscid singular flows

It is known that the energy of a weak solution to the Euler equation is conserved if it is slightly more regular than the Besov space $B^{1/3}_{3,\infty}$. When the singular set of the solution is (or belongs to) a smooth manifold, we derive various $L^p$-space regularity criteria dimensionally equivalent to the critical one. In particular, if the singular set is a hypersurface the energy of $u$ is conserved provided the one sided non-tangential limits to the surface exist and the non-tangential maximal function is $L^3$ integrable, while the maximal function of the pressure is $L^{3/2}$ integ…

Boundary stabilizing actuators for multi-phase fluids in a channel

This work is devoted to the problem of boundary stabilization of a mixture of two viscous and incompressible fluids in a three dimensional channel-like domain $(x,y,z)\in \mathbb{R}\times (0,1)\times\mathbb{R}$. The model consists of the Navier-Stokes equations, governing the fluid velocity, coupled with a convective Cahn-Hilliard equation for the relative density of the atoms of one of the fluids. The feedback controller, we design here, is with actuation only on the tangential component of the velocity field on both walls $y=0,1$. Moreover, it is given in a simple and explicit form, easy to …

三维半空间中各向异性Navier-Stokes方程解的大时间行为

Large time behavior for solutions to the anisotropic Navier-Stokes equations in a 3D half-space

本研究针对三维半空间各向异性Navier-Stokes方程解的大时间行为，通过分析线性化特性得到非线性解最优衰减估计，揭示速度场第三分量增强耗散机制，引入Besov型空间结合切空间Littlewood–Paley分析解决了非局部算子带来的L¹估计难题。