In this paper we deal with the stationary Navier-Stokes problem in a domain $\Omega$ with compact Lipschitz boundary $\partial \Omega$ and datum $a$ in Lebesgue spaces. We prove existence of a solution for arbitrary values of the fluxes through the connected components of $\partial \Omega$, with possible countable exceptional set, provided $a$ is the sum of the gradient of a harmonic function and a sufficiently small field, with zero total flux for $\Omega$ bounded.

We consider the Keller-Segel-Navier-Stokes system $$\left\{\begin{array}{cc}{n}_t+𝐮·\nabla n=\Delta n-\nabla ·\left(n\nabla v\right),\hfill & x\in \Omega ,t>0,\hfill \\ v_t+𝐮·\nabla v=\Delta v-v+w,\hfill & x\in \Omega ,t>0,\hfill \\ w_t+𝐮·\nabla w=\Delta w-w+n,\hfill & x\in \Omega ,t>0,\hfill \\ {𝐮}_t+(𝐮·\nabla )𝐮=\Delta 𝐮+\nabla P+n\nabla \phi ,\nabla ·𝐮=0,\hfill & x\in \Omega ,t>0,\hfill \end{array}\right.$$ which is considered in bounded domain $\Omega \subset {ℝ}^N$ $(N\in \{2,3\left\}\right)$ with smooth boundary, where $\phi \in C^{1+\delta }\left(\overline{\Omega }\right)$ with $\delta \in (0,1)$. We show that if the initial data $\parallel n_0{\parallel }_{L^{N/2}\left(\Omega \right)}$, $\parallel \nabla v_0{\parallel }_{L^N\left(\Omega \right)}$, $\parallel \nabla w_0{\parallel }_{L^N\left(\Omega \right)}$ and $\parallel {𝐮}_0{\parallel }_{L^N\left(\Omega \right)}$ is small enough, an associated initial-boundary value problem possesses a global classical solution which decays to the constant state $({\overline{n}}_0,{\overline{n}}_0,{\overline{n}}_0,0)$ exponentially with ${\overline{n}}_0:=(1/|\Omega \left|\right){\int }_{\Omega }{n}_0\left(x\right)\mathrm{d}x$.

We study higher local integrability of a weak solution to the steady Stokes problem. We consider the case of a pressure- and shear-rate-dependent viscosity, i.e., the elliptic part of the Stokes problem is assumed to be nonlinear and it depends on $p$ and on the symmetric part of a gradient of $u$, namely, it is represented by a stress tensor $T(Du,p):=\nu (p,|D{|}^2)D$ which satisfies $r$-growth condition with $r\in (1,2]$. In order to get the main result, we use Calderón-Zygmund theory and the method which was presented for...

We consider sequences of solutions of the Navier-Stokes equations in ${ℝ}^3$, associated with sequences of initial data bounded in ${\dot{H}}^{1/2}$. We prove, in the spirit of the work of H.Bahouri and P.Gérard (in the case of the wave equation), that they can be decomposed into a sum of orthogonal profiles, bounded in ${\dot{H}}^{1/2}$, up to a remainder term small in $L^3$; the method is based on the proof of a similar result for the heat equation, followed by a perturbation–type argument. If $𝒜$ is an “admissible” space (in...

We prove the existence of solutions to the evolutionary Stokes system in a bounded domain Ω ⊂ ℝ³. The main result shows that the velocity belongs either to $W_p^{2s+2,s+1}\left({\Omega }^T\right)$ or to $B_{p,q}^{2s+2,s+1}\left({\Omega }^T\right)$ with p > 3 and s ∈ ℝ₊ ∪ 0. The proof is divided into two steps. First the existence in $W_p^{2k+2,k+1}$ for k ∈ ℕ is proved. Next applying interpolation theory the existence in Besov spaces in a half space is shown. Finally the technique of regularizers implies the existence in a bounded domain. The result is generalized to the spaces...

We deal with the steady Stokes problem, associated with a flow of a viscous incompressible fluid through a spatially periodic profile cascade. Using the reduction to domain $\Omega$, which represents one spatial period, the problem is formulated by means of boundary conditions of three types: the conditions of periodicity on curves ${\Gamma }_{-}$ and ${\Gamma }_{+}$ (lower and upper parts of $\partial \Omega$), the Dirichlet boundary conditions on ${\Gamma }_{\mathrm{in}}$ (the inflow) and ${\Gamma }_0$ (boundary of the profile) and an artificial “do nothing”-type boundary...

One shows that the linearized Navier-Stokes equation in $𝒪\subset R^d,d\ge 2$, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller ${\displaystyle V(t,\xi )=\sum _{i=1}^N{V}_i\left(t\right){\psi }_i\left(\xi \right){\dot{\beta }}_i\left(t\right)}$, $\xi \in 𝒪$, where ${\left\{{\beta }_i\right\}}_{i=1}^N$ are independent Brownian motions in a probability space and ${\left\{{\psi }_i\right\}}_{i=1}^N$ is a system of functions on $𝒪$ with support in an arbitrary open subset ${𝒪}_0\subset 𝒪$. The stochastic control input ${\left\{V_i\right\}}_{i=1}^N$ is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability...

Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinder Ω = Σ × ℝ with $\Sigma \subset {ℝ}^{n-1}$, a bounded domain of class $C^{1,1}$, are obtained in the space $L^q(ℝ;L²\left(\Sigma \right))$, q ∈ (1,∞). As a preparation, spectral decompositions of vector-valued homogeneous Sobolev spaces are studied. The main theorem is proved using the techniques of Schauder decompositions, operator-valued multiplier functions and R-boundedness of operator families.

We examine the regularity of solutions to the Stokes system in a neighbourhood of the distinguished axis under the assumptions that the initial velocity v₀ and the external force f belong to some weighted Sobolev spaces. It is assumed that the weight is the (-μ )th power of the distance to the axis. Let $f\in L_{2,-\mu }$, $v₀\in H_{-\mu }¹$, μ ∈ (0,1). We prove an estimate of the velocity in the $H_{-\mu }^{2,1}$ norm and of the gradient of the pressure in the norm of $L_{2,-\mu }$. We apply the Fourier transform with respect to the variable along...

We establish the local-in-time existence of a solution to the non-resistive magneto-micropolar fluids with the initial data $u_0\in H^{s-1+\epsilon }$, $w_0\in H^{s-1}$ and $b_0\in H^s$ for $s>\frac{3}{2}$ and any $0<\epsilon <1$. The initial regularity of the micro-rotational velocity $w$ is weaker than velocity of the fluid $u$.

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...