This paper proves a Serrin’s type blow-up criterion for the 3D density-dependent Navier-Stokes-Korteweg equations with vacuum. It is shown that if the density $\rho$ and velocity field $u$ satisfy ${\parallel \nabla \rho \parallel }_{L^{\infty }(0,T;W^{1,q})}+{\parallel u\parallel }_{L^s(0,T;L_{\omega }^r)}<\infty$ for some $q>3$ and any $(r,s)$ satisfying $2/s+3/r\le 1$, $3<r\le \infty ,$ then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally over $[0,T]$. Here $L_{\omega }^r$ denotes the weak $L^r$ space.

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...

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.

Long time existence of solutions to the Navier-Stokes equations in cylindrical domains under boundary slip conditions is proved. Moreover, the existence of solutions with no restrictions on the magnitude of the initial velocity and the external force is shown. However, we have to assume that the quantity $I={\sum }_{i=1}^2\left(\right||{\partial }_{x₃}^i{v\left(0\right)\left|\right|}_{L₂\left(\Omega \right)}+\left|\right|{\partial }_{x₃}^i{f\left|\right|}_{L₂\left(\Omega \times \right(0,T\left)\right)})$ is sufficiently small, where x₃ is the coordinate along the axis parallel to the cylinder. The time of existence is inversely proportional to I. Existence of solutions is proved by...

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 consider the axisymmetric Navier-Stokes equations with non-zero swirl component. By invoking the Hardy-Sobolev interpolation inequality, Hardy inequality and the theory of $*A_{\beta }$ (1 < β < ∞) weights, we establish regularity criteria involving $u^r$, ${\omega }^z$ or ${\omega }^{\theta }$ in some weighted Lebesgue spaces. This improves many previous results.

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...

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 establish the existence of at least three weak solutions for the (p1,…,pₙ)-biharmonic system ⎧$\Delta \left(\right|\Delta u_i{|}^{p-2}\Delta u_i)=\lambda F_{u_i}(x,u₁,\dots ,uₙ)$ in Ω, ⎨ ⎩$u_i=\Delta u_i=0$ on ∂Ω, for 1 ≤ i ≤ n. The proof is based on a recent three critical points theorem.

We show time regularity of weak solutions for unsteady motion equations of generalized Newtonian fluids described by $p(x,t)$-power law for $p(x,t)\ge (3n+2)/(n+2)$, $n\ge 2,$ by using a higher integrability property and fractional difference method. Moreover, as its application we prove that every weak solution to the problem becomes a local in time strong solution and that it is unique.

We study the axisymmetric Navier-Stokes equations. In 2010, Loftus-Zhang used a refined test function and re-scaling scheme, and showed that $$|{\omega }^r(x,t)|+|{\omega }^z(r,t)|\le \frac{C}{r^{10}},0<r\le \frac{1}{2}.$$ By employing the dimension reduction technique by Lei-Navas-Zhang, and analyzing ${\omega }^r$, ${\omega }^z$ and ${\omega }^{\theta }/r$ on different hollow cylinders, we are able to improve it and obtain $$|{\omega }^r(x,t)|+|{\omega }^z(r,t)|\le \frac{C\left|\ln r\right|}{r^{17/2}},0<r\le \frac{1}{2}.$$

Numerical analysis of a model Stokes interface problem with the homogeneous Dirichlet boundary condition is considered. The interface condition is interpreted as an additional singular force field to the Stokes equations using the characteristic function. The finite element method is applied after introducing a regularization of the singular source term. Consequently, the error is divided into the regularization and discretization parts which are studied separately. As a result, error...

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 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...

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...

We study the Cauchy problem for the non-Newtonian incompressible fluid with the viscous part of the stress tensor ${\tau }^V\left(𝕖\right)=\tau \left(𝕖\right)-2{\mu }_1\Delta 𝕖$, where the nonlinear function $\tau \left(𝕖\right)$ satisfies ${\tau }_{ij}\left(𝕖\right)e_{ij}\ge c{\left|𝕖\right|}^p$ or ${\tau }_{ij}\left(𝕖\right)e_{ij}\ge {c\left(\right|𝕖|}^2+{\left|𝕖\right|}^p)$. First, the model for the bipolar fluid is studied and existence, uniqueness and regularity of the weak solution is proved for $p>1$ for both models. Then, under vanishing higher viscosity ${\mu }_1$, the Cauchy problem for the monopolar fluid is considered. For the first model the existence of the weak solution is proved for $p>\frac{3n}{n+2}$, its uniqueness...