The existence of global regular axially symmetric solutions to Navier-Stokes equations in a bounded cylinder and for boundary slip conditions is proved. Next, stability of these solutions is shown.

Motivated by [10], we prove that the upper bound of the density function $\rho$ controls the finite time blow up of the classical solutions to the 2-D compressible isentropic Navier-Stokes equations. This result generalizes the corresponding result in [3] concerning the regularities to the weak solutions of the 2-D compressible Navier-Stokes equations in the periodic domain.

This paper is devoted to the study of the incompressible Navier-Stokes equations with mass diffusion in a bounded domain in R³ with C³ boundary. We prove the existence of weak solutions, in the large, and the behavior of the solutions as the diffusion parameter λ → 0. Moreover, the existence of L²-strong solution, in the small, and in the large for small data, is proved. Asymptotic regularity (the regularity after a finite period) of a weak solution is studied. Finally, using the Dore-Venni theory,...

The existence for the Cauchy-Neumann problem for the Stokes system in a bounded domain $\Omega \subset {ℝ}^3$ is proved in a class such that the velocity belongs to $W_r^{2,1}\left(\Omega \times (0,T)\right)$, where r > 3. The proof is divided into three steps. First, the existence of solutions is proved in a half-space for vanishing initial data by applying the Marcinkiewicz multiplier theorem. Next, we prove the existence of weak solutions in a bounded domain and then we regularize them. Finally, the problem with nonvanishing initial data is considered....

We consider a two-dimensional Navier-Stokes shear flow with time dependent boundary driving and subject to Tresca law. We establish the existence of a unique global in time solution and then, using a recent method based on the concept of the Kuratowski measure of noncompactness of a bounded set, we prove the existence of the pullback attractor for the associated cocycle. This research is motivated by a problem from lubrication theory.

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 $W_p^{2s,s}\left({\Omega }^T\right)$ and $B_{p,q}^{2s,s}$ with...

We study the generalized Oldroyd model with viscosity depending on the shear stress behaving like ${\mu \left(D\right)\sim \left|D\right|}^{p-2}$ (p > 6/5), regularized by a nonlinear stress diffusion. Using the Lipschitz truncation method we prove global existence of a weak solution to the corresponding system of partial differential equations.

This paper deals with the global well-posedness of the $3$D axisymmetric Euler equations for initial data lying in critical Besov spaces $B_{p,1}^{1+\frac{3}{p}}$. In this case the BKM criterion is not known to be valid and to circumvent this difficulty we use a new decomposition of the vorticity .

We study the $N$-dimensional Boussinesq system with dissipation and diffusion generalized in terms of fractional Laplacians. In particular, we show that given the critical dissipation, a solution pair remains smooth for all time even with zero diffusivity. In the supercritical case, we obtain component reduction results of regularity criteria and smallness conditions for the global regularity in dimensions two and three.

We consider the spatial behavior of the velocity field $u(x,t)$ of a fluid filling the whole space ${ℝ}^n$ ($n\ge 2$) for arbitrarily small values of the time variable. We improve previous results on the spatial spreading by deducing the necessary conditions $\int u_h(x,t)u_k(x,t)\mathrm{d}x=c\left(t\right){\delta }_{h,k}$ under more general assumptions on the localization of $u$. We also give some new examples of solutions which have a stronger spatial localization than in the generic case.

We consider the spatial behavior of the velocity field u(x, t) of a fluid filling the whole space ${}^n$ ($n\ge 2$) for arbitrarily small values of the time variable. We improve previous results on the spatial spreading by deducing the necessary conditions $\int u_h(x,t)u_k(x,t)\mathrm{d}x=c\left(t\right){\delta }_{h,k}$ under more general assumptions on the localization of u. We also give some new examples of solutions which have a stronger spatial localization than in the generic case.

We establish regularity results up to the boundary for solutions to generalized Stokes and Navier–Stokes systems of equations in the stationary and evolutive cases. Generalized here means the presence of a shear dependent viscosity. We treat the case $p\ge 2$. Actually, we are interested in proving regularity results in $L^q\left(\Omega \right)$ spaces for all the second order derivatives of the velocity and all the first order derivatives of the pressure. The main aim of the present paper is to extend our previous scheme, introduced...

We consider a model for the viscoelastic fluid which has recently been studied in [4] and [1]. We show the local-in-time existence of a strong solution to the corresponding system of partial differential equations under less regularity assumptions on the initial data than in the above mentioned papers. The main difference in our approach is the use of the $L^p$ theory for the Stokes system.

We consider the problem of motion of several rigid bodies in a viscous fluid. Both compressible and incompressible fluids are studied. In both cases, the existence of globally defined weak solutions is established regardless possible collisions of two or more rigid objects.

This article deals with the solvability of the boundary-value problem for the Navier-Stokes equations with a direction-dependent Navier type slip boundary condition in a bounded domain. Such problems arise when steady flows of fluids in domains with rough boundaries are approximated as flows in domains with smooth boundaries. It is proved by means of the Galerkin method that the boundary-value problem has a unique weak solution when the body force and the variability of the surface friction are...