We prove long time existence of regular solutions to the Navier-Stokes equations coupled with the heat equation. We consider the system in a non-axially symmetric cylinder, with the slip boundary conditions for the Navier-Stokes equations, and the Neumann condition for the heat equation. The long time existence is possible because the derivatives, with respect to the variable along the axis of the cylinder, of the initial velocity, initial temperature and external force are assumed to be sufficiently...

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 the Leray-Schauder...

Global existence of regular solutions to the Navier-Stokes equations for (v,p) coupled with the heat convection equation for θ is proved in the two-dimensional case in a bounded domain. We assume the slip boundary conditions for velocity and the Neumann condition for temperature. First an appropriate estimate is shown and next the existence is proved by the Leray-Schauder fixed point theorem. We prove the existence of solutions such that $v,\theta \in W_s^{2,1}\left({\Omega }^T\right)$, $\nabla p\in L_s\left({\Omega }^T\right)$, s>2.

We study the long-time behavior of infinite-energy solutions to the incompressible Navier-Stokes equations in a two-dimensional exterior domain, with no-slip boundary conditions. The initial data we consider are finite-energy perturbations of a smooth vortex with small circulation at infinity, but are otherwise arbitrarily large. Using a logarithmic energy estimate and some interpolation arguments, we prove that the solution approaches a self-similar Oseen vortex as $t\to \infty$. This result was obtained in...

We study the first initial boundary value problem for the 2D non-autonomous g-Navier-Stokes equations in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality. The existence of a weak solution to the problem is proved by using the Galerkin method. We then show the existence of a unique minimal finite-dimensional pullback ${}_{\sigma }$-attractor for the process associated to the problem with respect to a large class of non-autonomous forcing terms. Furthermore, when the force is time-independent...

In this survey paper, we are concerned with the zero Mach number limit for compressible viscous flows. For the sake of (mathematical) simplicity, we restrict ourselves to the case of barotropic fluids and we assume that the flow evolves in the whole space or satisfies periodic boundary conditions. We focus on the case of ill-prepared data. Hence highly oscillating acoustic waves are likely to propagate through the fluid. We nevertheless state the convergence to the incompressible Navier-Stokes equations...

In this survey paper, we are concerned with the zero Mach number limit for compressible viscous flows. For the sake of (mathematical) simplicity, we restrict ourselves to the case of barotropic fluids and we assume that the flow evolves in the whole space or satisfies periodic boundary conditions. We focus on the case of ill-prepared data. Hence highly oscillating acoustic waves are likely to propagate through the fluid. We nevertheless state the convergence to the incompressible Navier-Stokes...

This paper presents a model based on spectral hyperviscosity for the simulation of 3D turbulent incompressible flows. One particularity of this model is that the hyperviscosity is active only at the short velocity scales, a feature which is reminiscent of Large Eddy Simulation models. We propose a Fourier–Galerkin approximation of the perturbed Navier–Stokes equations and we show that, as the cutoff wavenumber goes to infinity, the solution of the model converges (up to subsequences) to a weak solution...

This paper presents a model based on spectral hyperviscosity for the simulation of 3D turbulent incompressible flows. One particularity of this model is that the hyperviscosity is active only at the short velocity scales, a feature which is reminiscent of Large Eddy Simulation models. We propose a Fourier–Galerkin approximation of the perturbed Navier–Stokes equations and we show that, as the cutoff wavenumber goes to infinity, the solution of the model converges (up to subsequences) to a weak...

We consider a free interface problem for the Navier-Stokes equations. We obtain local in time unique existence of solutions to this problem for any initial data and external forces, and global in time unique existence of solutions for sufficiently small initial data. Thanks to global in time $L_p-L_q$ maximal regularity of the linearized problem, we can prove a global in time existence and uniqueness theorem by the contraction mapping principle.

Saccular aneurisms, swelling of a blood vessel, are investigated in order (i) to estimate the development risk of the wall lesion, before and after intravascular treatment, assuming that the pressure is the major factor, and (ii) to better plan medical interventions. Numerical simulations, using the finite element method, are performed in three-dimensional aneurisms. Computational meshes are derived from medical imaging data to take into account both between-subject and within-subject anatomical...

Saccular aneurisms, swelling of a blood vessel, are investigated in order (i) to estimate the development risk of the wall lesion, before and after intravascular treatment, assuming that the pressure is the major factor, and (ii) to better plan medical interventions. Numerical simulations, using the finite element method, are performed in three-dimensional aneurisms. Computational meshes are derived from medical imaging data to take into account both between-subject and within-subject anatomical...