This paper is devoted to the study of smooth flows of density-dependent fluids in ${ℝ}^N$ or in the torus ${𝕋}^N$. We aim at extending several classical results for the standard Euler or Navier-Stokes equations, to this new framework.Existence and uniqueness is stated on a time interval independent of the viscosity $\mu$ when $\mu$ goes to $0$. A blow-up criterion involving the norm of vorticity in $L^1(0,T;L^{\infty })$ is also proved. Besides, we show that if the density-dependent Euler equations have a smooth solution on a given time...

We formulate the Leray problem for inhomogeneous fluids in two dimensions and outline the proof of the existence of a solution. There are two kinds of results depending on whether the given value for the density is a continuous function or only an $L^{\infty }$ function. In the former case, the given densities are attained in the sense of uniform convergence and in the latter with respect to weak-* convergence.

We consider a free boundary value problem for a viscous, incompressible fluid contained in an uncovered three-dimensional rectangular channel, with gravity and surface tension, governed by the Navier-Stokes equations. We obtain existence results for the linear and nonlinear time-dependent problem. We analyse the qualitative behavior of the flow using tools of bifurcation theory. The main result is a Hopf bifurcation theorem with ${\mathbb{Z}}_k$-symmetry.

We formulate a boundary value problem for the Navier-Stokes equations with prescribed u·n, curl u·n and alternatively (∂u/∂n)·n or curl²u·n on the boundary. We deal with the question of existence of a steady weak solution.

We study the motion of a viscous incompressible fluid filling the whole three-dimensional space exterior to a rigid body, that is rotating with constant angular velocity ω, under the action of external force f. By using a frame attached to the body, the equations are reduced to (1.1) in a fixed exterior domain D. Given f = divF with $F\in BUC(ℝ;L_{3/2,\infty }\left(D\right))$, we consider this problem in D × ℝ and prove that there exists a unique solution $u\in BUC(ℝ;L_{3,\infty }\left(D\right))$ when F and |ω| are sufficiently small. If, in particular, the external force for...

In this paper we prove global existence and uniqueness for solutions of the 3-dimensional Navier-Stokes equations with small initial data in spaces which are Hδi in the i-th direction, δ1 + δ2 + δ3 = 1/2, -1/2 < δi < 1/2 and in a space which is L2 in the first two directions and B2,11/2 in the third direction, where H and B denote the usual homogeneous Sobolev and Besov spaces.

Corresponding to the wellposedness result [2] for the classical 3-D Navier-Stokes equations $\left(N{S}_{\nu }\right)$ with initial data in the scaling invariant Besov space, ${ℬ}_{p,\infty }^{-1+\frac{3}{p}},$ here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations $\left(AN{S}_{\nu }\right),$ where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces, ${ℬ}_4^{-\frac{1}{2},\frac{1}{2}}$ and ${ℬ}_4^{-\frac{1}{2},\frac{1}{2}}\left(T\right).$ Then with initial data in the scaling invariant space ${ℬ}_4^{-\frac{1}{2},\frac{1}{2}},$ we prove the global wellposedness for $\left(AN{S}_{\nu }\right)$ provided the norm of initial data is small enough compared...

The existence of a periodic solution of a nonlinear equation $z^{\text{'}}+A_{0}z+B_{0}z=F$ is proved. The theory developed may be used to prove the existence of a periodic solution of the variational formulation of the Navier-Stokes equations or the equations of magnetohydrodynamics. The proof of the main existence theorem is based on Rothe method in combination with the Galerkin method, using the Brouwer fixed point theorem.

This paper solves the scalar Oseen equation, a linearized form of the Navier-Stokes equation. Because the fundamental solution has anisotropic properties, the problem is set in a Sobolev space with isotropic and anisotropic weights. We establish some existence results and regularities in $L^p$ theory.

We consider the second-order projection schemes for the time-dependent natural convection problem. By the projection method, the natural convection problem is decoupled into two linear subproblems, and each subproblem is solved more easily than the original one. The error analysis is accomplished by interpreting the second-order time discretization of a perturbed system which approximates the time-dependent natural convection problem, and the rigorous error analysis of the projection schemes is...

The classical Stokes system is reconsidered and reformulated in a functional analytical setting allowing for low regularity of the data and the boundary. In fact the underlying domain can be any non-empty open subset Ω of ℝ³. A suitable solution concept and a corresponding solution theory is developed.

In this paper, we propose a topological sensitivity analysis for the Quasi-Stokes equations. It consists in an asymptotic expansion of a cost function with respect to the creation of a small hole in the domain. The leading term of this expansion is related to the principal part of the operator. The theoretical part of this work is discussed in both two and three dimensional cases. In the numerical part, we use this approach to optimize the locations of a fixed number of air injectors in an eutrophized...

In this paper, we propose a topological sensitivity analysis for the Quasi-Stokes equations. It consists in an asymptotic expansion of a cost function with respect to the creation of a small hole in the domain. The leading term of this expansion is related to the principal part of the operator. The theoretical part of this work is discussed in both two and three dimensional cases. In the numerical part, we use this approach to optimize the locations of a fixed number of air injectors in an eutrophized...