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For a bounded domain , we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system , , with , , and very general data classes for , , such that may have no differentiability property. For smooth data we get a large class of unique and regular solutions extending well known classical solution classes, and generalizing regularity results. Moreover, our results are closely related to those of a series of...
We study the local exponential stabilization of the 2D and 3D Navier-Stokes equations in a bounded domain, around a given steady-state flow, by means of a boundary control. We look for a control so that the solution to the Navier-Stokes equations be a strong solution. In the 3D case, such solutions may exist if the Dirichlet control satisfies a compatibility condition with the initial condition. In order to determine a feedback law satisfying such a compatibility condition, we consider an extended...
We study the local exponential stabilization of the 2D and 3D
Navier-Stokes equations in a bounded domain, around a given
steady-state flow, by means of a boundary control. We look for a
control so that the solution to the Navier-Stokes equations be a
strong solution. In the 3D case, such solutions may exist if the
Dirichlet control satisfies a compatibility condition with the
initial condition. In order to determine a feedback law satisfying
such a compatibility condition, we consider an extended...
We consider an implicit fractional step procedure for the time discretization of the non-stationary Stokes equations in smoothly bounded domains of ℝ³. We prove optimal convergence properties uniformly in time in a scale of Sobolev spaces, under a certain regularity of the solution. We develop a representation for the solution of the discretized equations in the form of potentials and the uniquely determined solution of some system of boundary integral equations. For the numerical computation of...
This paper is devoted to the global attractors of the tropical climate model. We first establish the global well-posedness of the system. Then by studying the existence of bounded absorbing sets, the global attractor is constructed. The estimates of the Hausdorff dimension and of the fractal dimension of the global attractor are obtained in the end.
We consider the time-periodic Oseen flow around a rotating body in ℝ³. We prove a priori estimates in -spaces of weak solutions for the whole space problem under the assumption that the right-hand side has the divergence form. After a time-dependent change of coordinates the problem is reduced to a stationary Oseen equation with the additional term -(ω ∧ x)·∇u + ω ∧ u in the equation of momentum where ω denotes the angular velocity. We prove the existence of generalized weak solutions in -space...
For robust discretizations of the Navier-Stokes equations with small viscosity, standard Galerkin schemes have to be augmented by stabilization terms due to the indefinite convective terms and due to a possible lost of a discrete inf-sup condition. For optimal control problems for fluids such stabilization have in general an undesired effect in the sense that optimization and discretization do not commute. This is the case for the combination of streamline upwind Petrov-Galerkin (SUPG) and pressure...
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 maximal regularity of the linearized problem, we can prove a global in time existence and uniqueness theorem by the contraction mapping principle.
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