In this paper we deal with the stationary Navier-Stokes problem in a domain with compact Lipschitz boundary and datum in Lebesgue spaces. We prove existence of a solution for arbitrary values of the fluxes through the connected components of , with possible countable exceptional set, provided is the sum of the gradient of a harmonic function and a sufficiently small field, with zero total flux for bounded.
On the complement of the unit disk we consider solutions of the equations describing the stationary flow of an incompressible fluid with shear dependent viscosity. We show that the velocity field is equal to zero provided and uniformly. For slow flows the latter condition can be replaced by uniformly. In particular, these results hold for the classical Navier-Stokes case.
We prove existence and a representation formula for solutions to the equations describing steady flows of an isothermal, viscous, compressible gas having a positive infimum for the density , moving in an exterior domain, when the speed of the obstacle and the external forces are sufficiently small.
In this paper, we consider the well-known Fattorini’s criterion for approximate controllability of infinite dimensional linear systems of type y′ = Ay + Bu. We precise the result proved by Fattorini in [H.O. Fattorini, SIAM J. Control 4 (1966) 686–694.] for bounded input B, in the case where B can be unbounded or in the case of finite-dimensional controls. More precisely, we prove that if Fattorini’s criterion is satisfied and if the set of geometric multiplicities of A is bounded then approximate...
We describe qualitative behaviour of solutions of the Gross-Pitaevskii equation in 2D in terms of motion of vortices and radiation. To this end we introduce the notion of the intervortex energy. We develop a rather general adiabatic theory of motion of well separated vortices and present the method of effective action which gives a fairly straightforward justification of this theory. Finally we mention briefly two special situations where we are able to obtain rather detailed picture of the vortex...
We study the generalized Oldroyd model with viscosity depending on the shear stress behaving like (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 D axisymmetric Euler equations for initial data lying in critical Besov spaces . 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 .
The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an maximum principle, in the form of a new “log” conservation law which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance . Previous results of this...
We consider the motion of a viscous compressible heat conducting fluid in ℝ³ bounded by a free surface which is under constant exterior pressure. Assuming that the initial velocity is sufficiently small, the initial density and the initial temperature are close to constants, the external force, the heat sources and the heat flow vanish, we prove the existence of global-in-time solutions which satisfy, at any moment of time, the properties prescribed at the initial moment.