These Lecture Notes are devoted to some aspects of the theory of the Navier-Stokes-Fourier system. We shall discuss 1) existence of weak solutions, 2) existence of suitable weak solutions and relative entropies, 3) weak strong uniqueness property in the class of weak solutions. For physical reasons, we shall limit ourselves to the three dimensional physical space, and for the sake of simplicity...

We show that molecular-dynamics simulations can furnish useful boundary conditions at a solid surface bounding a two-component fluid. In contrast to some previous reports, convective-diffusive flow is consistent with continuum equations down to atomic scales. However, concentration gradients can produce flow without viscous dissipation that is inconsistent with the commonly used Navier slip con...

We show that the k-th eigenvalue of the Dirichlet Laplacian is strictly less than the k-th eigenvalue of the classical Stokes operator (equivalently, of the clamped buckling plate problem) for a bounded domain in the plane having a locally Lipschitz boundary. For a C boundary, we show that eigenvalues of the Stokes operator with Navier slip (friction) boundary conditions interpolate continuousl...

We prove the existence and uniqueness of global strong solutions to the one dimensional, compressible Navier-Stokes system for the viscous and heat conducting ideal polytropic gas flow, when heat conductivity depends on temperature in power law of Chapman-Enskog. The results reported in this article is valid for initial boundary value problem with non-slip and heat insulated boundary along with...

An optimal control problem with quadratic cost functional for the steady-state Navier-Stokes equations with no-slip boundary condition is considered. Lipschitz stability of locally optimal controls with respect to certain perturbations of both the cost functional and the equation is proved provided a second-order sufficient optimality condition holds. For a sufficiently small Reynolds number, e...

7 We consider a set of equations modeling contact line dynamics. The model consists of the Navier–Stokes free surface flow with local slip-type boundary conditions and gradient surface ten-9 sion coupled with a mesoscopic local surface model (nonlinear degenerated equations) describes the surface tension variations. The dynamical contact angle and the local surface tension variations are 11 unk...

In order to predict the variations of the channel ionic resistance with time, we follow the framework proposed by Taylor and Aris to analyze our results. We consider here a channel of radius R, length L, and the diffusion coefficient of the salt in the solution is D. Solving the Navier-Stokes equation for a cylindrical channel, under the assumption of no-slip boundary condition, the velocity pr...

The Navier-Stokes equations for the motion of compressible, viscous flows with the no-slip boundary condition are considered. We study the problem of determining the evolution of the interface of discontinuity of a piece-wise smooth density when the interface is in transversal contact with the boundary of the domain. A unique global solution near a static constant equilibrium is constructed. Th...

Abstract We study the Stokes problem for incompressible fluid with mixed nonlinear boundary conditions of subdifferential type. The latter involve a unilateral condition, Navier slip nonmonotone version Navier–Fujita and threshold leak condition frictional weak form leads to new class variational–hemivariational inequalities on convex sets velocity field. Solution existence compactness solution...