The mathematical theory of low Mach number flows
The mathematical theory of the passage from compressible to incompressible fluid flow is reviewed.
The mathematics of suspensions: Kac walks and asymptotic analyticity.
The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section
We consider the Euler equations for compressible fluids in a nozzle whose cross-section is variable and may contain discontinuities. We view these equations as a hyperbolic system in nonconservative form and investigate weak solutions in the sense of Dal Maso, LeFloch and Murat [J. Math. Pures Appl.74 (1995) 483–548]. Observing that the entropy equality has a fully conservative form, we derive a minimum entropy principle satisfied by entropy solutions. We then establish the stability of a class...
The modulational regime of three-dimensional water waves and the Davey-Stewartson system
The Mortar finite element method for Bingham fluids
This paper deals with the flow problem of a viscous plastic fluid in a cylindrical pipe. In order to approximate this problem governed by a variational inequality, we apply the nonconforming mortar finite element method. By using appropriate techniques, we are able to prove the convergence of the method and to obtain the same convergence rate as in the conforming case.
The mortar finite element method for Bingham fluids
This paper deals with the flow problem of a viscous plastic fluid in a cylindrical pipe. In order to approximate this problem governed by a variational inequality, we apply the nonconforming mortar finite element method. By using appropriate techniques, we are able to prove the convergence of the method and to obtain the same convergence rate as in the conforming case.
The motion of a fluid in an open channel
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 -symmetry.
The motion of the free surface of a liquid
The nappe profile of a free overfall
The phenomenon of the free overfall at the sharp drop of a channel bed has been deeply investigated experimentally since the pioneering work of Rouse (1933). Its behaviour is well known at least in the usual case of a wide rectangular channel. However, no complete theoretical solution has yet been obtained. Assuming the steady flow to be two-dimensional, irrotational and frictionless, an analytical solution for the flow field is obtained accounting for the presence of two free boundaries. By applying...
The Navier-Stokes equation with inhomogeneous boundary conditions based on vorticity
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.
The Navier-Stokes equations in the critical Morrey-Campanato space.
THE Navier-stokes flow around a rotating obstacle with time-dependent body force
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 , we consider this problem in D × ℝ and prove that there exists a unique solution when F and |ω| are sufficiently small. If, in particular, the external force for...
The normal form of the Navier-Stokes equations in suitable normed spaces
The numerical interface coupling of nonlinear hyperbolic systems of conservation laws : II. The case of systems
We study the theoretical and numerical coupling of two hyperbolic systems of conservation laws at a fixed interface. As already proven in the scalar case, the coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model. We develop a detailed analysis of the coupling in the linear case. In the nonlinear case, we either use a linearized approach or a coupling method based on the solution of a Riemann problem. We...
The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: II. The case of systems
We study the theoretical and numerical coupling of two hyperbolic systems of conservation laws at a fixed interface. As already proven in the scalar case, the coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model. We develop a detailed analysis of the coupling in the linear case. In the nonlinear case, we either use a linearized approach or a coupling method based on the solution of a Riemann problem....
The optimum shape of an hydrofoil with no cavitation
The parabolic-parabolic Keller-Segel equation
I present in this note recent results on the uniqueness and stability for the parabolic-parabolic Keller-Segel equation on the plane, obtained in collaboration with S. Mischler in [11].
The penalty method for the equations of viscoelastic media
The Penetrable sphere with a soft Eccentric core within an Accoustic wave field
The performance of the higher-order radiation condition method for the penetrable cylinder.