On the uniqueness of weak solutions for the 3D Navier-Stokes equations
On the uniqueness problem for quasilinear elliptic equations involving measures.
We discuss the uniqueness of solutions to problems like⎧ λ |u|s-1u - div (|∇u|p-2∇u) = μ on Ω,⎨⎩ u = 0 in ∂Ω,where λ ≥ 0 and μ is a signed Radon measure.
On the validity of Chapman-Enskog expansions for shock waves with small strength.
On the validity of Friedrichs' inequalities.
On the validity of Huygens' principle for second order partial differential equations with four independent variables. II. A sixth necessary condition
On the validity of Huygens' principle for second order partial differential equations with four independent variables. Part I : derivation of necessary conditions
On the vanishing viscosity approximation to the Cauchy problem for a 2 × 2 system of conservation laws
On the vanishing viscosity method for first order differential-functional IBVP
We consider the initial-boundary value problem for first order differential-functional equations. We present the `vanishing viscosity' method in order to obtain viscosity solutions. Our formulation includes problems with a retarded and deviated argument and differential-integral equations.
On the variational inequality approach to compressible flows via hodograph method.
We study the flow of a compressible, stationary and irrotational fluid with wake, in a channel, around a convex symmetric profile, with assigned velocity q-infinity at infinity and q-s < q-infinity at the wake. In particular, we study the regularity of the free boundary (for a problem which has non-constant coefficients), in the hodograph plane.
On the variational principles for partial differential equations
On the Virasoro structure of symmetry algebras of nonlinear partial differential equations.
On the viscous Allen-Cahn and Cahn-Hilliard systems with Willmore regularization
We consider the viscous Allen-Cahn and Cahn-Hilliard models with an additional term called the nonlinear Willmore regularization. First, we are interested in the well-posedness of these two models. Furthermore, we prove that both models possess a global attractor. In addition, as far as the viscous Allen-Cahn equation is concerned, we construct a robust family of exponential attractors, i.e. attractors which are continuous with respect to the perturbation parameter. Finally, we give some numerical...
On the viscous Burgers equation in unbounded domain.
On the volume of intersection of three independent Wiener sausages
Let K be a compact, non-polar set in ℝm, m≥3 and let SKi(t)={Bi(s)+y: 0≤s≤t, y∈K} be Wiener sausages associated to independent brownian motions Bi, i=1, 2, 3 starting at 0. The expectation of volume of ⋂i=13SKi(t) with respect to product measure is obtained in terms of the equilibrium measure of K in the limit of large t.
On the von Neumann problem and the approximate controllability of Stackelberg-Nash strategies for some environmental problems.
Two problems arising in Environment are considered. The first one concerns a conjecture posed by von Neumann in 1955 on the possible modification of the albedo in order to control the Earth surface temperature. The second one is related to the approximate controllability of Stackelberg-Nash strategies for some optimization problems as, for instance, the pollution control in a lake. The results of the second part were obtained in collaboration with Jacques-Lois Lions.
On the -regularity of solutions to systems of differential equations in the case when the equations are constructed from discontinuous functions.
On the wave equation in curved spacetime
On the Wave Equation on a Compact Riemannian Manifold withour Conjugate Points.
On the wave equations in the spacetime of a cosmic string
On the wave equations with memory in noncylindrical domains.