Wladimir Neves

Scalar multidimensional conservation laws IBVP in noncylindrical Lipschitz domains

We study the initial-boundary-value problems for multidimensional scalar conservation laws in noncylindrical domains with Lipschitz boundary. We show the existenceuniqueness of this problem for initial-boundary data in L 1 and the flux-function in the class C 1 . In fact, first considering smooth boundary, we obtain the L 1 contraction property, discuss the existence problem and prove it by the Young measures theory. In the end we show how to pass the existence-uniqueness results on to some domains with Lipschitz boundary.

On a generalized Muskat–Brinkman type problem

The original Muskat problem was proposed in 1934 by Muskat [31] to study from Darcy’s law the encroachment of water into an oil sand. Due to applications to oil reservoir, this problem obtains a great practical interest and also, in view of the mathematical difficulties, the well-posedness for the Muskat problem takes attention of many mathematicians. In fact, many important results concerning the Muskat problem were obtained during the last 20 years. Most of existence and uniqueness results are related with the situation when there exists only one moving horizontal interface, that separate two different fluids. Two regimes were found for the Muskat problem: a stable regime, when this horizontal interface is stable under small deviations and an unstable one, which is to say, fingering occurs. The stable regime could be realized, if initially a horizontal interface separates the two fluids with a denser fluid from below and in the presence of gravity force. In the case of the stable regime we can mention the following results: Yi [41], Siegel, Caflisch, Howison [37] shown global-in–time existence for initial data, that is a small perturbation of a flat interface, which separate two fluids. Ambrose [3], Córdoba A., Córdoba D., Gancedo [17], Escher, Matioc [19] proved local-in-time existence and uniqueness of solutions for initial data in the Sobolev spaces. Further global well-posedness results were established by Constantin, Córdoba, Gancedo, Strain [15] for initial data smaller than an explicitly computable constant. Nevertheless the well-posedness of the Muskat problem for general initial data is not known. It is also interesting to mention the existence and non–uniqueness results of weak solutions for the Muskat problem obtained by Córdoba, Faraco, Gancedo [16] and Székelyhidi Jr. [39]. The number of constructed weak solutions, corresponding to the same initial data, is infinite. Indeed, the weak solutions considered have low regularity, and in particular, they do not satisfy a standard

Optimal boundary holes for the Sobolev trace constant

In this paper we study the problem of minimizing the Sobolev trace Rayleigh quotient ‖u‖W1,p(Ω)p/‖u‖Lq(∂Ω)p among functions that vanish in a set contained on the boundary ∂Ω of given boundary measure. We prove existence of extremals for this problem, and analyze some particular cases where information about the location of the optimal boundary set can be given. Moreover, we further study the shape derivative of the Sobolev trace constant under regular perturbations of the boundary set.

Some Remarks on the Lie Derivative and the Pullback Equation for Contact Forms

Abstract Given the contact forms f and g, and the 1-form h, we discuss the existence of a vector field u verifying ℒ u ⁢ ( f ) = d ⁢ ( u ⁢ ⌟ ⁢ f ) + u ⁢ ⌟ ⁢ d ⁢ f = h .

The incompleteness of the born-infeld model for non-linear multi-D Maxwell's equations

{\mathcal{L}_{u}(f\/)=d(u\,\lrcorner\,f\/)+u\,\lrcorner\,df=h.}

Multi-time systems of conservation laws

This is closely related to the pullback equation, where we seek for a diffeomorphism φ satisfying φ ∗ ⁢ ( f ) = g .

A Generalized Buckley-Leverett System

{\varphi^{\ast}(f\/)=g.}

Hyperbolic conservation laws on manifolds. An error estimate for finite volume schemes

We study the Born-Infeld system of conservation laws, which is the most famous model for non-linear Maxwells equations. This system is totally linear degenerated and there exists a conjecture, see Y. Brenier, Hydrodynamic structure of the augmented Born-Infeld equations, Arch. Rational Mech. Anal. 172 (2004), 65-91, that shocks are not allowed to form. In fact, we show that this conjecture is false and that the Born-Infeld model is not complete by itself. It means that a further theory is needed to complete the model.

Wellposedness for stochastic continuity equations with Ladyzhenskaya–Prodi–Serrin condition

Motivated by the work of P.L. Lions and J-C. Rochet [12], concerning multi-time Hamilton-Jacobi equations, we introduce the theory of multi-time systems of conservation laws. We show the existence and uniqueness of solution to the Cauchy problem for a system of multi-time conservation laws with two independent time variables in one space dimension. Our proof relies on a suitable generalization of the Lax-Oleinik formula.