nan Adimurthi

Optimal entropy solutions for conservation laws with discontinuous flux-functions

We deal with a single conservation law in one space dimension whose flux function is discontinuous in the space variable and we introduce a proper framework of entropy solutions. We consider a large class of fluxes, namely, fluxes of the convex-convex type and of the concave-convex (mixed) type. The alternative entropy framework that is proposed here is based on a two step approach. In the first step, infinitely many classes of entropy solutions are defined, each associated with an interface connection. We show that each of these class of entropy solutions form a contractive semigroup in

EXISTENCE AND STABILITY OF ENTROPY SOLUTIONS FOR A CONSERVATION LAW WITH DISCONTINUOUS NON-CONVEX FLUXES

L^1

Convergence of Godunov type methods for a conservation law with a spatially varying discontinuous flux function

and is hence unique. Godunov type schemes based on solutions of the Riemann problem are designed and shown to converge to each class of these entropy solutions. The second step is to choose one of these classes of solutions. This choice depends on the Physics of the problem being considered and we concentrate on the model of two-phase flows in a heterogeneous porous medium. We define an optimizationproblem on the set of admissible interface connections and show the existence of an unique optimal connection and its corresponding optimal entropy solution. The optimal entropy solution is consistent with the expected solutions for two-phase flows in heterogeneous porous media.

The DFLU flux for systems of conservation laws

We consider a scalar conservation law with a discontinuous flux function. The fluxes are non-convex, have multiple points of extrema and can have arbitrary intersections. We propose an entropy formulation based on interface connections and associated jump conditions at the interface. We show that the entropy solutions with respect to each choice of interface connection exist and form a contractive semi-group in

STRUCTURE OF ENTROPY SOLUTIONS TO SCALAR CONSERVATION LAWS WITH STRICTLY CONVEX FLUX

L^1

FINER REGULARITY OF AN ENTROPY SOLUTION FOR 1-D SCALAR CONSERVATION LAWS WITH NON UNIFORM CONVEX FLUX

. Existence is shown by proving convergence of a Godunov type scheme by a suitable modification of the singular mapping approach. This extends the results of [3] to the general case of non-convex flux geometries.

Optimal controllability for scalar conservation laws with convex flux

We deal with single conservation laws with a spatially varying and possibly discontinuous coefficient. This equation includes as a special case sin-gle conservation laws with conservative and possibly singular source terms. We extend the framework of optimal entropy solutions for these classes of equations based on a two-step approach. In the first step, an interface connection vector is used to define infinite classes of entropy solutions. We show that each of these classes of solutions is stable in L1 . This allows for the possibility of choosing one of these classes of solutions based on the physics of the problem. In the second step, we define optimal entropy solutions based on the solution of a certain optimization problem at the discontinuities of the coefficient. This method leads to optimal entropy solutions that are consistent with physically observed solutions in two-phase flows in heterogeneous porous media. Another central aim of this paper is to develop suitable numerical schemes for nthese equations. We develop and analyze a set of Godunov type finite volume methods that are based on exact solutions of the corresponding Riemann problem. Numerical experiments are shown comparing the performance of these nschemes on a set of test problems.

Conservation law with discontinuous flux

The DFLU numerical flux was introduced in order to solve hyperbolic scalar conservation laws with a flux function discontinuous in space. We show how this flux can be used to solve a certain class of systems of conservation laws such as systems modeling polymer flooding in oil reservoir engineering. Furthermore, these results are extended to the case where the flux function is discontinuous in the space variable, such a situation arises for example while dealing with oil reservoirs which are heterogeneous. Numerical experiments are presented to illustrate the efficiency of this new scheme compared to other standard schemes like upstream mobility, Lax-Friedrichs and FORCE schemes.

An improved Hardy-Sobolev inequality in W1,p and its application to Schrödinger operators

We consider scalar conservation laws in one space dimension with convex flux and we establish a new structure theorem for entropy solutions by identifying certain shock regions of interest, each of them representing a single shock wave at infinity. Using this theorem, we construct a smooth initial data with compact support for which the solution exhibits infinitely many shock waves asymptotically in time. Our proof relies on Lax–Oleinik explicit formula and the notion of generalized characteristics introduced by Dafermos.

Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux

Consider a scalar conservation law in one space dimension with initial data nin LI : If the flux f is in C 2 and locally uniformly convex, then for all t > 0, the nentropy solution is locally in BV (functions of bounded variation) in space variable. nIn this case it was shown in [5], that for all most every t > 0, locally, the solution is in nSBV (Special functions of bounded variations). Furthermore it was shown with an nexample that for almost everywhere in t > 0 cannot be removed. This paper deals nwith the regularity of the entropy solutions of the strict convex C 1 flux f which need nnot be in C 2 and locally uniformly convex. In this case, the entropy solution need not nbe locally in BV in space variable, but the composition with the derivative of the flux nfunction is locally in BV. Here we prove that, this composition is locally is in SBV on nall most every t > 0. Furthermore we show that this is optimal.