Accueil > Les trimestres du LMB > Printemps 2015 > Colloques, conférences et ateliers > CoToCoLa > **Résumés**

# Résumés

**Debora Amadori** (Université de l’Aquila) : $L^1$ error estimates for balance laws with space-dependent source

**Fabio Ancona** (Université de Padoue) : On quantitative compactness estimates for hyperbolic conservation laws and HJ equations

Inspired by a question posed by Lax, in recent years it has received an increasing attention the study of quantitative compactness estimates for the map $S_t$, $t>0$ that associates to every given initial data $u_0$ the corresponding solution $S_t u_0$ of a conservation law or of a first order Hamilton-Jacobi equation. Estimates of this type play a central roles in various areas of information theory and statistics as well as of ergodic and learning theory. In the present setting, this concept could provide a measure of the order of “resolution” of a numerical method for the corresponding equation. In this talk we shall first review the results obtained in collaboration with O. Glass and K.T. Nguyen, concerning the compactness estimates for solutions to conservation laws. Next, we shall turn to the more recent analysis of the Hamilton-Jacobi equation pursued in collaboration with P. Cannarsa and K.T. Nguyen. A control-type analysis of such equations turns out to be fundamental to establish some of these properties.

**Clément Cancès** (Université Pierre et Marie Curie-Paris VI) : Approximating the solution of Fokker-Planck equation in accordance with its gradient flows structure

**Giuseppe M. Coclite** (Université de Bari) : Hyperbolic-Elliptic models for two-phase flow in porous media

We consider the flow of two-phases, say oil and water, in a porous medium. The classical model of this flow involves a elliptic-hyperbolic system, based on the Darcy’s law. The saturation is governed by a hyperbolic conservation law and pressure obeys an elliptic equation. The problem of existence of global weak solutions for this model is still open. The main difficulty being the lack of regularity of the velocity field.

We propose two ways of modify the Darcy’s law. The resulting models are still hyperbolic-elliptic system but they have a more regular velocity field. We show the existence of global-in-time solutions for those models and compare them.

**Rinaldo M. Colombo** (Université de Brescia) : NonLocal Balance Laws

NonLocal Balance Laws are integro-differential partial differential equations, consisting in a balance law with flow, source, or boundary condition that depend on integrals of the unknown function. These equations naturally arise in mixed hyperbolic-parabolic problems, in applications to vehicular traffic, to crowd dynamics, to granular materials, to structured population dynamics and in other models of industrial interest.

The present talk overviews recent analytical results with reference to the applications that motivated them.

Works in collaboration with F. Marcellini and E. Rossi.

**Gianluca Crippa** (Université de Bâle) : Mixing and loss of regularity for two-dimensional flows

Consider a passive scalar which is advected by a smooth incompressible two-dimensional velocity field. The following question is of interest : Starting from a given initial distribution of the passive scalar, and given a certain energy budget, power budget, or palenstrophy budget, what velocity field best mixes the passive scalar ?

While it is easy to see that under energy bounds perfect mixing can be accomplished in finite time, it has been recently proven that under power bounds the mixing rate is at most exponential in time.

In the talk I will present a construction from a work in progress with Giovanni Alberti (Pisa) and Anna Mazzucato (Penn State), illustrating the optimality of the exponential bound on the mixing rate under power bounds. The velocity field can in fact be required to satisfy $W^{1,p}$ bounds uniformly in time, for any value of $1 \leq p \leq \infty$ : the Lipschitz case is also included. As a consequence, we deduce the existence of Sobolev velocity fields such that any fractional regularity of the initial datum is instantaneously destroyed.

**Olivier Delestre** (Université de Nice Sophia-Antipolis) : A shallow water model for blood flow simulations

We are interested in simulating blood flow in arteries with a one dimensional model. Thanks to recent developments in the analysis of hyperbolic system of conservation laws (in the shallow water equations context), we have adapted a well balanced scheme based on the hydrostatic reconstruction to solve this problem. Hence we have considered an enhanced model which takes into account a variable elasticity of the artery.

**Mauro Garavello** (Université de Milano Bicocca) : Control problems for balance laws

**Shyam S. Ghoshal** (GSSI Gran Sasso Science Institut-Infn) : Optimal Results On TV Bounds For Scalar Conservation Laws With Discontinuous Flux

**Paola Goatin** (Inria Sophia-Antipolis) : Conservation laws with non-local flux in traffic flow modeling

The talk will present some results obtained recently in collaboration with Sebastien Blandin (IBM Research Collaboratory, Singapore) and Sheila Scialanga (Università di Roma I - La Sapienza). We derive the well-posedness of entropy weak solutions of a scalar conservation law with non-local flux arising in traffic flow modeling. The result is obtained providing accurate $L^\infty$, BV and $L^1$ estimates for the sequence of approximate solutions constructed by an adapted Lax-Friedrichs scheme. A higher order scheme is also tested.

**Graziano Guerra** (Université de Milan-Bicocca) : A $1$D compressible—incompressible limit for the $p$-system in the non smooth case

**Piotr Gwiazda** (Université de Varsovie) : Regular Lagrangian flows and size-structure equations

We will consider the transport equation with Sobolev coefficients and a right-hand side in form of an integral operator. Such problem cannot be solved directly by means of renormalization techniques and the essential step is formulating the problem in terms of regular Lagrangian flows. The talk is based on common results with Camillo De Lellis and Agnieszka Swierczewska-Gwiazda.

**Cyril Imbert** (Université Paris-Est Créteil) : Recent results about Hamilton-Jacobi equations on networks

This talk is concerned with Hamilton-Jacobi equations on networks. After recalling the motivation from traffic flow modelling and the general theory developed with R. Monneau, I will present some results about homogenisation, singular perturbation and error estimates for monotone numerical schemes.

**Francois James** (Université d’Orléans) : Blow-up solutions for the aggregation equation

Existence and uniqueness of global in time measure solution for a one dimensional nonlinear aggregation equation are considered. When the selfconsistant interaction potential involved the velocity field is only Lipschitz continuous, blow up of regular solutions occurs. We introduce in this context the notion of duality solutions and investigate their links with gradient flow solutions.

**Grzegorz Karch** (Université de Wroclawski) : Blow up phenomena in conservation laws with fractional Laplacian and nonlocal fluxes

**Frédéric Lagoutière** (Université Paris-Sud) : Ghosts solutions to discretized transport equations

We explore the behaviour of the implicit centered scheme (which is "known" to be stable in the $L^2$ norm) for transport equations, in a bounded interval in dimension one, endowed with, for instance, homogeneous Neumann boundary conditions. The numerical solutions show some surprising, unexpected, periodical in time structures ("ghost solutions"). We provide a proof of this spectacular phenomenon involving the convergence of numerical solutions toward a very unexpected function. For other hyperbolic equations, such as the Burgers’ equation or the system of Euler equations of compressible gaz dynamics, numerical experiments show similar behaviors, for both smooth and discontinuous solutions. This is a joint work with Mélanie Inglard and Hans Henrik Rugh (Université Paris-Sud, Orsay).

**Andrea Marson** (Université de Padoue) : The method of characteristics and control problems for conservation laws

In this talk we want to illustrate some results and give some hints on the use of method of characteristics for determing the attainable set in control problems for conservation laws, involving both boundary and distributed controls.

**Siddhartha Mishra** (ETH Zurich) : Computing measure valued and statistical solutions of systems of conservation laws

**Darko Mitrovic** (Université de Montenegro) : Singular solutions concept for systems of conservation laws

**Evgeniy Yu. Panov** (Université de Novgorod State) : On the Cauchy problem for scalar conservation laws in the class of Besicovitch almost periodic functions

We study the Cauchy problem for a multidimensional scalar conservation law with merely continuous flux functions. In the case when the initial data is a bounded Besicovitch almost periodic function the Kruzhkov entropy solution $u(t,x)$ of this problem is proved to be Besicovitch almost periodic function (with respect to the spatial variables) as well. Moreover, this solution is unique in the Besicovitch space, and the additive subgroup generated by the spectrum of $u(t,\cdot)$ does not increase in time $t$. We also suggest a necessary and sufficient condition (linear nondegeneracy of the resonant flux components) for the decay property of almost periodic entropy solutions, which extends results of [1]. The obtained results admit an interpretation in the framework of conservation laws on the Bohr compactification of $\mathbb{R}^n$.

[1] E.Yu. Panov, On decay of periodic entropy solutions to a scalar conservation law, Annales de l’Institut Henri Poincare (C) Analyse Non Lineaire 30:6 (2013), 997-1007.

**Massimiliano D. Rosini** (Université de Varsovie) : Constrained conservation laws and their applications to traffics

In the first part of the talk we show the stability results obtained for constrained conservation laws. We start by considering the simplest case of a scalar conservation law with a local point constraint. Then we consider the non-local case and finally the case of a $2\times2$ system of conservation laws.

In the second part of the talk we highlight everyday real life experiences that require the theoretical set up of constrained conservation laws. We show then how to apply this theory to vehicular traffics and crowd dynamics. In particular, we show how it is possible to reproduce counter-intuitive phenomena such as Braess’s paradox, faster is slower effect and capacity drop.

**Jacques Sainte-Marie** (Inria Rocquencourt) : An energy-consistent depth-averaged Euler system : derivation, properties and numerical scheme

We present a non-hydrostatic shallow water-type model approximating the incompressible Euler and Navier-Stokes sytems with free surface. The closure relations are obtained by a minimal energy constraint instead of an asymptotic expansion. The model slightly differs from the well-known Green-Naghdi model and is confronted with stationnary and analytical solutions of the Euler system corresponding to rotational flows. In particular, we give time-dependent analytical solutions for the Euler system that are also analytical solutions for the proposed model but that are not solutions of the Green-Naghdi model.

For the proposed model, we derive a numerical scheme based on a projection-correction strategy. The scheme is endowed with properties such that positivity, consistency, well-balancing and discrete entropy inequality. The discrete model is confronted with analytical and experimental test cases.

**Nicolas Seguin** (Université Pierre et Marie Curie-Paris VI) : Systems of conservation laws with convex constraints

A basic notion of systems of conservation laws is the invariance of the set of admissible states. In some cases, physical motivations may lead to add a constraint on the unknown, leading to a new set, but which is not preserved by the semi-group. We present in this talk how to include these constraints in the definition of solution in order to obtain a well-posed problem in the case of Friedrichs systems. After, having in mind models of plasticity, we introduce a new formulation of boundary conditions for Friedrichs systems which we compare to the literature and prove that it leads to a well-posed problem.

**Laura Spinolo** (IMATI-CNR) : A counter-example concerning regularity properties for systems of conservation laws

In 1973 Schaeffer established a result that applies to scalar conservation laws with convex fluxes and can be loosely speaking formulated as follows : for a generic smooth initial datum, the admissible solution is smooth outside a locally finite number of curves in the $ (t, x) $ plane. Here "generic" should be interpreted in a suitable sense, related to the Baire Category Theorem. My talk will aim at discussing a recent counter-example that rules out the possibility of extending Schaeffer’s Theorem to systems of conservation laws. The talk will be based on a joint work with Laura Caravenna.

**Agnieszka Swierczewska-Gwiazda** (Université de Varsovie) : Conservation laws with fluxes discontinuous in the unknown and the spatial variable

We are interested in a scalar balance law with a discontinuous/multivalued flux and dissipative source term. The presented framework includes the fluxes which are discontinuous in the spatial variable and in the unknown function. Under some additional hypothesis on the structure of possible discontinuities, we formulate an appropriate notion of entropy weak solution and establish its existence and uniqueness. We partially follow the idea of entropy measure valued solutions tools and the method of doubling the variables, but on the level of measure valued solutions, introduced by R. DiPerna. The starting point in this framework is the definition of entropy measure valued solutions and the so-called contraction principle, which is satisfied by entropy measure valued solutions. An essential fact for showing existence of entropy weak solutions is the comparison principle and using the semi-Kruzhkov entropies.