# Registered Data

## [00084] Asymptotic approaches to multi-scale PDEs in mathematical physics

**Session Date & Time**:- 00084 (1/2) : 5B (Aug.25, 10:40-12:20)
- 00084 (2/2) : 5C (Aug.25, 13:20-15:00)

**Type**: Proposal of Minisymposium**Abstract**: Nonlinear PDEs play an important role in modelling many important phenomena observed in physics. One of the main challenges is that the physical problem at hand usually manifests its properties on a hierarchy of scales: the behaviour of the system at the large scale can only be understood by accessing a number of finer scales. Discovering the numerous scales in the governing equations and describing the singularities which appear in asymptotic processes give rise to exciting and difficult research problems (e.g. singular limits in fluid mechanics, macroscopic closures of kinetic models, or incompressible limits for tissue growth models).**Organizer(s)**: Tomasz Dębiec, Agnieszka Świerczewska-Gwiazda**Classification**:__35A01__,__35B40__,__35Q35__,__35Q92__**Speakers Info**:- Piotr Gwiazda (Polish Academy of Sciences)
- Noemi David (Sorbonne Université)
- Emil Wiedemann (Ulm University)
- Athanasios Tzavaras (KAUST)
- Piotr Mucha (University of Warsaw)
- Didier Bresch (Université Savoie Mont-Blanc)
- Eric Lars Hientzsch (University of Bielefeld)
- Slim Ibrahim (University of Victoria)

**Talks in Minisymposium**:**[03190] Strong Convergence of Vorticity in the Viscosity Limit****Author(s)**:**Emil Wiedemann**(Universität Erlangen-Nürnberg)

**Abstract**: Consider the 2D incompressible Navier-Stokes equations with initial vorticity in $L^p$ ($1

**[03337] On the asymptotic dynamics of point vortices for the lake equations****Author(s)**:**Lars Eric Hientzsch**(Bielefeld University)- Christophe Lacave (University Grenoble Alpes)
- Evelyne Miot (University Grenoble Alpes)

**Abstract**: The lake equations describe the evolution of the vertically averaged velocity field of an incompressible inviscid 3D fluid in a domain with spatially varying topography (depth). We derive the asymptotic dynamics of point vortices for the lake equations with positive depth, when the vorticity is initially sharply concentrated around $N$ points. More precisely, we show that the vorticity remains concentrated in suitable sense around $N$ points for all times, and that the trajectories follow the level lines of the depth function. This is joint work with Christophe Lacave and Evelyne Miot (Université Grenoble Alpes).

**[03628] Construction of weak solutions to Compressible Navier-Stokes equations****Author(s)**:**Piotr B. Mucha**(University of Warsaw)

**Abstract**: We prove the existence of the weak solutions to the compressible Navier--Stokes system with barotropic pressure $p(\rho)=\rho^\gamma$ for $\gamma \geq 9/5$ in three space dimension. The novelty of the paper is the approximation scheme that instead of the classical regularization of the continuity equation (based on the viscosity approximation $\epsilon \Delta \rho$) uses more direct truncation and regularisaton of nonlinear terms an the pressure. This scheme is compatible with the Bresch-Jabin compactness criterion for the density. We revisit this criterion and prove, in full rigour, that it can be applied in our approximation at any level.

**[04461] From compressible euler equation to porous media****Author(s)**:**Piotr Gwiazda**(Institute of Mathematics of Polish Academy of Sciences)

**Abstract**: We consider a combined system of Euler, Euler–Korteweg and Euler–Poisson equations. We show the existence of dissipative measure-valued solutions in the cases of repulsive and attractive potential in Euler–Poisson system. Furthermore we show that the strong solutions to the Cahn–Hillard–Keller–Segel system are a high-friction limit of the dissipative measure-valued solutions to Euler–Korteweg–Poisson equations.

**[04487] Incompressible limit for tumor growth models with convective effects****Author(s)**:**Noemi David**(Université de Lyon)- Tomasz Dębiec (University of Warsaw)
- Benoit Perthame (Sorbonne Université)
- Markus Schmidtchen (Technische Universität Dresden)

**Abstract**: Both compressible and incompressible models have been used in the literature to describe the mechanical aspects of living tissues. Using a stiff pressure law, it is possible to build a bridge between density-based models and free boundary problems where saturation holds. I will present the study of the incompressible limit for advection-porous medium equations and discuss the convergence rate of solutions of the compressible model to solutions of the limit Hele-Shaw problem.

**[04992] Relative entropy and application to asymptotic limits for bipolar Euler-Poisson systems.****Author(s)**:**Athanasios Tzavaras**(King Abdullah University of Science and Technology (KAUST))- Nuno Alves (King Abdullah University of Science and Technology (KAUST))

**Abstract**: The relative entropy method has been a very effective tool for describing asymptotic limit problems in mechanics and mathematical physics. A formalism for Hamiltonian systems can be easily extended to the system of bipolar Euler-Maxwell equations. We will describe here various results on asymptotic limits from bipolar Euler Poisson to models that are used for the description of plasmas, or (when combined with high-friction limits) to semi-conductors (joint work with Nuno Alves).

**[05460] Local smooth solvability for the Relativistic Vlasov-Maxwell system.****Author(s)**:**Slim IBRAHIM**(UNIVERSITY OF VICTORIA)- Christophe Cheverry (University of Rennes)

**Abstract**: This talk is devoted to the Relativistic Vlasov-Maxwell system in space dimension three. We prove the local smooth solvability for weak topologies (and its long time version for small data). This result is derived from a representation formula decoding how the momentum spreads, and showing that the domain of influence in momentum is controlled by mild information. We do so by developing a Radon Fourier analysis on the RVM system, leading to the study of a class of singular weighted integrals. In the end, we implement our method to construct smooth solutions to the RVM system in the regime of dense, hot and strongly magnetized plasmas. This is done by investigating the stability properties near a class of approximate solutions. This is a joint work with C. Cheverry.