## Université Paris Diderot |
## Investissements d'avenir, Université Sorbonne Paris Cité |
## Université Paris Descartes |

## Workshop on kinetic and fluid Partial Differential Equations## Univ. Paris Descartes, Wednesday 7th March 2018## Univ. Paris Diderot, Thursday 8th March and Friday 9th March 2018 |

**Abstracts**

**Claude
Bardos : **The Kolmogorov 1/3
Law, the Onsager Conjecture and the Kato Criteria for zero
viscosity limit

A report on ongoing work with Edriss Titi and Emile Wiedemann. The notion of weak convergence for Navier-Stokes and Euler equations presents many similarities with the notion of average in the statistical theory of turbulence. In particular the counterpart of Kolmogorov 1/3 law was the Onsager ``Hölder 1/3" conjecture. Around 1994 it was proven by Constantin, E and Titi that the 1/3 Hölder regularity impliesthe conservation of energy. The interest of such issue was recently emphasized by the construction, through convex integration, of wild solutions of Hölder regularity less than 1/3 that do not conserve the energy (Issett, Buckmaster, De Lellis, Szekelyhidi Jr., and Vicol). However one can also observe very easily the existence of some other type of solutions not regular at all that do conserve the energy. Hence the two points of view are not complementary. But in the zero viscosity limit in the presence of a no slip boundary one observes, thanks to a theorem of Kato, that the conservation of Hölder 1/3 regularity then is equivalent to the absence of anomalous energy dissipation. This will be the object of my talk which requires the treatement of the Onsager condition in the presence of boundary and several extensions of the theorem of Kato.

**Jacob
Bedrossian **: Landau damping and nonlinear echoes

In this talk we will discuss some of the intricacies of Landau damping in the collisionless Vlasov equations or the collisionless limits of Vlasov-Fokker-Planck equations. We will discuss the construction of solutions to the Vlasov-Poisson equations on S x R which are arbitrarily close to homogeneous equilibrium in Sobolev regularity but which display arbitrarily long sequences of nonlinear oscillations known as plasma echoes. In particular, these oscillations show that the collisionless linearization is not valid for long times in Sobolev regularity. Further, we show that the inclusion of weak collisional effects suppress these plasma echoes and make it possible to obtain Sobolev regularity results. We also prove that Debye shielding and dispersive effects can suppress such nonlinear oscillations (joint with Nader Masmoudi and Clement Mouhot). Combined with the existing infinite regularity results of Mouhot and Villani, these results together confirm and refute a variety of conjectures made by both mathematicians and physicists over the years regarding Landau damping near homogeneous equilibrium.

**Marc
Briant :** From
Boltzmann to Incompressible Navier-Stokes: Hydrodynamical
Limits and Speed of Convergence.

If one looks at the microscopic movements of particles in a gas or a fluid, they are ruled by Newton's laws. However, several different equations can be used to model the general behaviors : Euler, Navier-Stokes, Boltzmann among others.

This talk will begin with a discussion about the mathematical coherence that exists between the latter modellings. Then I will focus on the rigorous derivation of the Incompressible Navier-Stokes fluid equation from the Boltzmann equation for gases.

Along the way I will present a convergence result based on a hypocoercive property of the linear Boltzmann operator and I will conclude by presenting how one can obtain convergence rates between the two models thanks to Fourier transform and the spectrum of the linear operator.

**Michaël
Goldman : **A
variational approach to regularity theory for the Monge-Ampere
equation

In this talk I will present a new proof of the partial regularity of optimal transport maps. As opposed to the previous proof of Figalli and Kim which was using Caffarelli's approach to regularity of solutions of Monge-Ampere equations via maximum principles arguments, our proof is variational in nature. By using the fluid-dynamic formulation of optimal transportation (which usually goes by the name of Benamou-Brenier formulation) we prove that at every scale, the optimal transport map is close to the gradient of an harmonic function. This allows us to set up a Campanato iteration scheme to obtain the desired regularity.

This is joint work with F. Otto.

**Bérénice
Grec :** Diffusion
models for mixtures using a stiff dissipative hyperbolic
formalism

In this talk, we are interested in a system
of fluid equations for mixtures with a stiff relaxation term of
Maxwell-Stefan diffusion type. We use the formalism developed by
Chen, Levermore, Liu (CPAM, '94) to obtain a limit system of Fick
type where the species velocities get close to a bulk velocity when
the relaxation parameter remains small.

It is a joint work with Laurent Boudin and Vincent Pavan.

**Piotr
Gwiazda:** Dissipative measure valued
solutions for general hyperbolic conservation laws

In the last years measure-valued solutions started to be considered as a relevant notion of solutions if they satisfy the so-called measure-valued -- strong uniqueness principle. This means that they coincide with a strong solution emanating from the same initial data if this strong solution exists. Following result of Yann Brenier, Camillo De Lellis and Laszlo Szekelyhidi Jr. for incompresible Euler Equation, this property has been examined for many systems of mathematical physics, including incompressible and compressible Euler system, compressible Navier-Stokes system, polyconvex elastodynamics et al. One observes also some results concerning general hyperbolic systems. Our goal is to provide a unified framework for general systems, that would cover the most interesting cases of systems. Additionaly following result of Eduard Feireisl,Piotr Gwiazda, Agnieszka Swierczewska-Gwiazda and Emil Wiedemann for compresible Euler (and Navier-Stokes) system we introduce a new concept of dissipative measure valued solution to general hyperbolic system.

The talk is based on joint results with Ondrej Kreml and Agnieszka Swierczewska-Gwiazda.

References:

[1] Brenier, Y., De Lellis, C.,
Székelyhidi Jr., L., Weak-strong uniqueness for
measure-valued solutions. *Comm.
Math. Phys.,* **305**(2),
351-361 (2011)

[2] S. Demoulini, D. M. A Stuart, and
A. Tzavaras, Weak-strong uniqueness of
dissipative measure-valued solutions for polyconvex elastodynamics,
*Arch. Ration. Mech. Anal.,* **205** (2012),
no. 3, 927-961

[3] E. Feireisl, P. Gwiazda, A.
Swierczewska-Gwiazda and Emil Wiedemann, Dissipative measure-valued
solutions to the compressible Navier-Stokes system, *Calc.
Var. Partial Differential Equations*,
**55**
(2016), no. 6,
55-141

[4] P. Gwiazda, A. Swierczewska-Gwiazda
and E. Wiedemann, Weak-Strong Uniqueness
for Measure-Valued Solutions of Some Compressible Fluid Models,
*Nonlinearity*
**28**
(2015), no. 11,
3873-3890.

[5] T. Debiec, P. Gwiazda, K. Lyczek, A. Swierczewska-Gwiazda, A tribute to conservation of energy for weak solutions, to appear in Topol. Methods Nonlinear Anal.

**Frédéric
Hérau :** hypocoercive
schemes for thediscrete Fokker-Planck equation.

So-called
hypocoercive methods give e.g. Exponential time return to the
equilibrium for a large class of inhomogeneous kinetic equations.
They are based on commutator identities of microlocal inspiration. We
show in this talk that these methods are suffiently robust to get
also exponential return to the equilibrium for semi-discrete or fully
discrete kinetic schemes associated to the Fokker-Planck equation,
although the notions of commutators or even of equilibrium are then
ambiguous.

This is a joint work with Pauline
Laffite (Centrale-Supelec) et Guillaume Dujardin (INRIA and
University of Lille)

**Shi
Jin **: Stochastic Asymptotic-Preserving Schemes and Hypocoercivity
Based Sensitivity Analysis for Multiscale Kinetic Equations with
Random Inputs

In this talk we will study kinetic equations with multiple scales and random uncertainties from initial data and/or collision kernel. Here the multiple scales, characterized by the Knudsen number, will lead the kinetic equations to hydrodynamic (Euler, incompressible Navier-Stokes or diffusion) equations as the Knudsen number goes to zero. Asymptotic-preserving schemes, which minic the asymptotic transitions from the microscopic to the macroscopic scales at the discrete level, have been shown to be effective to deal with multiscale problems in the deterministic setting.

We first extend the prodigm of asymptotic-preserving schemes to the random kinetic equations, and show how it can be constructed in the setting of the stochastic Galerkin approximations. We then extend the hypocoercivity theory, developed for deterministic kinetic equations, to the random case, and establish in the random space regularity, long-time sensitivity analysis, and uniform (in Knudsen number) spectral convergence of the stochastic Galerkin methods, for general linear and nonlinear random kinetic equations in various asymptotic-including the diffusion, incompressible Navier-Stokes, high-field, and acoustic regimes.

**Clément
Mouhot :** TBA

**Sárka
Necasová : **Viscous
compressible fluids in time dependent domain

We consider the compressible Navier-Stokes system on time-dependent domains, supplemented with slip boundary conditions. First the global-in-time weak solutions are obtained, see [1]. For both the no-slip boundary conditions as well as slip boundary conditions we prove local-in-time existence of strong solutions. These results are obtained using a transformation of the problem to a fixed domain and an existence theorem for Navier-Stokes like systems with lower order terms and perturbed boundary conditions. We also show the weak-strong uniqueness principle for slip boundary conditions which remained so far open question see [2].

References:

[1] E. Feireisl, O. Kreml, S. Necasová,
J. Neustupa, J. Stebel: Weak solutions to the barotropic
Navier-Stokes system with slip boundary conditions in time dependent
domains. *J. Differential Equations*, **254**, (2013), no. 1,
125--140.

[2] O. Kreml, S. Necasová, T. Piasecki: Local existence of strong solutions and weak-strong uniqueness for the compressible Navier-Stokes system on moving domains, Submitted.

**Charlotte
Perrin:** Macroscopic
systems with maximum packing constraint

This talk addresses the mathematical analysis of fluid models including a maximum packing constraint. These equations arise naturally for instance in the modeling of mixtures like suspensions. I will present recent results on two classes of PDEs systems which correspond to two modeling approaches: the "soft" approach based on compressible equations with singular constitutive laws (pressure and/or viscosities) close to the maximal constraint; and the "hard" approach based on a free boundary problem between a congested domain with incompressible dynamics and a free domain with compressible dynamics.

**Mario
Pulvirenti :** Propagation
of chaos for a model with topological interaction

We
consider an alignment model with a topological interaction. It is
possible to consider a kinetic description arising from a mean-field
scaling. Starting from the microscopic dynamics of the multiagent
system, we want to prove propagation of chaos to derive rigorously
the kinetic equation which was heuristically found by A. Blanchet and
P. Degond.

This is a research in collaboration
with P. Degond.

**Michael
Renardy :** On
controllability of linear viscoelastic flows

Controllability
is the ability to steer a system from a given initial state to a
desired final state. In this talk, we consider linear viscoelastic
flow in a bounded domain. The control is a body force acting in a
subdomain. We consider Maxwell or Jeffreys models with several,
possibly infinitely many, relaxation modes. Results on approximate
null controllability of the stresses as well as the motion are
established. We also show that exact null controllability is not
possible.

Nonlinear viscoelastic flows are
not controllable. It then becomes a challenging and mostly unsolved
problem how to characterize the states to which a flow can be
controlled.

**Jose
Rodrigo**: Smooth Solutions to a family of Prandtl-like solutions
arising from fluid mechanics

In this talk I will describe joint works with Charles Fefferman and Calvin Khor (independently) concerning families of equations arising from the Surface Quasi-Geostrophic equation and other more singular models. These systems and the particular type of solutions we are interested in are the analogue of arbitrarily thin vortex tubes for 3D Euler equations. In studying the equations with a prescribed geometry we need to study Prantdl-like equations, with more singular kernels.

In the talk I will describe how can solve the equations in the smooth case (where ill-posedness would be expected), under some monotonicity assumptions.

**Laure
Saint-Raymond:** Internal
waves in a domain with topography

Stratification
of the density in an incompressible fluid is responsible for the
propagation of internal waves.

In domains with topography, these
waves exhibit interesting properties.

In particular, numerical
and lab experiments show that in 2D these waves concentrate on
attractors for some generic frequencies of the forcing (see Dauxois
et al).

At the mathematical level, this behavior can be analyzed
with tools from spectral theory and microlocal analysis.

This is a research in collaboration with Y. Colin de Verdière

**Sergio
Simonella: **Size
of chaos in collisional dynamics

We present a strategy to obtain optimal estimates of correlations in large particle systems, covering simultaneously approximating modelsfor the homogeneous (classical) Boltzmann equation, the Povzner kinetic equation, and the Hartree equation in the quantum mean field. We make use of a common hierarchical structure which can be described in an abstract formalism.

This is joint work with T. Paul and M. Pulvirenti.

**Agnieszka
Swierczewska-Gwiazda : **Energy
conservation for some compressible fluid models

A common feature of systems of conservation laws of continuum physics is that they are endowed with natural companion laws which are in such case most often related to the second law of thermodynamics. This observation easily generalizes to any symmetrizable system of conservation laws. They are endowed with nontrivial companion conservation laws, which are immediately satisfied by classical solutions.

Not surprisingly, weak solutions may fail to satisfy companion laws, which are then often relaxed from equality to inequality and overtake a role of a physical admissibility condition for weak solutions. We want to discuss what is a critical regularity of weak solutions to a general system of conservation laws to satisfy an associated companion law as an equality. An archetypal example of such result was derived for the incompressible Euler system by Constantin et al. ([1]) in the context of the seminal Onsager's conjecture. This general result can serve as a simple criterion to numerous systems of mathematical physics to prescribe the regularity of solutions needed for an appropriate companion law to be satisfied.

References:

[1] P. Constantin, W. E, and E. S.
Titi. Onsager's conjecture on the energy conservation for solutions
of Euler's equation., *Comm. Math. Phys.*, **165 **(1):
207-209, 1994.

[2] Feireisl, Eduard; Gwiazda, Piotr;
Swierczewska-Gwiazda, Agnieszka; Wiedemann, Emil; Regularity and
Energy Conservation for the Compressible Euler Equations, *Arch.
Ration. Mech. Anal.*, **223** (2017), no. 3, 1375-1395

[3] P. Gwiazda, M. Michálek, A. Swierczewska-Gwiazda. A note on weak solutions of conservation laws and energy/entropy conservation, arXiv:1706.10154

**Isabelle
Tristani :** On linear
Fokker-Planck equations

We will talk about several
homogeneous Fokker-Planck equations with classical, fractional or
discrete diffusion. We present a result of convergence to equilibrium
in a unified framework for those equations. One key point of this
analysis is the study of regularisation properties of the
Fokker-Planck operators. In a second part, we will also address the
problem of regularisation for the classical and fractional
Fokker-Planck operators in the inhomogeneous case.

It is joint work with Stéphane Mischler and Frédéric Hérau, Daniela Tonon.

**Juan
Velazquez : **Long
time asymptotics of homoenergetic solutions for the Boltzmann
equation.Homoenergetic
solutions are a particular class of solutions of the Boltzmann
equation which were introduced in the 1950's by Galkin
and Truesdell. They are useful to describe the dynamics of Boltzmann
gases under shear, expansion or compression in nonequilibrium
situations. Homoenergetic solutions are much simpler than the general
solutions of the Boltzmann equation. Their well posedness theory,
which has many similarities with the theory of homogeneous solutions
of the Boltzmann equation was studied by Cercignani in the 1980's.
However, the corresponding long time asymptotics theory differs much
of the analogous theory for homogeneous solutions. Actually, in the
case of homoenergetic solutions, the long time asymptotics cannnot
always be described using Maxwellian distributions. For several
collision kernels the long time behaviour of homoenergetic solutions
is given by particle distributions which do not satisfy the detailed
balance condition. In this talk I will describe different possible
long time asymptotics of homoenergetic solutions of the Boltzmann
equation as well as some open problems in this direction. (Joint work
with R.D. James and A. Nota).

**Tong
Yang :** Some
Mathematical Theories of MHD Boundary Layers

We will first present a work about justification of the Prandtl ansatz for the MHD system when the initial tangential magnetic field is not degenerate on the boundary. And then we will discuss the life span and global existence of smooth solutions to some boundary layer systems derived from MHD system. The talk includes some recent joint work with Chengjie Liu and Feng Xie.

update: Feb. 1st 2017