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In this talk, after reviewing the work on global well-posedness of the Boltzmann equation without angular cutoff with algebraic decay tails, we will present a recent work on the global weighted $L^{\infty}$-solutions to the Boltzmann equation without angular cutoff in the regime close to equilibrium. A De Giorgi type argument, well developed for diffusion equations, is crafted in this kinetic context with the help of the averaging lemma. More specifically, we use a strong averaging lemma to obtain suitable $L^{p}$ estimates for level-set functions. These estimates are crucial for constructing an appropriate energy functional to carry out the De Giorgi argument. Then we extend local solutions to global by using the spectral gap of the linearized Boltzmann operator with the convergence to the equilibrium state obtained as a byproduct. This result fill in the gap of well-posedness theory for the Boltzmann equation without angular cutoff in the $L^{\infty}$ framework. The talk is based on the joint works with Ricardo Alonso, Yoshinori Morimoto and Weiran Sun.
In this talk, after reviewing the work on global well-posedness of the Boltzmann equation without angular cutoff with algebraic decay tails, we will present a recent work on the global weighted $L^{\infty}$-solutions to the Boltzmann equation without angular cutoff in the regime close to equilibrium. A De Giorgi type argument, well developed for diffusion equations, is crafted in this kinetic context with the help of the averaging lemma. More ...

76P05 ; 35Q35 ; 47H20

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Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past decade, the Fourier-Galerkin spectral method has become a popular deterministic method for solving the Boltzmann equation, manifested by its high accuracy and potential of being further accelerated by the fast Fourier transform. Albeit its practical success, the stability of the method is only recently proved by Filbet, F. & Mouhot, C. in [Trans.Amer.Math.Soc. 363, no. 4 (2011): 1947-1980.] by utilizing the”spreading” property of the collision operator. In this work, we provide anew proof based on a careful L2 estimate of the negative part of the solution. We also discuss the applicability of the result to various initial data, including both continuous and discontinuous functions. This is joint work with Kunlun Qi and Tong Yang.
Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past decade, the Fourier-Galerkin spectral method has become a popular deterministic method for solving the Boltzmann equation, manifested by its high accuracy and potential of being further accelerated by the fast Fourier transform. Albeit its practical success, the stability of the method ...

35Q20 ; 65M12 ; 65M70 ; 45G10

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We develop a numerical method for the Levy-Fokker-Planck equation with the fractional diffusive scaling. There are two main challenges. One comes from a two-fold non locality, that is, the need to apply the fractional Laplacian operator to a power law decay distribution. The other comes from long-time/small mean-free-path scaling, which calls for a uniform stable solver. To resolve the first difficulty, we use a change of variable to convert the unbounded domain into a bounded one and then apply Chebyshev polynomial based pseudo-spectral method. To resolve the second issue, we propose an asymptotic preserving scheme based on a novel micro-macro decomposition that uses the structure of the test function in proving the fractional diffusion limit analytically.
We develop a numerical method for the Levy-Fokker-Planck equation with the fractional diffusive scaling. There are two main challenges. One comes from a two-fold non locality, that is, the need to apply the fractional Laplacian operator to a power law decay distribution. The other comes from long-time/small mean-free-path scaling, which calls for a uniform stable solver. To resolve the first difficulty, we use a change of variable to convert the ...

82C40 ; 45K05 ; 65M70 ; 82C80

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The viewpoint proposed in this lecture is that uncertainty quantification for kinetic equations does not always represent the viewpoint of the experimentalists. Instead they want to determine the uncertain coefficient in a PDE by measuring the solution at the parts of the boundary, given data on other parts of the boundary. In other words experimentalists are interested in solving the inverse problem in a Baysian setting.
We shall give examples and results to this end. In particular we shall consider a model from mathematical biology, namely the motion of cells, as described by the kinetic chemotaxis equations. The corresponding macroscopic Keller-Segel type model will be a diffusion equation. Our aim is to study the inverse problems for these two settings. We shall analytically study the convergence of the inverse problem of the kinetic equation to the inverse problem of the corresponding diffusion equation.
This is joint work with Kathrin Hellmuth, Qin Li and Min Tang
The viewpoint proposed in this lecture is that uncertainty quantification for kinetic equations does not always represent the viewpoint of the experimentalists. Instead they want to determine the uncertain coefficient in a PDE by measuring the solution at the parts of the boundary, given data on other parts of the boundary. In other words experimentalists are interested in solving the inverse problem in a Baysian setting.
We shall give examples ...

35Qxx

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At the individual scale, bacteria as E. coli move by performing so-called run-and-tumble movements. This means that they alternate a jump (run phase) followed by fast re-organization phase (tumble) in which they decide of a new direction for run. For this reason, the population is described by a kinetic-Botlzmann equation of scattering type. Nonlinearity occurs when one takes into account chemotaxis, the release by the individual cells of a chemical in the environment and response by the population.

These models can explain experimental observations, fit precise measurements and sustain various scales. They also allow to derive, in the diffusion limit, macroscopic models (at the population scale), as the Flux-Limited Keller-Segel system, in opposition to the traditional Keller-Segel system, this model can sustain robust traveling bands as observed in Adler’s famous experiment.

Furthermore, the modulation of the tumbles, can be understood using intracellular molecular pathways. Then, the kinetic-Boltzmann equation can be derived with a fast reaction scale. Long runs at the individual scale and abnormal diffusion at the population scale, can also be derived mathematically.
At the individual scale, bacteria as E. coli move by performing so-called run-and-tumble movements. This means that they alternate a jump (run phase) followed by fast re-organization phase (tumble) in which they decide of a new direction for run. For this reason, the population is described by a kinetic-Botlzmann equation of scattering type. Nonlinearity occurs when one takes into account chemotaxis, the release by the individual cells of a ...

35B25 ; 35Q20 ; 35Q84 ; 35Q92 ; 92C17

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It is possible to model dissipation effects subjected by a particle by interactions between the particle and its environment. This seminal idea dates back to Caldeira-Leggett in the ’80ies. The specific case of a particle interacting with vibrational degrees of freedom has been thoroughsly investigated by S. De Bièvre and his collaborators. We will go back to these issues in the framework of kinetic equations, and also consider quantum versions of the problem based on couplings with the Schrödinger equation. We are particularly interested in stability issues. We will describe ; through rigorous statements and numerical experiments, analogies and differences with the case of a single classical particle and with the standard coupling with the Poisson equation.
It is possible to model dissipation effects subjected by a particle by interactions between the particle and its environment. This seminal idea dates back to Caldeira-Leggett in the ’80ies. The specific case of a particle interacting with vibrational degrees of freedom has been thoroughsly investigated by S. De Bièvre and his collaborators. We will go back to these issues in the framework of kinetic equations, and also consider quantum versions ...

35Q40 ; 35Q51 ; 35Q55

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Quantized vortices have been experimentally observed in type-II superconductors, superfluids, nonlinear optics, etc. In this talk, I will review different mathematical equations for modeling quantized vortices in superfluidity and superconductivity, including the nonlinear Schrödinger/Gross-Pitaevskii equation, Ginzburg-Landau equation, nonlinear wave equation, etc. Asymptotic approximations on single quantized vortex state and the reduced dynamic laws for quantized vortex interaction are reviewed and solved approximately in several cases. Collective dynamics of quantized vortex interaction based on the reduced dynamic laws are presented. Extension to bounded domains with different boundary conditions are discussed.
Quantized vortices have been experimentally observed in type-II superconductors, superfluids, nonlinear optics, etc. In this talk, I will review different mathematical equations for modeling quantized vortices in superfluidity and superconductivity, including the nonlinear Schrödinger/Gross-Pitaevskii equation, Ginzburg-Landau equation, nonlinear wave equation, etc. Asymptotic approximations on single quantized vortex state and the reduced ...

34A05 ; 65N30 ; 35Q40

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The dynamical low-rank approximation is a low-rank factorization updating technique. It leads to differential equations for factors in a decomposition of the solution, which need to be solved numerically. The dynamical low-rank method seems particularly suitable for solving kinetic equations, because in many relevant cases the effective dynamics takes place on a lower-dimensional manifold and thus the solution has low rank. In this way, the 5-dimensional (3 space, 2 angle) radiation transport problem is reduced, both in computational cost as well as in memory footprint. We show several numerical examples.
The dynamical low-rank approximation is a low-rank factorization updating technique. It leads to differential equations for factors in a decomposition of the solution, which need to be solved numerically. The dynamical low-rank method seems particularly suitable for solving kinetic equations, because in many relevant cases the effective dynamics takes place on a lower-dimensional manifold and thus the solution has low rank. In this way, the ...

65M08 ; 76M12

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Usual numerical methods become inefficient when they are applied to highly oscillatory evolution problems (order reduction or complete loss of accuracy). The numerical parameters must indeed be adapted to the high frequencies that come into play to correctly capture the desired information, and this induces a prohibitive computational cost. Furthermore, the numerical resolution of averaged models, even at high orders, is not sufficient to capture low frequencies and transition regimes. We present (very briefly) two strategies allowing to remove this obstacle for a large class of evolution problems : a 2-scale method and a micro/macro method. Two different frameworks will be considered : constant frequency, and variable - possibly vanishing - frequency. The result of these approaches is the construction of numerical schemes whose order of accuracy no longer depends on the frequency of oscillation, one then speaks of uniform accuracy (UA) for these schemes. Finally, a new technique for systematizing these two methods will be presented. Its purpose is to reduce the number of inputs that the user must provide to apply the method in practice. In other words, only the values of the field defining the evolution equation (and not its derivatives) are used.These methods have been successfully applied to solve a number of evolution models: non-linear Schrödinger and Klein-Gordon equations, Vlasov-Poisson kinetic equation with strong magnetic field, quantum transport in graphene.
Usual numerical methods become inefficient when they are applied to highly oscillatory evolution problems (order reduction or complete loss of accuracy). The numerical parameters must indeed be adapted to the high frequencies that come into play to correctly capture the desired information, and this induces a prohibitive computational cost. Furthermore, the numerical resolution of averaged models, even at high orders, is not sufficient to ...

65L05 ; 35Q55 ; 37L05

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We propose a modulated free energy which combines of the method previously developed by the speaker together with the modulated energy introduced by S. Serfaty. This modulated free energy may be understood as introducing appropriate weights in the relative entropy to cancel the more singular terms involving the divergence of the flow. This modulated free energy allows to treat singular interactions of gradient-flow type and allows potentials with large smooth part, small attractive singular part and large repulsive singular part. As an example, a full rigorous derivation (with quantitative estimates) of some chemotaxis models, such as Patlak-Keller Segel system in the subcritical regimes, is obtained. This is joint work with D. Bresch and Z. Wang.
We propose a modulated free energy which combines of the method previously developed by the speaker together with the modulated energy introduced by S. Serfaty. This modulated free energy may be understood as introducing appropriate weights in the relative entropy to cancel the more singular terms involving the divergence of the flow. This modulated free energy allows to treat singular interactions of gradient-flow type and allows potentials ...

35Q70 ; 60H30 ; 60F10 ; 82C22

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This lecture is devoted to the characterization of convergence rates in some simple equations with mean field nonlinear couplings, like the Keller-Segel and Nernst-Planck systems, Cucker-Smale type models, and the Vlasov-Poisson-Fokker-Planck equation. The key point is the use of Lyapunov functionals adapted to the nonlinear version of the model to produce a functional framework adapted to the asymptotic regime and the corresponding spectral analysis.
This lecture is devoted to the characterization of convergence rates in some simple equations with mean field nonlinear couplings, like the Keller-Segel and Nernst-Planck systems, Cucker-Smale type models, and the Vlasov-Poisson-Fokker-Planck equation. The key point is the use of Lyapunov functionals adapted to the nonlinear version of the model to produce a functional framework adapted to the asymptotic regime and the corresponding spectral ...

82C40 ; 35H10 ; 35P15 ; 35Q84 ; 35R09 ; 47G20 ; 82C21 ; 82D10 ; 82D37 ; 76P05 ; 35K65 ; 35Q84 ; 46E35 ; 35K55 ; 35Q70

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The aim of this talk is the rigorous derivation of crossdiffusion systems from stochastic, moderately interacting many-particle systems for multiple species. Applications include animal populations and neuronal ensembles. The mean-field limit leads to nonlocal cross-diffusion systems, while the limit of vanishing interaction radius gives local cross-diffusion equations. This allows for the derivation of fluid-type models that can be found in neuronal networks and of Shigesada-Kawasaki-Teramoto population models. The derivation uses the techniques of Oehlschläger. The entropy structure of the limiting models is discussed and some numerical experiments are presented.
The aim of this talk is the rigorous derivation of crossdiffusion systems from stochastic, moderately interacting many-particle systems for multiple species. Applications include animal populations and neuronal ensembles. The mean-field limit leads to nonlocal cross-diffusion systems, while the limit of vanishing interaction radius gives local cross-diffusion equations. This allows for the derivation of fluid-type models that can be found in ...

35Q92 ; 35K45 ; 60J70 ; 82C22

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We are concerned with deriving sharp exponential decay estimates (i.e. with maximum rate and minimum multiplicative constant) for linear, hypocoercive evolution equations. Using a modal decomposition of the model allows to assemble a Lyapunov functional using Lyapunov matrix inequalities for each Fourier mode.
We shall illustrate the approach on the 1D Goldstein-Taylor model, a2-velocity transport-relaxation equation. On the torus the lowest Fourier modes determine the spectral gap of the whole equation in $L^{2}$. By contrast, on the whole real line the Goldstein-Taylor model does not have a spectral gap, since the decay rate of the Fourier modes approaches zero in the small mode limit. Hence, the decay is reduced to algebraic.
In the final part of the talk we consider the Goldstein-Taylor model with non-constant relaxation rate, which is hence not amenable to a modal decomposition. In this case we construct a Lyapunov functional of pseudodifferential nature, one that is motivated by the modal analysis in the constant case.The robustness of this approach is illustrated on a multi-velocity GoldsteinTaylor model, yielding explicit rates of convergence to the equilibrium.
This is joint work with J. Dolbeault, A. Einav, C. Schmeiser, B. Signorello, and T. Wöhrer.
We are concerned with deriving sharp exponential decay estimates (i.e. with maximum rate and minimum multiplicative constant) for linear, hypocoercive evolution equations. Using a modal decomposition of the model allows to assemble a Lyapunov functional using Lyapunov matrix inequalities for each Fourier mode.
We shall illustrate the approach on the 1D Goldstein-Taylor model, a2-velocity transport-relaxation equation. On the torus the lowest ...

82C40 ; 35B40 ; 35Q82 ; 35S05

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We model, simulate and control the guiding problem for a herd of evaders under the action of repulsive drivers. The problem is formulated in an optimal control framework, where the drivers (controls) aim to guide the evaders (states) to a desired region of the Euclidean space.
Classical control methods allow to build coordinated strategies so that the drivers successfully drive the evaders to the desired final destination.
But the computational cost quickly becomes unfeasible when the number of interacting agents is large.
We present a method that combines the Random Batch Method (RBM) and Model Predictive Control (MPC) to significantly reduce the computational cost without compromising the efficiency of the control strategy.
This talk is based on joint work with Dongnam Ko, from the Catholic University of Korea.
We model, simulate and control the guiding problem for a herd of evaders under the action of repulsive drivers. The problem is formulated in an optimal control framework, where the drivers (controls) aim to guide the evaders (states) to a desired region of the Euclidean space.
Classical control methods allow to build coordinated strategies so that the drivers successfully drive the evaders to the desired final destination.
But the computational ...

93D20 ; 93B52 ; 49N75

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Whether there is global regularity or finite time blow-up for the space homogeneous Landau equation with Coulomb potential is a longstanding open problem in the mathematical analysis of kinetic models. This talk shows that the Hausdorff dimension of the set of singular times of the global weak solutions obtained by Villanis procedure is at most 1/2.
(Work in collaboration with M.P. Gualdani, C. Imbert and A. Vasseur)

35Q20 ; 35B65 ; 35K15 ; 35B44

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We consider hypoelliptic equations of kinetic Fokker-Planck type, also sometimes called of Kolmogorov or Langevin type, with rough coefficients in the drift-diffusion operator in velocity. We present novel short quantitative proofs of the De Giorgi intermediate-value Lemma as well as weak Harnack and Harnack inequalities (which imply Hölder continuity with quantitative estimates).
This is a joint work with Jessica Guerand.

35Q84 ; 35B45 ; 35B65

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This talk is devoted to the quasi linear approximation for solutions of the Vlasov equation a very popular tool in Plasma Physic cf. [4] which proposes, for the quantity:
(1)
$$
q(t,\ v)=\int_{\mathbb{R}_{v}^{d}}f(x,\ v,\ t)dx)\ ,
$$
the solution of a parabolic, linear or non linear evolution equation
(2)
$$
\partial_{t}q(t,\ v)-\nabla_{v}(D(q,\ t;v)\nabla_{v}q)=0
$$
Since the Vlasov equation is an hamiltonian reversible dynamic while (2) is not reversible whenever $D(q,\ t,\ v)\# 0$ the problem is subtle. Hence I did the following things :

1. Give some sufficient conditions, in particular in relation with the Landau damping that would imply $D(q,\ t,\ v)\simeq 0$. a situation where the equation (2) with $D(q,\ t;v)=0$ does not provides a meaning full approximation.

2. Building on contributions of [7] and coworkers show the validity of the approximation (2) for large time and for a family of convenient randomized solutions. This is justified by the fact that the assumed randomness law is in agreement which what is observed by numerical or experimental observations (cf. [1]).

3. In the spirit of a Chapman Enskog approximation formalize the very classical physicist approach (cf. [6] pages 514-532) one can show [3] that under analyticity assumptions this approximation is valid for short time. As in [6] one of the main ingredient of this construction is based on the spectral analysis of the linearized equation and as such it makes a link with a classical analysis of instabilities in plasma physic.

Remarks

In some sense the two approaches are complementary The short time is purely deterministic and the stochastic is based on the intuition that over longer time the randomness will take over of course the transition remains from the first regime to the second remains a challenging open problem. The similarity with the transition to turbulence in fluid mechanic is striking It is underlined by the fact that the tensor
$$
\lim_{\epsilon\rightarrow 0}\mathbb{D}^{\epsilon}(t,\ v)=\lim_{\epsilon\rightarrow 0}\int dx\int_{0}^{\frac{t}{\epsilon^{2}}}d\sigma E^{\epsilon}(t,\ x+\sigma v)\otimes E^{\epsilon}(t-\epsilon^{2}\sigma,\ x)
$$
which involves the electric fields here plays the role of the Reynolds stress tensor.

2 Obtaining, for some macroscopic description, a space homogenous equation for the velocity distribution is a very natural goal. Here the Vlasov equation is used as an intermediate step in the derivation. And more generally it appears as an example of weak turbulence. In particular defining what would be the physical natural probability seems related to the derivation of $\mathrm{e}$ of the Lenard-Balescu equation as done in [5].
This talk is devoted to the quasi linear approximation for solutions of the Vlasov equation a very popular tool in Plasma Physic cf. [4] which proposes, for the quantity:
(1)
$$
q(t,\ v)=\int_{\mathbb{R}_{v}^{d}}f(x,\ v,\ t)dx)\ ,
$$
the solution of a parabolic, linear or non linear evolution equation
(2)
$$
\partial_{t}q(t,\ v)-\nabla_{v}(D(q,\ t;v)\nabla_{v}q)=0
$$
Since the Vlasov equation is an hamiltonian reversible dynamic while (2) is not ...

35Q83 ; 82C70

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We are interested in the stabilisation of linear kinetic equations for applications in e.g. closed-loop feedback control. Progress has been made in recent years on stabilisation of hyperbolic balance equations using special Lyapunov functions. However, those are not necessarily suitable for the kinetic equation. We present results on kinetic equations under uncertainties and closed loop feedback control.

35B35 ; 93D20 ; 37L45 ; 35B30 ; 35R60

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The Hamiltonian Mean-Field (HMF) model is a 1D simplified version of the gravitational Vlasov-Poisson system. I will present two recent works in collaboration with Mohammed Lemou and Ana Maria Luz. In the first one, we proved the nonlinear stability of steady states for this model, using a technique of generalized Schwarz rearrangements. To be stable, the steady state has to satisfy a criterion. If this criterion is not satisfied, some instabilities can occur: this is the topic of the second work that I will present.
The Hamiltonian Mean-Field (HMF) model is a 1D simplified version of the gravitational Vlasov-Poisson system. I will present two recent works in collaboration with Mohammed Lemou and Ana Maria Luz. In the first one, we proved the nonlinear stability of steady states for this model, using a technique of generalized Schwarz rearrangements. To be stable, the steady state has to satisfy a criterion. If this criterion is not satisfied, some ...

35Q83 ; 35B35 ; 35Q60

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