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Documents : Virtualconference  | enregistrements trouvés : 200

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In the late 70s, Fathi showed that the group of compactly supported volume-preserving homeomorphisms of the ball is simple in dimensions greater than 2. We present our recent article which proves that the remaining group, that is area-preserving homeomorphisms of the disc, is not simple. This settles what is known as the simplicity conjecture in the affirmative. This is joint work with Dan Cristofaro-Gardiner and Vincent Humiliere.

53DXX ; 53D40 ; 37E30

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Symplectic embedding problems are at the heart of the study of symplectic topology, and are closely related with Hamiltonian dynamics. In this talk we discuss how to compute the symplectic inner and outer radii of certain convex domains using the theory of integrable systems.
The talk is based on a joint work with Vinicius Ramos.

53DXX

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Let G be a compact Lie group and let M = G/H be a G-homogeneous space, equipped with an invariant metric. We prove that the spectral norm of any compact exact Lagrangian submanifold of the cotangent bundle T*M is bounded in terms of the diameter and dimension of G. Our proof is by sheaf theoretical methods; it recovers some results of Shelukhin and gives some other cases. This is a joint work in progress with Nicolas Vichery.

53DXX ; 70Hxx ; 54B40

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For a given symplectic manifold and a finite set of Lagrangian submanifolds which intersect transversally each other we can construct an $A_{\infty }$-category. When we consider Lagrangian submanifolds which are not necessary intersect transversally there are issues to perform such a construction. In this talk I want to explain a way to study this situation. It can be used to compose Lagrangian correspondences without assuming transversality.

53D40 ; 18G99 ; 57R99

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This partly expository talk focuses on the notion of ”symplectic Landau-Ginzburg models”, i.e. symplectic manifolds equipped with maps to the complex plane, ”stops”, or both, as they naturally arise in the context of mirror symmetry. We describe several viewpoints on these spaces and their Fukaya categories, their monodromy, and the functors relating them to other flavors of Fukaya categories. (This touches on work of Abouzaid, Seidel, Ganatra, Hanlon, Sylvan, Jeffs, and others).
This partly expository talk focuses on the notion of ”symplectic Landau-Ginzburg models”, i.e. symplectic manifolds equipped with maps to the complex plane, ”stops”, or both, as they naturally arise in the context of mirror symmetry. We describe several viewpoints on these spaces and their Fukaya categories, their monodromy, and the functors relating them to other flavors of Fukaya categories. (This touches on work of Abouzaid, Seidel, Ganatra, ...

53D37 ; 14J33 ; 53D40

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Recent years have seen the development of a range of modified gravity theories to tackle the dark energy and cosmological constant problem. An interesting class of such models are those in which the graviton effectively becomes massive, modifying gravity at large (cosmological) distances without changing physics at solar system and smaller scales. Such effective field theories have the Vainshtein mechanism built in, and are closely associated with Galileon theories.

In these lectures I will give a brief general overview of both soft and hard massive gravity theories, their origin from extra dimensional models, and the special class of ghost-free massive gravity models and their extensions with multiple massive (and massless) spin-2 states. I will review the Vainshtein mechanism, and the decoupling limits of these theories and how they are related to Galileons. I will then discuss a variety of implications of such theories.

• lecture 1: I will introduce the description of massless and massive spin-2 states, and explain how theseemerge naturally in extra dimensional models, and go on to give the description of so-called ghost-freeor $\Lambda_{3}$ theories of interacting massive spin-2 fields, aka massive gravity and its multi-gravity extensions.

• lecture 2: I will review some aspects of the phenomenology of the general class of massive gravity theories: screening - cosmology - black hole solutions, and if time go on to discuss more recent extensions which attempt to raise the cutoff, connections with UV completions such as TTbar deformations, and the significance of ‘positivity bounds’ applied to these effective theories.
Recent years have seen the development of a range of modified gravity theories to tackle the dark energy and cosmological constant problem. An interesting class of such models are those in which the graviton effectively becomes massive, modifying gravity at large (cosmological) distances without changing physics at solar system and smaller scales. Such effective field theories have the Vainshtein mechanism built in, and are closely associated ...

85A40 ; 83C47 ; 83D05 ; 83E15 ; 83F05

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In this course I will give an overview of different line of sight and environmental effects that are expected to modify/distort a gravitational wave (GW) signal emitted by an astrophysical source at cosmological distance. After an overview of the state of the art of GW observations, I will review the derivation of the waveform emitted by a binary system of compact objects in a flat background, in the newtonian approximation. I will then consider a cosmological context and introduce the concept of standard siren. Finally I will derive which is the expected distortion of an emitted waveform induced by the presence of peculiar velocities and (strong) gravitational lensing.

Lecture 1 — Gravitational waves-introduction
Lecture 2 — Gravitational waves-propagation effects in waveform modeling
In this course I will give an overview of different line of sight and environmental effects that are expected to modify/distort a gravitational wave (GW) signal emitted by an astrophysical source at cosmological distance. After an overview of the state of the art of GW observations, I will review the derivation of the waveform emitted by a binary system of compact objects in a flat background, in the newtonian approximation. I will then consider ...

83C35 ; 83F05

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In these lectures we give a general introduction to the theory of gravitational waves and the analytic approximation methods in general relativity. More precisely we focus on the theory which is necessary to accurately and reliably predict the gravitational waves generated by compact binary systems, made of black holes or neutron stars. The predictions are used in the form of gravitational-wave “templates” in the data analysis of the detectors LIGO, VIRGO, ... LISA. In particular we present the state-of-the-art on the post-Newtonian approximation in general relativity, which is the main tool for describing the famous gravitational wave “chirp” of compact binary systems. The outline of the lectures is :
1. Gravitational wave events
2. Methods to compute gravitational waves
3. Einstein quadrupole formalism
4. Post-Newtonian parameters
5. Finite size effects in compact binaries
6. Synergy with the effective field theory
7. Radiation reaction and balance equations.
In these lectures we give a general introduction to the theory of gravitational waves and the analytic approximation methods in general relativity. More precisely we focus on the theory which is necessary to accurately and reliably predict the gravitational waves generated by compact binary systems, made of black holes or neutron stars. The predictions are used in the form of gravitational-wave “templates” in the data analysis of the detectors ...

83C35 ; 83C57 ; 83F05

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Virtualconference  Model-independent cosmology
Amendola, Luca (Auteur de la Conférence) | CIRM (Editeur )

Current estimations of cosmological parameters depend often on assumptions about the cosmological model itself. The estimates of $H_{0}$ or $\Omega_{m}$ from CMB surveys, for instance, are valid only assuming a particular model, typically the standard $\Lambda$CDM. Similar model-dependent results are obtained also from analyses of large-scale structure. In some case it is however possible to combine observations in such a way to get estimates of physical quantities that are valid regardless (to some extent) of the underlying model. Here I will discuss how one can determine the cosmological expansion rate $H(z)$ and the deviation from Einstein gravity $\eta$ by combining in a model-independent way several observational probes, from redshift distortions, to lensing, to matter clustering.
Current estimations of cosmological parameters depend often on assumptions about the cosmological model itself. The estimates of $H_{0}$ or $\Omega_{m}$ from CMB surveys, for instance, are valid only assuming a particular model, typically the standard $\Lambda$CDM. Similar model-dependent results are obtained also from analyses of large-scale structure. In some case it is however possible to combine observations in such a way to get estimates of ...

83F05

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We study the regularity of the Gaussian random measures $(-\Delta)^{-s}W$ on the Sierpiński gasket where $W$ is a white noise and $\Delta$ the Laplacian with respect to the Hausdorff measure. Along the way we prove sharp global Hölder regularity estimates for the fractional Riesz kernels on the gasket which are new and of independent interest.
This is a joint work with Celine Lacaux.

60G60 ; 28A80 ; 60G15

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Robotic design involves modeling the behavior of a robot mechanism when it moves along potential paths set by the users. In this lecture, we will first give an overview of different approaches to design a set of kinematic equations associated with a robot mechanism. In particular, these equations can be used to solve the forward and the backward kinematics problems associated with a robot mechanism or to model its singularity locus. Then we will review methods to solve those equations, and notably methods to draw with guarantees the real solutions of an under-constrained system of equations modeling the singularities of a robot.
Robotic design involves modeling the behavior of a robot mechanism when it moves along potential paths set by the users. In this lecture, we will first give an overview of different approaches to design a set of kinematic equations associated with a robot mechanism. In particular, these equations can be used to solve the forward and the backward kinematics problems associated with a robot mechanism or to model its singularity locus. Then we will ...

68T40 ; 65G20 ; 68W30 ; 65Dxx

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Virtualconference  The structure group revisited
Otto, Felix (Auteur de la Conférence) | CIRM (Editeur )

Following the treatment of a class of quasi-linear SPDE with Sauer, Smith, and Weber, we approach Hairer's regularity structure $(\mathrm{A},\ \mathrm{T},\ \mathrm{G})$ from a different angle. In this approach, the model space $\mathrm{T}$ is a direct sum over an index set that corresponds to specific linear combination of (decorated) trees, and thus amounts to a more parsimonious parameterization of the solution manifold. Moreover, the same structure group $\mathrm{G}$ captures different classes of equations; depending on the class, different (sub)spaces $\mathrm{T}$ matter, which correspond to linear combinations of different types of trees.

In our approach to $\mathrm{G}$, we start from the space of tuples $(a,p)$ of (polynomial) nonlinearities $a$ and space-time polynomials $p$, which we think of parameterizing the entire manifold of solutions $u$ (satisfying the equation up to space-time polynomials) via re-centering. We consider the actions of a shift by a space-time vector $h\in \mathbb{R}^{d+1}$ and of tilt by space-time polynomial $q$ on $(a,p)$-space, where, crucially, the tilt by a constant is treated as a shift of the (one-dimensional) $u$-space. We consider the infinitesimal generators of these actions, and pull them back as derivations on the algebra of formal power series $\mathbb{R}[[\mathrm{z}_{k},\ \mathrm{z}_{\mathrm{n}}]]$ in the natural coordinates $\{\mathrm{z}_{k}\}_{k\in \mathbb{N}_{0}}$ and $\{\mathrm{z}_{\mathrm{n}}\}_{\mathrm{n}\in \mathbb{N}_{0}^{d+1}-\{\mathrm{O}\}}$ of $(a,\ p)$-space. This defines a Lie algebra $\mathrm{L}\subset \mathrm{D}\mathrm{e}\mathrm{r}(\mathbb{R}[[\mathrm{z}_{k},\ \mathrm{z}_{\mathrm{n}}]])$ . Loosely speaking, the corresponding Lie group coincides with $\mathrm{G}^{*}$, but we follow an algebraic path to construct G.

As a group, $\mathrm{G}\subset(\mathrm{T}^{+})^{*}$ arises in the standard way from the Hopf algebra $\mathrm{T}^{+}$ that is obtained from dualizing the universal enveloping algebra $\mathrm{U}(\mathrm{L})$ . Here, gradedness and finiteness properties are needed for the well-posedness of the co-product $\triangle^{+}:\mathrm{T}^{+}\rightarrow \mathrm{T}^{+}\otimes \mathrm{T}^{+}$ and the antipode. The passage from $\mathbb{R}[[\mathrm{z}_{k},\ \mathrm{z}_{\mathrm{n}}]]$ to a smaller (linear) subspace $\mathrm{T}^{*}$ is needed for dualizing the module defined through $\mathrm{L}\subset$ End(T$*$) to obtain the co-module $\triangle:\mathrm{T}\rightarrow \mathrm{T}^{+}\otimes \mathrm{T}$. This yields the representation $\mathrm{G}\subset$ End(T). Both $\triangle$ and $\triangle^{+}$ satisfy the postulates of regularity structures, in particular the properties that intertwine $\triangle, \triangle^{+}$, and the family of re-centering maps $\mathcal{J}_{\mathrm{n}}:\mathrm{T}\rightarrow \mathrm{T}^{+}$. The latter relies on choosing a natural basis of $\mathrm{U}(\mathrm{L})$ , different from the standard Poincar\'{e}-Birkhoff-Witt basis, for dualization.

This is joint work with P. Linares and M. Tempelmayr.
Following the treatment of a class of quasi-linear SPDE with Sauer, Smith, and Weber, we approach Hairer's regularity structure $(\mathrm{A},\ \mathrm{T},\ \mathrm{G})$ from a different angle. In this approach, the model space $\mathrm{T}$ is a direct sum over an index set that corresponds to specific linear combination of (decorated) trees, and thus amounts to a more parsimonious parameterization of the solution manifold. Moreover, the same ...

60H17 ; 35k59 ; 16T05

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The reconstruction theorem, a cornerstone of Martin Hairer’s theory of regularity structures,appears in this article as the unique extension of the explicitly given reconstruction operatoron the set of smooth models due its inherent Lipschitz properties. This new proof is a directconsequence of constructions of mollification procedures on spaces of models and modelled distributions: more precisely, for an abstract model Z of a given regularity structure, a mollifiedmodel is constructed, and additionally, any modelled distribution f can be approximated byelements of a universal subspace of modelled distribution spaces. These considerations yield inparticular a non-standard approximation results for rough path theory. All results are formulatedin a generic (p, q) Besov setting. There are also implications on learning solution maps from amachine learning perspective.Joint work with Harprit Singh.
The reconstruction theorem, a cornerstone of Martin Hairer’s theory of regularity structures,appears in this article as the unique extension of the explicitly given reconstruction operatoron the set of smooth models due its inherent Lipschitz properties. This new proof is a directconsequence of constructions of mollification procedures on spaces of models and modelled distributions: more precisely, for an abstract model Z of a given regularity ...

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We present a novel framework for the study of a large class of non-linear stochastic PDEs, which is inspired by the algebraic approach to quantum field theory. The main merit is that, by realizing random fields within a suitable algebra of functional-valued distributions, we are able to use techniques proper of microlocal analysis which allow us to discuss renormalization and its associated freedom without resorting to any regularization scheme and to the subtraction of infinities. As an example of the effectiveness of the approach we apply it to the perturbative analysis of the stochastic $\Phi _{d}^{3}$ model.
We present a novel framework for the study of a large class of non-linear stochastic PDEs, which is inspired by the algebraic approach to quantum field theory. The main merit is that, by realizing random fields within a suitable algebra of functional-valued distributions, we are able to use techniques proper of microlocal analysis which allow us to discuss renormalization and its associated freedom without resorting to any regularization scheme ...

81T05 ; 60H17

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I will start from reviewing Gröbner bases and their connection to polynomial system solving. The problem of solving a polynomial system of equations over a finite field has relevant applications to cryptography and coding theory. For many of these applications, being able to estimate the complexity of computing a Gröbner basis is crucial. With these applications in mind, I will review linear-algebra based algorithms, which are currently the most efficient algorithms available to compute Gröbner bases. I will define and compare several invariants, that were introduced with the goal of providing an estimate on the complexity of computing a Gröbner basis, including the solving degree, the degree of regularity, and the last fall degree. Concrete examples will complement the theoretical discussion.
I will start from reviewing Gröbner bases and their connection to polynomial system solving. The problem of solving a polynomial system of equations over a finite field has relevant applications to cryptography and coding theory. For many of these applications, being able to estimate the complexity of computing a Gröbner basis is crucial. With these applications in mind, I will review linear-algebra based algorithms, which are currently the most ...

13P10

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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 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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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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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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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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