Recent zbMATH articles in MSC 35Q20https://zbmath.org/atom/cc/35Q202021-05-28T16:06:00+00:00WerkzeugNon-cutoff Boltzmann equation with polynomial decay perturbations.https://zbmath.org/1459.353052021-05-28T16:06:00+00:00"Alonso, Ricardo"https://zbmath.org/authors/?q=ai:alonso.ricardo-j"Morimoto, Yoshinori"https://zbmath.org/authors/?q=ai:morimoto.yoshinori"Sun, Weiran"https://zbmath.org/authors/?q=ai:sun.weiran"Yang, Tong"https://zbmath.org/authors/?q=ai:yang.tongSummary: The Boltzmann equation without the angular cutoff is considered when the initial data is a small perturbation of a global Maxwellian and decays algebraically in the velocity variable. We obtain a well-posedness theory in the perturbative framework for both mild and strong angular singularities. The three main ingredients in the proof are the moment propagation, the spectral gap of the linearized operator, and the regularizing effect of the linearized operator when the initial data is in a Sobolev space with a negative index. A carefully designed pseudo-differential operator plays a central role in capturing the regularizing effect. In addition, some intrinsic symmetry with respect to the collision operator and an intrinsic functional in the coercivity estimate are essentially used in the commutator estimates for the collision operator with velocity weights.From uncertainty propagation in transport equations to kinetic polynomials.https://zbmath.org/1459.652062021-05-28T16:06:00+00:00"Després, Bruno"https://zbmath.org/authors/?q=ai:despres.brunoSummary: In view of the modeling of uncertainties which propagate in non linear transport equations and general hyperbolic systems, we review some recent alternatives to the classical moment method. These approaches are obtained by reconsidering the non linear structure with entropy considerations. It is shown that the entropy variable and the kinetic formulation of conservation laws yield new approaches with strong control of the maximum principle. A general minimization principle is proposed for these kinetic polynomials, together with an original reformulation as an optimal control problem. Basic numerical illustrations show the properties of these new techniques. A surprising linked to quaternion algebras is evoked in relation with kinetic polynomials. Natural limitations are discussed in the conclusion.
For the entire collection see [Zbl 1393.35003].On the kinetic equation in Zakharov's wave turbulence theory for capillary waves.https://zbmath.org/1459.821522021-05-28T16:06:00+00:00"Nguyen, Toan T."https://zbmath.org/authors/?q=ai:nguyen.toan-trong"Tran, Minh-Binh"https://zbmath.org/authors/?q=ai:tran.minh-binhUniqueness of the non-equilibrium steady state for a 1d BGK model in kinetic theory.https://zbmath.org/1459.353062021-05-28T16:06:00+00:00"Carlen, E."https://zbmath.org/authors/?q=ai:carlen.eric-anders"Esposito, R."https://zbmath.org/authors/?q=ai:esposito.raffaele"Lebowitz, J."https://zbmath.org/authors/?q=ai:lebowitz.joel-louis"Marra, R."https://zbmath.org/authors/?q=ai:marra.rossana"Mouhot, C."https://zbmath.org/authors/?q=ai:mouhot.clementSummary: We continue our investigation of kinetic models of a one-dimensional gas in contact with homogeneous thermal reservoirs at different temperatures. Nonlinear collisional interactions between particles are modeled by a so-called BGK dynamics which conserves local energy and particle density. Weighting the nonlinear BGK term with a parameter \(\alpha\in [0,1]\), and the linear interaction with the reservoirs by \((1-\alpha)\), we prove that for some \(\alpha\) close enough to zero, the explicit spatially uniform non-equilibrium steady state (NESS) is \textit{unique}, and there are no spatially non-uniform NESS with a spatial density \(\rho\) belonging to \(L^p\) for any \(p> 1\). We also show that for all \(\alpha\in [0,1]\), the spatially uniform NESS is \textit{dynamically stable}, with small perturbation converging to zero exponentially fast.Regularity for the Boltzmann equation conditional to macroscopic bounds.https://zbmath.org/1459.353072021-05-28T16:06:00+00:00"Imbert, Cyril"https://zbmath.org/authors/?q=ai:imbert.cyril"Silvestre, Luis"https://zbmath.org/authors/?q=ai:silvestre.luis-eSummary: The Boltzmann equation is a nonlinear partial differential equation that plays a central role in statistical mechanics. From the mathematical point of view, the existence of global smooth solutions for arbitrary initial data is an outstanding open problem. In the present article, we review a program focused on the type of particle interactions known as non-cutoff. It is dedicated to the derivation of a priori estimates in \(C^\infty\), depending only on physically meaningful conditions. We prove that the solution will stay uniformly smooth provided that its mass, energy and entropy densities remain bounded, and away from vacuum.Transport distances for PDEs: the coupling method.https://zbmath.org/1459.350372021-05-28T16:06:00+00:00"Fournier, Nicolas"https://zbmath.org/authors/?q=ai:fournier.nicolas-g"Perthame, Benoît"https://zbmath.org/authors/?q=ai:perthame.benoitSummary: We informally review a few PDEs for which some transport cost between pairs of solutions, possibly with some judicious cost function, decays: heat equation, Fokker-Planck equation, heat equation with varying coefficients, fractional heat equation with varying coefficients, homogeneous Boltzmann equation for Maxwell molecules, and some nonlinear integro-differential equations arising in neurosciences. We always use the same method, that consists in building a coupling between two solutions. This means that we double the variables and solve, globally in time, a well-chosen PDE posed on the Euclidean square of the physical space. Finally, although the above method fails, we recall a simple idea to treat the case of the porous media equation. We also introduce another method based on the dual Monge-Kantorovich problem.On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases.https://zbmath.org/1459.822602021-05-28T16:06:00+00:00"Baranger, Céline"https://zbmath.org/authors/?q=ai:baranger.celine"Bisi, Marzia"https://zbmath.org/authors/?q=ai:bisi.marzia"Brull, Stéphane"https://zbmath.org/authors/?q=ai:brull.stephane"Desvillettes, Laurent"https://zbmath.org/authors/?q=ai:desvillettes.laurentSummary: In this paper, we propose a formal Chapman-Enskog expansion in the context of mixtures of monoatomic and polyatomic gases. We start from a Boltzmann model that is based on the [\textit{C. Borgnakke} and \textit{P. S. Larsen}, ``Statistical collision model for Monte-Carlo simulation of polyatomic mixtures'', Journ. Comput. Phys. 18, 405--420 (1975)] and we derive a compressible Navier-Stokes system. In a last part, we perform some explicit computations of the transport coefficients in the case of Maxwell molecules for diatomic gases.
For the entire collection see [Zbl 1453.35003].Reconstruction of the collision kernel in the nonlinear Boltzmann equation.https://zbmath.org/1459.354002021-05-28T16:06:00+00:00"Lai, Ru-Yu"https://zbmath.org/authors/?q=ai:lai.ru-yu"Uhlmann, Gunther"https://zbmath.org/authors/?q=ai:uhlmann.gunther-a"Yang, Yang"https://zbmath.org/authors/?q=ai:yang.yang.3