Kinetic theory and PDE s

Transcription

1 University of Novi Sad Faculty of Sciences Department of Mathematics and Informatics Problems on Kinetic theory and PDE s Book of Abstracts Novi Sad, September 2014

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3 Organizers Department of Mathematics and Informatics, Faculty of Sciences, Novi Sad Department of Mechanics, Faculty of Technical Sciences, Novi Sad Laboratoire MAP5, Université Paris Descartes & CNRS, Paris Organizing Committee Marko Nedeljkov, Department of Mathematics and Informatics, Novi Sad Bérénice Grec, Laboratoire MAP5, Université Paris Descartes, Paris Milana Pavić, Department of Mathematics and Informatics, Novi Sad Srboljub Simić, Department of Mechanics, Novi Sad

4 Supported by Ministry of Education, Science and Technological Development of the Republic of Serbia Provincial Secretariat for Science and Technological Development, Province of Vojvodina Centre National de la Recherche Scientifique (CNRS), France Bilateral project (France-Serbia) CNRS/MSTD No Fluid and kinetic models for gaseous mixtures PICS France-Serbia CNRS No Kinetic and macroscopic modelling of gaseous mixtures Project ON Methods of functional and harmonic analysis and PDEs with singularities Project ON Mechanics of nonlinear and dissipative systems - contemporary models, analysis and applications

5 Program Thursday, 25 September Morning session DMI, Amphitheater Opening Francesco Salvarani: Transport phenomena in evolutionary domains Coffee break Jelena Aleksić: Strong traces of ultra-parabolic equation via averaged traces of ultra-parabolic transport equation Afternoon session University of Novi Sad, Central building Milana Pavić: Mathematical modelling and analysis of polyatomic gases and mixtures in the context of kinetic theory of gases and fluid mechanics Cocktail Dinner

6 Friday, 26 September Morning session DMI, Amphitheater Tommaso Ruggeri: Molecular Extended Thermodynamics of Rarefied Polyatomic Gases and Wave Velocities for Increasing Number of Moments Maria Groppi: Kinetic relaxation models for reacting gas mixtures Coffee break Laurent Desvillettes: The incompressible Navier- Stokes limit of the Boltzmann equation for mixtures of gases Klemens Fellner: Convergence to Equilibrium for a Coagulation-Fragmentation Model with Degenerate Spatial Diffusion Lunch break

7 Afternoon session DMI, Amphitheater Marzia Bisi: Recent advances in kinetic theory for mixtures of polyatomic gases Laurent Boudin: Kinetic modelling for respiratory aerosols, numerical treatment Coffee break Valeria Ricci: About the validation of models for multicomponent systems Dušan Zorica: Generalizations of the classical wave equation within the theory of fractional calculus

8 Saturday, 27 September Morning session DMI, Amphitheater Stéphane Brull: Derivation of BGK models for gas mixtures Damir Madjarević: Shock structure for macroscopic multi-temperature model of binary mixtures: comparison with kinetic models Coffee break Marko Nedeljkov: Shadow waves Bérénice Grec: A diffusion limit for gaseous mixtures Closing

9 Abstracts

10 Strong traces of ultra-parabolic equation via averaged traces of ultra-parabolic transport equation Jelena ALEKSIĆ Department of Mathematics and Informatics University of Novi Sad, Serbia Our aim is to prove the existence of strong traces for entropy solutions to ultraparabolic equations in heterogeneous media, u t + div x f(x, u) = k x 2 ix j b ij (t, x, u), k d. i,j=1 We obtain this as consequence of the following result: Namely, we prove that if traceability conditions are fulfilled then a weak solution h L (IR + IR d IR) to the ultra-parabolic transport equation t h + div x (F (t, x, λ)h) = k x 2 ix j (b ij (t, x, λ)h) + λ γ(t, x, λ), i,j=1 is such that for every ρ C 1 c (IR), the velocity averaged quantity IR h(t, x, λ) ρ(λ)dλ admits the strong L1 loc (IRd )-limit as t 0,

11 i.e. there exist h 0 (x, λ) L 1 loc (IRd IR) and set E IR + of full measure such that for every ρ Cc 1 (IR), L 1 loc(ir d ) lim h(t, x, λ)ρ(λ)dλ = h 0 (x, λ)ρ(λ)dλ. t 0, t E IR IR

12 Recent advances in kinetic theory for mixtures of polyatomic gases Marzia BISI Department of Mathematics and Computer Sciences University of Parma, Italy It is well known that gas mixtures involved in physical applications are usually composed also of polyatomic species, for instance in simple dissociation and recombination problems or in the evolution of powders in the atmosphere. In kinetic approaches, each polyatomic gas is endowed with a (discrete or continuous) internal energy variable, to mimic non-translational degrees of freedom. The mathematical properties of the Boltzmann operator allowing energy transfer in each interaction are still under investigation even in absence of chemical reactions (implying also transfer of mass). In this talk we present some recent generalizations to the polyatomic frame of models of BGK type and of hydrodynamic limits well established for monatomic gas mixtures. Since Boltzmann-like equations are quite awkward to deal with, a consistent BGK relaxation model is proposed for inert or reactive polyatomic gases, determining in a unique way parameters of the Maxwellian attractors in terms of species number densities, mass velocities and temperatures; correct collision invariants and Maxwellian equilibria are recovered,

13 as well as fulfillment of Boltzmann H-theorem. Then, we investigate the asymptotic limit of the Boltzmann equations leading to the incompressible Navier-Stokes system; analogies and differences with respect to monatomic mixtures are presented, and contributions due to inelastic scattering are explicitly computed.

14 Kinetic modelling for respiratory aerosols, numerical treatment Laurent BOUDIN Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie, Paris, France In this talk, we first deal with the modelling and the discretization of an aerosol evolving in the air, in the respiration framework, within a domain which can be fixed or moving. The model consists in strongly coupling a Vlasov-type equation for the aerosol with the incompressible Navier-Stokes equations for the air, through a drag term between aerosol and air. We also discuss some basic numerical properties of the numerical code which was developped, and focus on the influence of the aerosol on the air flow through the drag term, which implies a stability condition. This set of works was partially funded by the ANR-08-JCJC and ANR-10-BLAN-1119 projects of the French research agency.

15 Derivation of BGK models for gas mixtures Stéphane BRULL Institut de Mathématiques de Bordeaux Université Bordeaux, France This paper is devoted to the construction of a BGK operator for gas mixtures. The construction is based as in introduced in some previous works on the introduction of relaxation coeficients and a principle of minimization of the entropy under constraints of moments. These free parameters are com pared with the free parameters introduced in the Thermodynamics of Irreversible Processes approach of the Navier-Stokes system. At the end the BGK model is proved to satisfy Fick and Newton laws.

16 The incompressible Navier-Stokes limit of the Boltzmann equation for mixtures of gases Laurent DESVILLETTES Centre de Mathématiques et Leurs Applications École normale supérieure de Cachan, France We present a work in common with Marzia Bisi, in which we extend the now classical paper by Claude Bardos, Francois Golse and Dave Levermore on the formal limit from the Boltzmann equation (for one monoatomic gas) towards the incompressible Navier-Stokes equation. Our starting point is the Boltzmann equation for a mixture of monoatomic gases, and we develop a formal asymptotics which leads to a system of incompressible Navier-Stokes equations for a mixture which includes the diffusion between different species (Fick s law), the viscosity terms, and the diffusion of temperature (Fourier s law). The parabolic character of the obtained equations is observed, and a comparison with the asymptotics from compressible to incompressible equations is described.

17 Convergence to Equilibrium for a Coagulation-Fragmentation Model with Degenerate Spatial Diffusion Klemens FELLNER Institute for Mathematics and Scientific Computing University of Graz, Austria klemens.fellner@uni-graz.at We prove explicit convergence to equilibrium for a coagulationfragmentation model with spatial diffusion. In particular, we study the continuous-in-size Smoluchowski s equation with constant coefficients. The main difficulties arise from considering realistic diffusion coefficients, which degenerate for large size clusters. The main techniques include a-priori estimates based on the dissipation of an entropy functional, entropy entropydissipation estimates, moment bounds and duality methods.

18 A diffusion limit for gaseous mixtures Bérénice GREC Laboratoire MAP5 Université Paris Descartes, France Many works dealing with the derivation of macroscopic equations starting from kinetic theory considered a mono-species, monatomic and ideal gas. However, common physical situations can be far more intricate, e.g. multi-species mixtures. In the case of mixtures, it is of interest to derive the macroscopic equations -even formally- from kinetic models, in order to link different modelling levels and identify the range of validity of the equations. On the other hand, the time evolution of diffusive phenomena for mixtures at a macroscopic level is well described by the Maxwell-Stefan equations. The mathematical study of the Maxwell-Stefan system is, however, very recent and solid results on the subject appeared only in the last years. In this talk, we present the Maxwell-Stefan equations and some of their properties, and we establish formally the relationship between the kinetic description of a mixture and the diffusion phenomena governed by the Maxwell-Stefan equations. This is a joint work with Laurent Boudin and Francesco Salvarani.

19 Kinetic relaxation models for reacting gas mixtures Maria GROPPI Department of Mathematics and Computer Sciences University of Parma, Italy Recent relaxation time-approximation models of BGK type for the kinetic description of chemically reacting gas mixtures are briefly reviewed [1, 5]. In spite of their simplicity, their capability in retaining the most significant mathematical and physical properties of the Boltzmann-type kinetic equations made them useful and tractable tools of investigation of chemical reactions in rarefied gas dynamics. For the numerical approximation of these BGK-type models, high order numerical methods, based on a semi-lagrangian formulation, have been studied and implemented [4]. As well known, the main drawback of the BGK approach is an uncorrect prediction of transport coefficients in the continuum limit. To overcome this problem, ellipsoidal (ES) BGK models for inert mixtures have been investigated [2,3]. Moving towards this direction, in this talk we present an ES-BGK model for a slowly reacting binary gas mixture, which is able to correctly reproduce, in the hydrodynamic limit, Fick s law for diffusion velocities and Newton s law for the viscous stress.

20 References [1] M. Bisi, M. Groppi, G. Spiga, Kinetic Bhatnagar-Gross- Krook model for fast reactive mixtures and its hydrodynamic limit, Phys. Rev. E 81 (2010) (pp. 1 9). [2] S. Brull, V. Pavan, J. Schneider, Derivation of a BGK model for mixtures, Eur. J. Mech. B Fluids 33 (2012) [3] M. Groppi, S. Monica, G. Spiga, A kinetic ellipsoidal BGK model for a binary gas mixture, Europhys. Lett. 96 (2011) (pp. 1 6). [4] M. Groppi, G. Russo, G. Stracquadanio, High order semilagrangian methods for the BGK equation, preprint July 2014, submitted. [5] M. Groppi, G. Spiga, A Bhatnagar-Gross-Krook type approach for chemically reacting gas mixtures, Phys. Fluids 16 (2004)

21 Shock structure for macroscopic multi-temperature model of binary mixtures: comparison with kinetic models Damir MADJAREVIĆ Department of Mechanics University of Novi Sad, Serbia The present study deals with the shock wave profiles in the macroscopic multi-temperature (MT) model of binary gaseous mixtures. For that purpose we have adopted the hyperbolic model developed within the framework of extended thermodynamics, with diffusion as only dissipative mechanism. Diffusivity and relaxation times are taken from kinetic theory for the mixture of monatomic gases. Recently, this model proved to give good agreement with experimental data in the case of helium and argon mixture. However, systematic study ought be restricted to shock structures propagating at speeds lower than the highest characteristic speed of the system. In the present study we include extra dissipation to eliminate restriction on the shock speed. This allows as to compare our results with more sophisticated kinetic solutions which were computed for hypothetic mixtures of gases. Numerical implementation of the MT model is considerably simpler than the one for Boltzmann equations for mixtures or the direct simulation Monte Carlo method (DSMC).

22 Shadow waves Marko NEDELJKOV Department of Mathematics and Informatics University of Novi Sad, Serbia Approximate solutions to conservation law systems called Shadow Waves (SDWs) have been used in lot of situations when classical solutions do not exist. More precisely, they appears in situations when one expect delta function to appear in solution ( infinite concentration of some variable). SDWs are represented by piecewise constant functions for a fixed time so they are well adopted to the following important issues: - one can easily check an entropy condition when (semi-)convex entropyentropy flux pair exists, and - wave interactions can be relatively easy investigated (simple use of the ideas from Wave Front Tracking algorithm) The idea of this talk is to show some strong points of SDW use, but also show some limitations ( blow up of SDW-solutions). Finally, we will present some preliminary results about multidimensional cases obtained with the collaboration with Michael Oberguggenberger, Lucas Neumann, and Manas Sahoo.

23 Mathematical modelling and analysis of polyatomic gases and mixtures in the context of kinetic theory of gases and fluid mechanics Milana PAVIĆ Department of Mathematics and Informatics University of Novi Sad, Serbia This talk is dedicated to the problems arising in the mathematical modelling of polyatomic gases, and mixtures of monatomic and polyatomic gases, in the context of the kinetic theory of gases and fluid mechanics. The kinetic theory of gases (Boltzmann equation and its variants) is a very active field of applied mathematics. At the same time, continuum theories of physics have quite similar aims and very often treat the same problems as kinetic theory, although from a different point of view. The issues related to their mutual relationships are rather involved and call for the application of mathematical techniques, as well as elaborate physical explanations of modeling problems. Considering polyatomic gases, our aim is to derive a macroscopic model for 14 moments starting from kinetic theory. At the microscopic level, one single parameter is introduced and it becomes an additional argument of the distribution function that enables to recover the proper equation of state at the macroscopic level. We first propose two independent hierarchies of the moment equations for polyatomic gases, which allow to

24 obtain conservation laws for mass density, momentum and total energy of a gas. Such hierarchies are usually truncated at some order. A method which provides an appropriate solution to the closure problem when one performs such a truncation is the maximization of entropy method. We formulate a variational problem for polyatomic gases, and give the solution for any number of moments. We explore in detail the physical case of 14 moments, in which the appropriate approximative distribution function yields the closed system, that is further compared with the model arising from extended thermodynamics. In particular, we compute production terms, and obtain the explicit expressions for relaxation times in terms of two parameters that can be fitted in order to obtain a correct value of the Prandtl number and/or temperature dependence of viscosity. When dealing with mixtures of polyatomic gases, the hydrodynamic approximation in which collisions between molecules of the same component of a mixture are much more frequent than collisions between the molecules of different components is studied. It leads to the so-called maxwellization of a distribution function: the distribution function of each species converges towards a Maxwellian distribution function, each with its own bulk velocity and temperature. With the help of this specified distribution function, balance laws for mass density, momentum and energy can be obtained for each component of the mixture, that can be compared with the multitemperature models for mixtures of Eulerian fluids coming out of extended thermodynamics. In particular, if we restrict the attention to processes which occur in the neighborhood of the average velocity and temperature of the mixture, the phenomenological coefficients

25 of extended thermodynamics can be determined from the source terms provided by the kinetic theory. Regarding mixtures of monatomic gases, we discuss the diffusion asymptotics of the Boltzmann equations. It amounts to scale the macroscopic arguments of the distribution function - time and space position - with the help of a small parameter interpreted as the mean free path. This asymptotics corresponds to a slow dynamics in space and an even slower one in time. The Hilbert expansion of each distribution function yields two equations. The first equation allows to state that the mixture is close to equilibrium. The second equation is a linear functional equation in the velocity variable. We prove the existence of a solution to this equation. On the one hand, when molecular masses are equal, the techniques introduced by Grad in order to prove the compactness of one part of the kernel can be extended to the multispecies case. On the other hand, we propose a new approach based on a change of variables in velocities for the same issue, which only holds when molecular masses are different.

26 About the validation of models for multicomponent systems Valeria RICCI Dipartimento di Metodi e Modelli Matematici Università di Palermo, Italy valeria.ricci@unipa.it We shall discuss the derivation of systems of partial differential equations, where hydrodynamic equations are coupled to mean field (Vlasov type) kinetic equations, as the asymptotic limit of suitable models on a smaller scale. This kind of PDE systems can be considered as simply modelling multicomponent flows in mixtures containing a dispersed phase, such as sprays or aerosols. In this talk we shall give an overview of results of various type obtained in cooperation with E. Bernard, L. Desvillettes, F. Golse.

27 Molecular Extended Thermodynamics of Rarefied Polyatomic Gases and Wave Velocities for Increasing Number of Moments Tommaso RUGGERI Department of Mathematics University of Bologna, Italy Molecular Extended Thermodynamics of rarefied polyatomic gases is characterized by two hierarchies of equations for moments of a suitable distribution function in which the internal degrees of freedom of a molecule is taken into account [1]. On the basis of physical relevance the truncation orders of the two hierarchies are proven to be not independent on each other, and the closure procedures based on the maximum entropy principle (MEP) and on the entropy principle (EP) are proven to be equivalent [2],[3]. The characteristic velocities of the emerging symmetric hyperbolic system of differential equations are compared to those obtained for monatomic gases and the lower bound estimate for the maximum equilibrium characteristic velocity established for monatomic gases (characterized by only one hierarchy for moments with truncation order of moments N) by Boillat and Ruggeri [4] E, max λ(n) c 0 ( 6 N 1 ), 5 2 ( ) 5 k c 0 = 3 m T

28 is proven to hold also for rarefied polyatomic gases independently from the degrees of freedom of a molecule [2]. References [1] M. Pavić, T. Ruggeri and S. Simić, Physica A 392, (2013). [2] T. Arima, A. Mentrelli and T. Ruggeri, Annals of Physics, (2014). [3] T. Arima, A. Mentrelli and T. Ruggeri, Rend. Lincei Mat. Appl (2014). [4] G. Boillat and T. Ruggeri, Cont. Mech. Thermodyn. 9, (1997).

29 Transport phenomena in evolutionary domains Francesco SALVARANI Dipartimento di matematica F. Casorati Università degli Studi di Pavia, Italia We study the transport equation in a time-dependent vessel with absorbing boundary, in any space dimension. We first prove existence and uniqueness, and subsequently we consider the problem of the time-asymptotic convergence to equilibrium. We show that the convergence towards equilibrium heavily depends on the initial data and on the evolution law of the vessel. Subsequently, we describe a numerical strategy to simulate the problem, based on a particle method implemented on generalpurpose graphics processing units (GPGPU). We observe that the parallelization procedure on GPGPU allows for a marked improvement of the performances when compared with the standard approach on CPU.

30 Generalizations of the classical wave equation within the theory of fractional calculus Dušan ZORICA Mathematical Institute Serbian Academy of Sciences and Arts, Belgrade, Serbia dusan The classical wave equation, containing integer order partial derivatives with respect to spatial and time coordinate, can be written in the form of system of equations consisting of: equation of motion of deformable body, constitutive equation (Hooke s law) and strain. The equation of motion, being the consequence of the Second Newton s law, is not generalized. Constitutive equation describes the response of the material to the applied force and it is generalized within the theory of fractional calculus, so that a class of linear constitutive equations describing viscoelastic materials is obtained. Constitutive equations containing fractional derivatives with respect to time proved to successfully model hereditary properties of viscoelastic materials. Spatial non-locality can be modeled not only by introducing derivatives (of integer, or fractional order) in the constitutive equation, but also generalizing strain to a non-local strain measure provided that it satisfies certain physical requirement. Laplace and Fourier transform methods are used in order to solve system of partial differential equations of integer and

31 fractional order. The solution is obtained as the convolution of initial conditions and solution kernel. Illustrative numerical examples are presented as well. Results are obtained in collaboration with Prof. Teodor Atanacković, Prof. Stevan Pilipović, dr Sanja Konjik, dr Ljubica Oparnica and dr Marko Janev. The joint work of Prof. Teodor Atanacković and Prof. Bogoljub Stanković is also addressed.