{"title":"Dissipative solutions to Hamiltonian systems","authors":"S. Bianchini, G. M. Leccese","doi":"10.3934/krm.2023019","DOIUrl":"https://doi.org/10.3934/krm.2023019","url":null,"abstract":"","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"80 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82728153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marzia Bisi, Maria Groppi, Enrico Lucchin, Giorgio Martalò
We propose a mixed Boltzmann–BGK model for mixtures of monatomic gases, where some kinds of collisions are described by bi–species Boltzmann operators and the others by the binary BGK terms given in [Bobylev et al., Kinetic and Related Models 11 (2018)], that is the relaxation model for mixtures with the closest structure to the Boltzmann one. At first, we assume that collisions occurring within the same species (intra-species) are modelled by Boltzmann operators, while interactions between different constituents (inter-species) are described by BGK terms. This option allows us to rigorously derive hydrodynamic equations not only in the classical collision dominated regime, but also in situations with intra–species collisions playing the dominant role (as in mixtures with very disparate particle masses). Then, we present a general form of this mixed Boltzmann–BGK model, characterized by further parameters allowing us to select which binary interactions have to be described by Boltzmann integrals or by BGK operators. We prove that this model preserves conservations of global momentum and energy, positivity of all temperatures and the validity of Boltzmann H-theorem, allowing us to conclude that the unique admissible equilibrium state is the expected Maxwellian distribution with all species sharing a common mean velocity and a common temperature.
{"title":"A mixed Boltzmann–BGK model for inert gas mixtures","authors":"Marzia Bisi, Maria Groppi, Enrico Lucchin, Giorgio Martalò","doi":"10.3934/krm.2023037","DOIUrl":"https://doi.org/10.3934/krm.2023037","url":null,"abstract":"We propose a mixed Boltzmann–BGK model for mixtures of monatomic gases, where some kinds of collisions are described by bi–species Boltzmann operators and the others by the binary BGK terms given in [Bobylev et al., Kinetic and Related Models 11 (2018)], that is the relaxation model for mixtures with the closest structure to the Boltzmann one. At first, we assume that collisions occurring within the same species (intra-species) are modelled by Boltzmann operators, while interactions between different constituents (inter-species) are described by BGK terms. This option allows us to rigorously derive hydrodynamic equations not only in the classical collision dominated regime, but also in situations with intra–species collisions playing the dominant role (as in mixtures with very disparate particle masses). Then, we present a general form of this mixed Boltzmann–BGK model, characterized by further parameters allowing us to select which binary interactions have to be described by Boltzmann integrals or by BGK operators. We prove that this model preserves conservations of global momentum and energy, positivity of all temperatures and the validity of Boltzmann H-theorem, allowing us to conclude that the unique admissible equilibrium state is the expected Maxwellian distribution with all species sharing a common mean velocity and a common temperature.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135319258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the fractional diffusion limit of a Linear Boltzmann equation with heavy-tailed velocity equilibrium in a half-space with Maxwell boundary conditions. We derive a new confined version of the fractional Laplacian in a spatially bounded domain and show uniqueness of weak solution to the associated non-local diffusion equation.
{"title":"Fractional diffusion limit of a linear Boltzmann model with reflective boundaries in a half-space","authors":"Ludovic Cesbron","doi":"10.3934/krm.2023033","DOIUrl":"https://doi.org/10.3934/krm.2023033","url":null,"abstract":"We investigate the fractional diffusion limit of a Linear Boltzmann equation with heavy-tailed velocity equilibrium in a half-space with Maxwell boundary conditions. We derive a new confined version of the fractional Laplacian in a spatially bounded domain and show uniqueness of weak solution to the associated non-local diffusion equation.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784802","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An internal state kinetic model for chemically reacting mixtures of monatomic and polyatomic gases","authors":"M. Bisi, Thomas Borsoni, M. Groppi","doi":"10.3934/krm.2023023","DOIUrl":"https://doi.org/10.3934/krm.2023023","url":null,"abstract":"","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"14 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74945612","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
F. Deluzet, J. Narski, M. Ndiaye, G. Hagelaar, J. Boeuf
{"title":"Numerical methods and macroscopic models of magnetically confined low temperature plasmas","authors":"F. Deluzet, J. Narski, M. Ndiaye, G. Hagelaar, J. Boeuf","doi":"10.3934/krm.2023002","DOIUrl":"https://doi.org/10.3934/krm.2023002","url":null,"abstract":"","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"52 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73022787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nouamane Bakhdil, Abdelghani El Mousaoui, A. Hakim
{"title":"A kinetic theory approach to model pedestrian social groups behavior in bounded domain","authors":"Nouamane Bakhdil, Abdelghani El Mousaoui, A. Hakim","doi":"10.3934/krm.2023017","DOIUrl":"https://doi.org/10.3934/krm.2023017","url":null,"abstract":"","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"20 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87730383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study fragmentation equations with power-law fragmentation rates and polynomial daughter fragments distribution function $p(s)$. The corresponding selfsimillar solutions are analysed and their exponentially decaying asymptotic behaviour and $C^{infty }$ regularity deduced. Stability of selfsimilar solutions (under smooth exponentially decaying perturbations), with sharp exponential decay rates in time are proved, as well as $C^{infty }$ regularity of solutions for $t>0$. The results are based on explicit expansion in terms of generalized Laguerre polynomials and the analysis of such expansions. For perturbations with power-law decay at infinity stability is also proved. Finally, we consider real analytic $p(s)$.
{"title":"Stability of selfsimilar solutions to the fragmentation equation with polynomial daughter fragments distribution","authors":"M. Fontelos","doi":"10.3934/krm.2023016","DOIUrl":"https://doi.org/10.3934/krm.2023016","url":null,"abstract":"We study fragmentation equations with power-law fragmentation rates and polynomial daughter fragments distribution function $p(s)$. The corresponding selfsimillar solutions are analysed and their exponentially decaying asymptotic behaviour and $C^{infty }$ regularity deduced. Stability of selfsimilar solutions (under smooth exponentially decaying perturbations), with sharp exponential decay rates in time are proved, as well as $C^{infty }$ regularity of solutions for $t>0$. The results are based on explicit expansion in terms of generalized Laguerre polynomials and the analysis of such expansions. For perturbations with power-law decay at infinity stability is also proved. Finally, we consider real analytic $p(s)$.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"59 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89062840","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Entropic property of the Ellipsoidal Statistical model with the Prandtl number Pr below 2/3 is discussed. Although 2/3 is the lower bound of Pr for the H theorem to hold unconditionally, it is shown that the theorem still holds even for begin{document}$ mathrm{Pr} , provided that anisotropy of stress tensor satisfies a certain criterion. The practical tolerance of that criterion is assessed numerically by the strong normal shock wave and the Couette flow problems. A couple of moving plate tests are also conducted.
{"title":"On the entropic property of the Ellipsoidal Statistical model with the prandtl number below 2/3","authors":"S. Takata, Masanari Hattori, Takumu Miyauchi","doi":"10.3934/krm.2020041","DOIUrl":"https://doi.org/10.3934/krm.2020041","url":null,"abstract":"Entropic property of the Ellipsoidal Statistical model with the Prandtl number Pr below 2/3 is discussed. Although 2/3 is the lower bound of Pr for the H theorem to hold unconditionally, it is shown that the theorem still holds even for begin{document}$ mathrm{Pr} , provided that anisotropy of stress tensor satisfies a certain criterion. The practical tolerance of that criterion is assessed numerically by the strong normal shock wave and the Couette flow problems. A couple of moving plate tests are also conducted.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"13 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81973939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the four waves spatial homogeneous kinetic equation arising in wave turbulence theory. We study the long-time behaviour and existence of solutions around the Rayleigh-Jeans equilibrium solutions. For cut-off'd frequencies, we show that for dispersion relations weakly perturbed around the quadratic case, the linearized operator around the Rayleigh-Jeans equilibria is coercive. We then pass to the fully nonlinear operator, showing an $L^2$ - stability for initial data close to Rayleigh-Jeans.
{"title":"$ L^2 $-stability near equilibrium for the 4 waves kinetic equation","authors":"A. Menegaki","doi":"10.3934/krm.2023031","DOIUrl":"https://doi.org/10.3934/krm.2023031","url":null,"abstract":"We consider the four waves spatial homogeneous kinetic equation arising in wave turbulence theory. We study the long-time behaviour and existence of solutions around the Rayleigh-Jeans equilibrium solutions. For cut-off'd frequencies, we show that for dispersion relations weakly perturbed around the quadratic case, the linearized operator around the Rayleigh-Jeans equilibria is coercive. We then pass to the fully nonlinear operator, showing an $L^2$ - stability for initial data close to Rayleigh-Jeans.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"126 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72508804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the backward problem for the one-dimensional Vlasov-Poisson system with massless electrons, and we show the Landau damping by fixing the asymptotic behaviour of our solution.
{"title":"Backward problem for the 1D ionic Vlasov-Poisson equation","authors":"A. Gagnebin","doi":"10.3934/krm.2023024","DOIUrl":"https://doi.org/10.3934/krm.2023024","url":null,"abstract":"In this paper, we study the backward problem for the one-dimensional Vlasov-Poisson system with massless electrons, and we show the Landau damping by fixing the asymptotic behaviour of our solution.","PeriodicalId":49942,"journal":{"name":"Kinetic and Related Models","volume":"17 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75556489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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