SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5386-5408, August 2024. Abstract. This paper deals with the stability analysis for steady-states perturbed by the full cross-diffusion limit of the SKT model with Dirichlet boundary conditions. Our previous result showed that positive steady-states consist of the branch of small coexistence type bifurcating from the trivial solution and the branches of segregation type bifurcating from points on the branch of small coexistence type. This paper shows the Morse index of steady-states on the branches and constructs the local unstable manifold around each steady-state of which the dimension is equal to the Morse index.
{"title":"Morse Index of Steady-States to the SKT Model with Dirichlet Boundary Conditions","authors":"Kousuke Kuto, Homare Sato","doi":"10.1137/23m1627705","DOIUrl":"https://doi.org/10.1137/23m1627705","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5386-5408, August 2024. <br/> Abstract. This paper deals with the stability analysis for steady-states perturbed by the full cross-diffusion limit of the SKT model with Dirichlet boundary conditions. Our previous result showed that positive steady-states consist of the branch of small coexistence type bifurcating from the trivial solution and the branches of segregation type bifurcating from points on the branch of small coexistence type. This paper shows the Morse index of steady-states on the branches and constructs the local unstable manifold around each steady-state of which the dimension is equal to the Morse index.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":"22 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141864225","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5409-5444, August 2024. Abstract. In this work, we generalize Kac’s original many-particle binary stochastic model to derive a space homogeneous Boltzmann equation that includes a linear combination of higher-order collisional terms. First, we prove an abstract theorem about convergence from a finite hierarchy to an infinite hierarchy of coupled equations. We apply this convergence theorem on hierarchies for marginals corresponding to the generalized Kac model mentioned above. As a corollary, we prove propagation of chaos for the marginals associated to the generalized Kac model. In particular, the first marginal converges towards the solution of a Boltzmann equation including interactions up to a finite order and whose collision kernel is of Maxwell type with cut-off.
{"title":"Derivation of a Boltzmann Equation with Higher-Order Collisions from a Generalized Kac Model","authors":"Esteban Cárdenas, Nataša Pavlović, William Warner","doi":"10.1137/23m1606150","DOIUrl":"https://doi.org/10.1137/23m1606150","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5409-5444, August 2024. <br/> Abstract. In this work, we generalize Kac’s original many-particle binary stochastic model to derive a space homogeneous Boltzmann equation that includes a linear combination of higher-order collisional terms. First, we prove an abstract theorem about convergence from a finite hierarchy to an infinite hierarchy of coupled equations. We apply this convergence theorem on hierarchies for marginals corresponding to the generalized Kac model mentioned above. As a corollary, we prove propagation of chaos for the marginals associated to the generalized Kac model. In particular, the first marginal converges towards the solution of a Boltzmann equation including interactions up to a finite order and whose collision kernel is of Maxwell type with cut-off.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":"8 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141864328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5285-5329, August 2024. Abstract. We study several different aspects of the energy equipartition principle for water waves. We prove a virial identity that implies that the potential energy is equal, on average, to a modified version of the kinetic energy. This is an exact identity for the complete nonlinear water-wave problem, which is valid for arbitrary solutions. As an application, we obtain nonperturbative results about the free-surface Rayleigh–Taylor instability, for any nonzero initial data. We also derive exact virial identities involving higher order energies. We illustrate these results by an explicit computation for standing waves. As an aside, we prove trace inequalities for harmonic functions in Lipschitz domains which are optimal with respect to the dependence in the Lipschitz norm of the graph.
{"title":"Virial Theorems and Equipartition of Energy for Water Waves","authors":"Thomas Alazard, Claude Zuily","doi":"10.1137/23m1574312","DOIUrl":"https://doi.org/10.1137/23m1574312","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5285-5329, August 2024. <br/> Abstract. We study several different aspects of the energy equipartition principle for water waves. We prove a virial identity that implies that the potential energy is equal, on average, to a modified version of the kinetic energy. This is an exact identity for the complete nonlinear water-wave problem, which is valid for arbitrary solutions. As an application, we obtain nonperturbative results about the free-surface Rayleigh–Taylor instability, for any nonzero initial data. We also derive exact virial identities involving higher order energies. We illustrate these results by an explicit computation for standing waves. As an aside, we prove trace inequalities for harmonic functions in Lipschitz domains which are optimal with respect to the dependence in the Lipschitz norm of the graph.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":"26 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5330-5349, August 2024. Abstract. The relative entropy for two different degenerate diffusion processes is estimated by using the Wasserstein distance of initial distributions and the difference between coefficients. As applications, the entropy-cost inequality and exponential ergodicity in entropy are derived for distribution dependent stochastic Hamiltonian systems associated with nonlinear kinetic Fokker–Planck equations.
{"title":"Entropy Estimate for Degenerate SDEs with Applications to Nonlinear Kinetic Fokker–Planck Equations","authors":"Zhongmin Qian, Panpan Ren, Feng-Yu Wang","doi":"10.1137/24m1634473","DOIUrl":"https://doi.org/10.1137/24m1634473","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5330-5349, August 2024. <br/> Abstract. The relative entropy for two different degenerate diffusion processes is estimated by using the Wasserstein distance of initial distributions and the difference between coefficients. As applications, the entropy-cost inequality and exponential ergodicity in entropy are derived for distribution dependent stochastic Hamiltonian systems associated with nonlinear kinetic Fokker–Planck equations.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":"41 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5262-5284, August 2024. Abstract. We consider a translation-invariant system of [math] bosons in [math] that interact through a repulsive two-body potential with scattering length of order [math] in the limit [math]. We derive second order expressions for the one- and two-particle reduced density matrix matrices of the Gibbs state at fixed positive temperatures, thus obtaining a justification of Bogoliubov’s prediction on the fluctuations around the condensate.
{"title":"Second Order Expansion of Gibbs State Reduced Density Matrices in the Gross–Pitaevskii Regime","authors":"Christian Brennecke, Jinyeop Lee, Phan Thành Nam","doi":"10.1137/23m1608215","DOIUrl":"https://doi.org/10.1137/23m1608215","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5262-5284, August 2024. <br/> Abstract. We consider a translation-invariant system of [math] bosons in [math] that interact through a repulsive two-body potential with scattering length of order [math] in the limit [math]. We derive second order expressions for the one- and two-particle reduced density matrix matrices of the Gibbs state at fixed positive temperatures, thus obtaining a justification of Bogoliubov’s prediction on the fluctuations around the condensate.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":"18 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5209-5261, August 2024. Abstract. We study well-posedness for the stochastic transport equation with transport noise, as introduced by Flandoli, Gubinelli, and Priola [Invent. Math., 180 (2010), pp. 1–53]. We consider periodic solutions in [math] for divergence-free drifts [math] for a large class of parameters. We prove local-in-time pathwise nonuniqueness and compare them to uniqueness results by Beck et al. [Electron. J. Probab., 24 (2019), 136], addressing a conjecture made by these authors, in the case of bounded-in-time drifts for a large range of spatial parameters. To this end, we use convex integration techniques to construct velocity fields [math] for which several solutions [math] exist in the classes mentioned above. The main novelty lies in the ability to construct deterministic drift coefficients, which makes it necessary to consider a convex integration scheme with a constraint, which poses a series of technical difficulties.
{"title":"Local Nonuniqueness for Stochastic Transport Equations with Deterministic Drift","authors":"Stefano Modena, Andre Schenke","doi":"10.1137/23m1589104","DOIUrl":"https://doi.org/10.1137/23m1589104","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5209-5261, August 2024. <br/> Abstract. We study well-posedness for the stochastic transport equation with transport noise, as introduced by Flandoli, Gubinelli, and Priola [Invent. Math., 180 (2010), pp. 1–53]. We consider periodic solutions in [math] for divergence-free drifts [math] for a large class of parameters. We prove local-in-time pathwise nonuniqueness and compare them to uniqueness results by Beck et al. [Electron. J. Probab., 24 (2019), 136], addressing a conjecture made by these authors, in the case of bounded-in-time drifts for a large range of spatial parameters. To this end, we use convex integration techniques to construct velocity fields [math] for which several solutions [math] exist in the classes mentioned above. The main novelty lies in the ability to construct deterministic drift coefficients, which makes it necessary to consider a convex integration scheme with a constraint, which poses a series of technical difficulties.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":"61 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5175-5208, August 2024. Abstract. This paper focuses on systems of nonlinear second-order stochastic differential equations with multiscales. The motivation for our study stems from mathematical physics and statistical mechanics, for example, Langevin dynamics and stochastic acceleration in a random environment. Our aim is to carry out asymptotic analysis to establish large deviations principles. Our focus is on obtaining the desired results for systems under weaker conditions. When the fast-varying process is a diffusion, neither Lipschitz continuity nor linear growth needs to be assumed. Our approach is based on combinations of the intuition from Smoluchowski–Kramers approximation and the methods initiated in [A. A. Puhalskii, Ann. Probab., 44 (2016), pp. 3111–3186] relying on the concepts of relatively large deviations compactness and the identification of rate functions. When the fast-varying process is under a general setup with no specified structure, the paper establishes the large deviations principle of the underlying system under the assumption on the local large deviations principles of the corresponding first-order system.
{"title":"Second-Order Fast-Slow Stochastic Systems","authors":"Nhu N. Nguyen, George Yin","doi":"10.1137/23m1567382","DOIUrl":"https://doi.org/10.1137/23m1567382","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5175-5208, August 2024. <br/> Abstract. This paper focuses on systems of nonlinear second-order stochastic differential equations with multiscales. The motivation for our study stems from mathematical physics and statistical mechanics, for example, Langevin dynamics and stochastic acceleration in a random environment. Our aim is to carry out asymptotic analysis to establish large deviations principles. Our focus is on obtaining the desired results for systems under weaker conditions. When the fast-varying process is a diffusion, neither Lipschitz continuity nor linear growth needs to be assumed. Our approach is based on combinations of the intuition from Smoluchowski–Kramers approximation and the methods initiated in [A. A. Puhalskii, Ann. Probab., 44 (2016), pp. 3111–3186] relying on the concepts of relatively large deviations compactness and the identification of rate functions. When the fast-varying process is under a general setup with no specified structure, the paper establishes the large deviations principle of the underlying system under the assumption on the local large deviations principles of the corresponding first-order system.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":"16 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5144-5174, August 2024. Abstract. In this paper, we study the Lipschitz continuous dependence of conservative Hölder continuous weak solutions to a variational wave system derived from a model for nematic liquid crystals. Since the solution of this system generally forms a finite time cusp singularity, the solution flow is not Lipschitz continuous under the Sobolev metric used in the existence and uniqueness theory. We establish a Finsler type optimal transport metric, and show the Lipschitz continuous dependence of the solution on the initial data under this metric. This kind of Finsler type optimal transport metrics was first established in [A. Bressan and G. Chen, Arch. Ration. Mech. Anal., 226 (2017), pp. 1303-1343] for the scalar variational wave equation. This equation can be used to describe the unit direction [math] of mean orientation of nematic liquid crystals, when [math] is restricted on a circle. The model considered in this paper describes the propagation of [math] without this restriction, i.e. [math], takes any value on the unit sphere. So we need to consider a wave system instead of a scalar equation.
SIAM 数学分析期刊》，第 56 卷第 4 期，第 5144-5174 页，2024 年 8 月。 摘要本文研究了由向列液晶模型导出的变分波系统的保守霍尔德连续弱解的 Lipschitz 连续依赖性。由于该系统的解一般会形成一个有限时间的尖顶奇点，因此解流在存在性和唯一性理论中使用的 Sobolev 度量下不是 Lipschitz 连续的。我们建立了一种 Finsler 型最优传输度量，并证明了在此度量下解对初始数据的 Lipschitz 连续依赖性。Bressan and G. Chen, Arch.Ration.Mech.Anal., 226 (2017), pp. 1303-1343]中首次建立了标量变分波方程。当[math]被限制在一个圆上时，该方程可用于描述向列液晶平均取向的单位方向[math]。本文所考虑的模型描述的是没有这种限制的[math]传播，即[math]在单位球面上取任意值。因此，我们需要考虑一个波系，而不是标量方程。
{"title":"Lipschitz Optimal Transport Metric for a Wave System Modeling Nematic Liquid Crystals","authors":"Hong Cai, Geng Chen, Yannan Shen","doi":"10.1137/24m1629547","DOIUrl":"https://doi.org/10.1137/24m1629547","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5144-5174, August 2024. <br/> Abstract. In this paper, we study the Lipschitz continuous dependence of conservative Hölder continuous weak solutions to a variational wave system derived from a model for nematic liquid crystals. Since the solution of this system generally forms a finite time cusp singularity, the solution flow is not Lipschitz continuous under the Sobolev metric used in the existence and uniqueness theory. We establish a Finsler type optimal transport metric, and show the Lipschitz continuous dependence of the solution on the initial data under this metric. This kind of Finsler type optimal transport metrics was first established in [A. Bressan and G. Chen, Arch. Ration. Mech. Anal., 226 (2017), pp. 1303-1343] for the scalar variational wave equation. This equation can be used to describe the unit direction [math] of mean orientation of nematic liquid crystals, when [math] is restricted on a circle. The model considered in this paper describes the propagation of [math] without this restriction, i.e. [math], takes any value on the unit sphere. So we need to consider a wave system instead of a scalar equation.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":"157 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5099-5143, August 2024. Abstract. We study statistical solutions of the incompressible Navier–Stokes equation and their vanishing viscosity limit. We show that a formulation using correlation measures as in [U. S. Fjordholm, S. Lanthaler, and S. Mishra, Arch. Ration. Mech. Anal., 226 (2017), pp. 809–849] and moment equations is equivalent to statistical solutions in the Foiaş–Prodi sense. Under the assumption of weak scaling, a weaker version of Kolmogorov’s self-similarity at small scales hypothesis that allows for intermittency corrections, we show that the limit is a statistical solution of the incompressible Euler equations. To pass to the limit, we derive a Kármán–Howarth–Monin relation for statistical solutions and combine it with the weak scaling assumption and a compactness theorem for correlation measures from [U. S. Fjordholm et al., Math. Models Methods Appl. Sci., 30 (2020), pp. 539–609].
SIAM 数学分析期刊》，第 56 卷第 4 期，第 5099-5143 页，2024 年 8 月。 摘要。我们研究不可压缩纳维-斯托克斯方程的统计解及其粘度消失极限。我们证明了使用相关测量法的公式 [U. S. Fjordholm, J. M.S. Fjordholm, S. Lanthaler, and S. Mishra, Arch.Ration.Mech.Anal., 226 (2017), pp. 809-849] 和矩方程等价于 Foiaş-Prodi 意义上的统计解。在允许间歇性修正的弱缩放假设（即柯尔莫戈罗夫小尺度自相似性假设的弱化版本）下，我们证明了极限是不可压缩欧拉方程的统计解。为了达到极限，我们推导出了统计解的 Kármán-Howarth-Monin 关系，并将其与弱比例假设和相关量的紧凑性定理结合起来[U. S. Fjordholm et al.S. Fjordholm 等人，Math.模型方法应用科学》，30 (2020)，第 539-609 页]。
{"title":"On the Vanishing Viscosity Limit of Statistical Solutions of the Incompressible Navier–Stokes Equations","authors":"Ulrik Skre Fjordholm, Siddhartha Mishra, Franziska Weber","doi":"10.1137/23m1566261","DOIUrl":"https://doi.org/10.1137/23m1566261","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5099-5143, August 2024. <br/> Abstract. We study statistical solutions of the incompressible Navier–Stokes equation and their vanishing viscosity limit. We show that a formulation using correlation measures as in [U. S. Fjordholm, S. Lanthaler, and S. Mishra, Arch. Ration. Mech. Anal., 226 (2017), pp. 809–849] and moment equations is equivalent to statistical solutions in the Foiaş–Prodi sense. Under the assumption of weak scaling, a weaker version of Kolmogorov’s self-similarity at small scales hypothesis that allows for intermittency corrections, we show that the limit is a statistical solution of the incompressible Euler equations. To pass to the limit, we derive a Kármán–Howarth–Monin relation for statistical solutions and combine it with the weak scaling assumption and a compactness theorem for correlation measures from [U. S. Fjordholm et al., Math. Models Methods Appl. Sci., 30 (2020), pp. 539–609].","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":"4 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5079-5098, August 2024. Abstract. We consider deterministic mean field games (MFGs) in all Euclidean space with a cost functional continuous with respect to the distribution of the agents and attaining its minima in a compact set. We first show that the static MFG with such a cost has an equilibrium, and we build from it a solution of the ergodic MFG system of first order PDEs with the same cost. Next we address the long-time limit of the solutions to finite horizon MFGs with cost functional satisfying various additional assumptions, but not the classical Lasry–Lions monotonicity condition. Instead we assume that the cost has the same set of minima for all measures describing the population. We prove the convergence of the distribution of the agents and of the value function to a solution of the ergodic MFG system as the horizon of the game tends to infinity, extending to this class of MFGs some results of weak KAM theory.
{"title":"Long-Time Behavior of Deterministic Mean Field Games with Nonmonotone Interactions","authors":"Martino Bardi, Hicham Kouhkouh","doi":"10.1137/23m1608100","DOIUrl":"https://doi.org/10.1137/23m1608100","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5079-5098, August 2024. <br/> Abstract. We consider deterministic mean field games (MFGs) in all Euclidean space with a cost functional continuous with respect to the distribution of the agents and attaining its minima in a compact set. We first show that the static MFG with such a cost has an equilibrium, and we build from it a solution of the ergodic MFG system of first order PDEs with the same cost. Next we address the long-time limit of the solutions to finite horizon MFGs with cost functional satisfying various additional assumptions, but not the classical Lasry–Lions monotonicity condition. Instead we assume that the cost has the same set of minima for all measures describing the population. We prove the convergence of the distribution of the agents and of the value function to a solution of the ergodic MFG system as the horizon of the game tends to infinity, extending to this class of MFGs some results of weak KAM theory.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":"19 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141722213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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