Pub Date : 2023-11-08DOI: 10.1080/03605302.2023.2276564
Charles Bertucci
AbstractWe present the notion of monotone solution of mean field games master equations in the case of a continuous state space. We establish the existence, uniqueness and stability of such solutions under standard assumptions. This notion allows us to work with solutions which are merely continuous in the measure argument, in the case of first order master equations. We study several structures of common noises, in particular ones in which common jumps (or aggregate shocks) can happen randomly, and ones in which the correlation of randomness is carried by an additional parameter.KEYWORDS: Mean Field GamesMaster equationWeak solutions Notes1 C−k is the topological dual set of C k while we understand C1,1 in the sense that the usual differential is a Lipschitz function.
{"title":"Monotone solutions for mean field games master equations: continuous state space and common noise","authors":"Charles Bertucci","doi":"10.1080/03605302.2023.2276564","DOIUrl":"https://doi.org/10.1080/03605302.2023.2276564","url":null,"abstract":"AbstractWe present the notion of monotone solution of mean field games master equations in the case of a continuous state space. We establish the existence, uniqueness and stability of such solutions under standard assumptions. This notion allows us to work with solutions which are merely continuous in the measure argument, in the case of first order master equations. We study several structures of common noises, in particular ones in which common jumps (or aggregate shocks) can happen randomly, and ones in which the correlation of randomness is carried by an additional parameter.KEYWORDS: Mean Field GamesMaster equationWeak solutions Notes1 C−k is the topological dual set of C k while we understand C1,1 in the sense that the usual differential is a Lipschitz function.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":" 46","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135292959","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}
Pub Date : 2023-09-02DOI: 10.1080/03605302.2023.2263205
Xuantao Chen, Hans Lindblad
AbstractWe study the coupled wave-Klein-Gordon systems, introduced by LeFloch-Ma and then Ionescu-Pausader, to model the nonlinear effects from the Einstein-Klein-Gordon equation in harmonic coordinates. We first go over a slightly simplified version of global existence based on LeFloch-Ma, and then derive the asymptotic behavior of the system. The asymptotics of the Klein-Gordon field consist of a modified phase times a homogeneous function, and the asymptotics of the wave equation consist of a radiation field in the wave zone and an interior homogeneous solution coupled to the Klein-Gordon asymptotics. We then consider the inverse problem, the scattering from infinity. We show that given the type of asymptotic behavior at infinity, there exist solutions of the system that present the exact same behavior.KEYWORDS: AsymptoticsScattering from infinityWave-Klein-Gordon systems Notes1 In fact, u3 would be zero if we only consider the term ϕ02. The presence of u3 comes from the lower order terms in (∂tϕ0)2.2 We use the Einstein summation convention. Also, when the repeated index is spatial, we define the expression to be the sum regardless of whether it is upper or lower, as the spatial part of the Minkowski metric is Euclidean.3 Here we use the decay |△yϕ|≲ερ−32+δ(1−|y|2)74−δ, for which we only have it with 47 replaced by 45 at this stage, but this can also be shown by dealing with commutators like the existence proof.4 Recall in Lemma 4.5, we require that α cannot be too worse. i.e. close to –1. However, once α satisfies this condition, the value of α does not affect the outcome of the estimate.5 Note that here we take the contribution from (∂tϕ)2 into account, which we did not consider in the introduction part for simplicity.Additional informationFundingH.L. was supported in part by Simons Collaboration Grant 638955. X.C. thanks Junfu Yao for helpful discussions.
{"title":"Asymptotics and scattering for wave Klein-Gordon systems","authors":"Xuantao Chen, Hans Lindblad","doi":"10.1080/03605302.2023.2263205","DOIUrl":"https://doi.org/10.1080/03605302.2023.2263205","url":null,"abstract":"AbstractWe study the coupled wave-Klein-Gordon systems, introduced by LeFloch-Ma and then Ionescu-Pausader, to model the nonlinear effects from the Einstein-Klein-Gordon equation in harmonic coordinates. We first go over a slightly simplified version of global existence based on LeFloch-Ma, and then derive the asymptotic behavior of the system. The asymptotics of the Klein-Gordon field consist of a modified phase times a homogeneous function, and the asymptotics of the wave equation consist of a radiation field in the wave zone and an interior homogeneous solution coupled to the Klein-Gordon asymptotics. We then consider the inverse problem, the scattering from infinity. We show that given the type of asymptotic behavior at infinity, there exist solutions of the system that present the exact same behavior.KEYWORDS: AsymptoticsScattering from infinityWave-Klein-Gordon systems Notes1 In fact, u3 would be zero if we only consider the term ϕ02. The presence of u3 comes from the lower order terms in (∂tϕ0)2.2 We use the Einstein summation convention. Also, when the repeated index is spatial, we define the expression to be the sum regardless of whether it is upper or lower, as the spatial part of the Minkowski metric is Euclidean.3 Here we use the decay |△yϕ|≲ερ−32+δ(1−|y|2)74−δ, for which we only have it with 47 replaced by 45 at this stage, but this can also be shown by dealing with commutators like the existence proof.4 Recall in Lemma 4.5, we require that α cannot be too worse. i.e. close to –1. However, once α satisfies this condition, the value of α does not affect the outcome of the estimate.5 Note that here we take the contribution from (∂tϕ)2 into account, which we did not consider in the introduction part for simplicity.Additional informationFundingH.L. was supported in part by Simons Collaboration Grant 638955. X.C. thanks Junfu Yao for helpful discussions.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134969703","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}
Pub Date : 2023-09-02DOI: 10.1080/03605302.2023.2264611
Rene Carmona, Quentin Cormier, H. Mete Soner
AbstractThe classical Kuramoto model is studied in the setting of an infinite horizon mean field game. The system is shown to exhibit both synchronization and phase transition. Incoherence below a critical value of the interaction parameter is demonstrated by the stability of the uniform distribution. Above this value, the game bifurcates and develops self-organizing time homogeneous Nash equilibria. As interactions get stronger, these stationary solutions become fully synchronized. Results are proved by an amalgam of techniques from nonlinear partial differential equations, viscosity solutions, stochastic optimal control and stochastic processes.KEYWORDS: Mean field gamesKuramoto modelsynchronizationviscosity solutions2020 MATHEMATICS SUBJECT CLASSIFICATION: 35Q8935D4039N8091A1692B25 Additional informationFundingResearch of Carmona was partially supported by AFOSR FA9550-19-1-0291 and ARPA-E DE-AR0001289. Research of Soner was partially supported by the National Science Foundation grant DMS 2106462.
{"title":"Synchronization in a Kuramoto mean field game","authors":"Rene Carmona, Quentin Cormier, H. Mete Soner","doi":"10.1080/03605302.2023.2264611","DOIUrl":"https://doi.org/10.1080/03605302.2023.2264611","url":null,"abstract":"AbstractThe classical Kuramoto model is studied in the setting of an infinite horizon mean field game. The system is shown to exhibit both synchronization and phase transition. Incoherence below a critical value of the interaction parameter is demonstrated by the stability of the uniform distribution. Above this value, the game bifurcates and develops self-organizing time homogeneous Nash equilibria. As interactions get stronger, these stationary solutions become fully synchronized. Results are proved by an amalgam of techniques from nonlinear partial differential equations, viscosity solutions, stochastic optimal control and stochastic processes.KEYWORDS: Mean field gamesKuramoto modelsynchronizationviscosity solutions2020 MATHEMATICS SUBJECT CLASSIFICATION: 35Q8935D4039N8091A1692B25 Additional informationFundingResearch of Carmona was partially supported by AFOSR FA9550-19-1-0291 and ARPA-E DE-AR0001289. Research of Soner was partially supported by the National Science Foundation grant DMS 2106462.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134948596","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}
Pub Date : 2023-09-02DOI: 10.1080/03605302.2023.2263208
Roland Donninger, Matthias Ostermann
This paper is concerned with the Cauchy problem for an energy-supercritical nonlinear wave equation in odd space dimensions that arises in equivariant Yang-Mills theory. In each dimension, there is a self-similar finite-time blowup solution to this equation known in closed form. It will be proved that this profile is stable in the whole space under small perturbations of the initial data. The blowup analysis is based on a recently developed coordinate system called hyperboloidal similarity coordinates and depends crucially on growth estimates for the free wave evolution, which will be constructed systematically for odd space dimensions in the first part of this paper. This allows to develop a nonlinear stability theory beyond the singularity.
{"title":"A Globally Stable Self-Similar Blowup Profile in Energy Supercritical Yang-Mills Theory","authors":"Roland Donninger, Matthias Ostermann","doi":"10.1080/03605302.2023.2263208","DOIUrl":"https://doi.org/10.1080/03605302.2023.2263208","url":null,"abstract":"This paper is concerned with the Cauchy problem for an energy-supercritical nonlinear wave equation in odd space dimensions that arises in equivariant Yang-Mills theory. In each dimension, there is a self-similar finite-time blowup solution to this equation known in closed form. It will be proved that this profile is stable in the whole space under small perturbations of the initial data. The blowup analysis is based on a recently developed coordinate system called hyperboloidal similarity coordinates and depends crucially on growth estimates for the free wave evolution, which will be constructed systematically for odd space dimensions in the first part of this paper. This allows to develop a nonlinear stability theory beyond the singularity.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134948594","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}
Pub Date : 2023-08-03DOI: 10.1080/03605302.2023.2247467
Anna Logioti, B. Niethammer, Matthias Röger, J. Velázquez
Abstract We consider a parabolic non-local free boundary problem that has been derived as a limit of a bulk-surface reaction-diffusion system which models cell polarization. We have justified the well-posedness of this problem and have further proved uniqueness of solutions and global stability of steady states. In this paper we investigate qualitative properties of the free boundary. We present necessary and sufficient conditions for the initial data that imply continuity of the support at t = 0. If one of these assumptions fail, then jumps of the support take place. In addition we provide a complete characterization of the jumps for a large class of initial data.
{"title":"Qualitative properties of solutions to a mass-conserving free boundary problem modeling cell polarization","authors":"Anna Logioti, B. Niethammer, Matthias Röger, J. Velázquez","doi":"10.1080/03605302.2023.2247467","DOIUrl":"https://doi.org/10.1080/03605302.2023.2247467","url":null,"abstract":"Abstract We consider a parabolic non-local free boundary problem that has been derived as a limit of a bulk-surface reaction-diffusion system which models cell polarization. We have justified the well-posedness of this problem and have further proved uniqueness of solutions and global stability of steady states. In this paper we investigate qualitative properties of the free boundary. We present necessary and sufficient conditions for the initial data that imply continuity of the support at t = 0. If one of these assumptions fail, then jumps of the support take place. In addition we provide a complete characterization of the jumps for a large class of initial data.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"1065 - 1101"},"PeriodicalIF":1.9,"publicationDate":"2023-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48106982","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}
Pub Date : 2023-05-31DOI: 10.1080/03605302.2023.2215296
Hui Chen, Tai-Peng Tsai, Ting Zhang
{"title":"Correction to: Remarks on local regularity of axisymmetric solutions to the 3D Navier–Stokes equations","authors":"Hui Chen, Tai-Peng Tsai, Ting Zhang","doi":"10.1080/03605302.2023.2215296","DOIUrl":"https://doi.org/10.1080/03605302.2023.2215296","url":null,"abstract":"","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"987 - 988"},"PeriodicalIF":1.9,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43029604","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}
Pub Date : 2023-05-04DOI: 10.1080/03605302.2023.2202732
S. Biesenbach, R. Schubert, Maria G. Westdickenberg
Abstract A dissipation estimate from the original article is corrected. The main result carries through unchanged, as explained below.
摘要对原文章中的损耗估计进行了更正。主要结果保持不变，如下所述。
{"title":"Erratum to “optimal relaxation of bump-like solutions of the one-dimensional Cahn–Hilliard equation”","authors":"S. Biesenbach, R. Schubert, Maria G. Westdickenberg","doi":"10.1080/03605302.2023.2202732","DOIUrl":"https://doi.org/10.1080/03605302.2023.2202732","url":null,"abstract":"Abstract A dissipation estimate from the original article is corrected. The main result carries through unchanged, as explained below.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"792 - 793"},"PeriodicalIF":1.9,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43259338","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}
Pub Date : 2023-04-03DOI: 10.1080/03605302.2023.2190526
S. Ervedoza, K. L. Balc'h
Abstract The goal of this article is to obtain observability estimates for non-homogeneous elliptic equations in the presence of a potential, posed on a smooth bounded domain Ω in and observed from a non-empty open subset More precisely, for our main result shows that, when has a finite number of holes, the observability constant of the elliptic operator with domain is of the form where C is a positive constant depending only on Ω and ω. Our methodology of proof is crucially based on the one recently developed by Logunov, Malinnikova, Nadirashvili, and Nazarov [1], in the context of the Landis conjecture on exponential decay of solutions to homogeneous elliptic equations in the plane The main difference and additional difficulty compared to [1] is that the zero set of the solutions to elliptic equations with source term can be very intricate and should be dealt with carefully. As a consequence of these new observability estimates, we obtain new results concerning control of semi-linear elliptic equations in the spirit of Fernández-Cara, Zuazua’s open problem concerning small-time global null-controllability of slightly super-linear heat equations.
{"title":"Cost of observability inequalities for elliptic equations in 2-d with potentials and applications to control theory","authors":"S. Ervedoza, K. L. Balc'h","doi":"10.1080/03605302.2023.2190526","DOIUrl":"https://doi.org/10.1080/03605302.2023.2190526","url":null,"abstract":"Abstract The goal of this article is to obtain observability estimates for non-homogeneous elliptic equations in the presence of a potential, posed on a smooth bounded domain Ω in and observed from a non-empty open subset More precisely, for our main result shows that, when has a finite number of holes, the observability constant of the elliptic operator with domain is of the form where C is a positive constant depending only on Ω and ω. Our methodology of proof is crucially based on the one recently developed by Logunov, Malinnikova, Nadirashvili, and Nazarov [1], in the context of the Landis conjecture on exponential decay of solutions to homogeneous elliptic equations in the plane The main difference and additional difficulty compared to [1] is that the zero set of the solutions to elliptic equations with source term can be very intricate and should be dealt with carefully. As a consequence of these new observability estimates, we obtain new results concerning control of semi-linear elliptic equations in the spirit of Fernández-Cara, Zuazua’s open problem concerning small-time global null-controllability of slightly super-linear heat equations.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"623 - 677"},"PeriodicalIF":1.9,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42972923","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}
Pub Date : 2023-02-21DOI: 10.1080/03605302.2023.2183408
J. Arbunich, J. Faupin, F. Pusateri, I. Sigal
Abstract We prove maximal speed estimates for nonlinear quantum propagation in the context of the Hartree equation. More precisely, under some regularity and integrability assumptions on the pair (convolution) potential, we construct a set of energy and space localized initial conditions such that, up to time-decaying tails, solutions starting in this set stay within the light cone of the corresponding initial datum. We quantify precisely the light cone speed, and hence the speed of nonlinear propagation, in terms of the momentum of the initial state.
{"title":"Maximal speed of quantum propagation for the Hartree equation","authors":"J. Arbunich, J. Faupin, F. Pusateri, I. Sigal","doi":"10.1080/03605302.2023.2183408","DOIUrl":"https://doi.org/10.1080/03605302.2023.2183408","url":null,"abstract":"Abstract We prove maximal speed estimates for nonlinear quantum propagation in the context of the Hartree equation. More precisely, under some regularity and integrability assumptions on the pair (convolution) potential, we construct a set of energy and space localized initial conditions such that, up to time-decaying tails, solutions starting in this set stay within the light cone of the corresponding initial datum. We quantify precisely the light cone speed, and hence the speed of nonlinear propagation, in terms of the momentum of the initial state.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"542 - 575"},"PeriodicalIF":1.9,"publicationDate":"2023-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41607919","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}
Pub Date : 2023-02-03DOI: 10.1080/03605302.2023.2246194
William Cooperman
Abstract Capuzzo-Dolcetta–Ishii proved that the rate of periodic homogenization for coercive Hamilton–Jacobi equations is . We complement this result by constructing examples of coercive nonconvex Hamiltonians whose rate of periodic homogenization is .
{"title":"Slow periodic homogenization for Hamilton–jacobi equations","authors":"William Cooperman","doi":"10.1080/03605302.2023.2246194","DOIUrl":"https://doi.org/10.1080/03605302.2023.2246194","url":null,"abstract":"Abstract Capuzzo-Dolcetta–Ishii proved that the rate of periodic homogenization for coercive Hamilton–Jacobi equations is . We complement this result by constructing examples of coercive nonconvex Hamiltonians whose rate of periodic homogenization is .","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"1056 - 1064"},"PeriodicalIF":1.9,"publicationDate":"2023-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42808957","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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