Pub Date : 2021-10-07DOI: 10.1080/03605302.2022.2109173
M'at'e Gerencs'er, Martin Hairer
Abstract We consider the continuum parabolic Anderson model (PAM) and the dynamical equation on the 3-dimensional cube with boundary conditions. While the Dirichlet solution theories are relatively standard, the case of Neumann/Robin boundary conditions gives rise to a divergent boundary renormalisation. Furthermore for a ‘boundary triviality’ result is obtained: if one approximates the equation with Neumann boundary conditions and the usual bulk renormalisation, then the limiting process coincides with the one obtained using Dirichlet boundary conditions.
{"title":"Boundary renormalisation of SPDEs","authors":"M'at'e Gerencs'er, Martin Hairer","doi":"10.1080/03605302.2022.2109173","DOIUrl":"https://doi.org/10.1080/03605302.2022.2109173","url":null,"abstract":"Abstract We consider the continuum parabolic Anderson model (PAM) and the dynamical equation on the 3-dimensional cube with boundary conditions. While the Dirichlet solution theories are relatively standard, the case of Neumann/Robin boundary conditions gives rise to a divergent boundary renormalisation. Furthermore for a ‘boundary triviality’ result is obtained: if one approximates the equation with Neumann boundary conditions and the usual bulk renormalisation, then the limiting process coincides with the one obtained using Dirichlet boundary conditions.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"2070 - 2123"},"PeriodicalIF":1.9,"publicationDate":"2021-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42732883","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 : 2021-09-30DOI: 10.1080/03605302.2022.2047725
Myrto Galanopoulou, Andreas Vikelis, K. Koumatos
Abstract This article studies the equations of adiabatic thermoelasticity endowed with an internal energy satisfying an appropriate quasiconvexity assumption which is associated to the symmetrisability condition for the system. A Gårding-type inequality for these quasiconvex functions is proved and used to establish a weak-strong uniqueness result for a class of dissipative measure-valued solutions.
{"title":"Weak-strong uniqueness for measure-valued solutions to the equations of quasiconvex adiabatic thermoelasticity","authors":"Myrto Galanopoulou, Andreas Vikelis, K. Koumatos","doi":"10.1080/03605302.2022.2047725","DOIUrl":"https://doi.org/10.1080/03605302.2022.2047725","url":null,"abstract":"Abstract This article studies the equations of adiabatic thermoelasticity endowed with an internal energy satisfying an appropriate quasiconvexity assumption which is associated to the symmetrisability condition for the system. A Gårding-type inequality for these quasiconvex functions is proved and used to establish a weak-strong uniqueness result for a class of dissipative measure-valued solutions.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"1133 - 1175"},"PeriodicalIF":1.9,"publicationDate":"2021-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44289749","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 : 2021-09-30DOI: 10.1080/03605302.2023.2169936
M. Alfaro, Dongyuan Xiao
Abstract We consider the reaction-diffusion competition system in the so-called critical competition case. The associated ODE system then admits infinitely many equilibria, which makes the analysis intricate. We first prove the nonexistence of ultimately monotone traveling waves by applying the phase plane analysis. Next, we study the large time behavior of the solution of the Cauchy problem with a compactly supported initial datum. We not only reveal that the “faster” species excludes the “slower” one (with a known spreading speed), but also provide a sharp description of the profile of the solution, thus shedding light on a new bump phenomenon.
{"title":"Lotka–Volterra competition-diffusion system: the critical competition case","authors":"M. Alfaro, Dongyuan Xiao","doi":"10.1080/03605302.2023.2169936","DOIUrl":"https://doi.org/10.1080/03605302.2023.2169936","url":null,"abstract":"Abstract We consider the reaction-diffusion competition system in the so-called critical competition case. The associated ODE system then admits infinitely many equilibria, which makes the analysis intricate. We first prove the nonexistence of ultimately monotone traveling waves by applying the phase plane analysis. Next, we study the large time behavior of the solution of the Cauchy problem with a compactly supported initial datum. We not only reveal that the “faster” species excludes the “slower” one (with a known spreading speed), but also provide a sharp description of the profile of the solution, thus shedding light on a new bump phenomenon.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"48 1","pages":"182 - 208"},"PeriodicalIF":1.9,"publicationDate":"2021-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43031516","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 : 2021-09-23DOI: 10.1080/03605302.2022.2068425
V'aclav M'acha, B. Muha, Š. Nečasová, Arnab Roy, Srđan Trifunović
Abstract In this paper, we study a nonlinear interaction problem between a thermoelastic shell and a heat-conducting fluid. The shell is governed by linear thermoelasticity equations and encompasses a time-dependent domain which is filled with a fluid governed by the full Navier-Stokes-Fourier system. The fluid and the shell are fully coupled, giving rise to a novel nonlinear moving boundary fluid-structure interaction problem involving heat exchange. The existence of a weak solution is obtained by combining three approximation techniques – decoupling, penalization and domain extension. In particular, the penalization and the domain extension allow us to use the methods already developed for compressible fluids on moving domains. In such a way, the proof is more elegant and the analysis is drastically simplified. Let us stress that this is the first time the heat exchange in the context of fluid-structure interaction problems is considered.
{"title":"Existence of a weak solution to a nonlinear fluid-structure interaction problem with heat exchange","authors":"V'aclav M'acha, B. Muha, Š. Nečasová, Arnab Roy, Srđan Trifunović","doi":"10.1080/03605302.2022.2068425","DOIUrl":"https://doi.org/10.1080/03605302.2022.2068425","url":null,"abstract":"Abstract In this paper, we study a nonlinear interaction problem between a thermoelastic shell and a heat-conducting fluid. The shell is governed by linear thermoelasticity equations and encompasses a time-dependent domain which is filled with a fluid governed by the full Navier-Stokes-Fourier system. The fluid and the shell are fully coupled, giving rise to a novel nonlinear moving boundary fluid-structure interaction problem involving heat exchange. The existence of a weak solution is obtained by combining three approximation techniques – decoupling, penalization and domain extension. In particular, the penalization and the domain extension allow us to use the methods already developed for compressible fluids on moving domains. In such a way, the proof is more elegant and the analysis is drastically simplified. Let us stress that this is the first time the heat exchange in the context of fluid-structure interaction problems is considered.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"1591 - 1635"},"PeriodicalIF":1.9,"publicationDate":"2021-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48594644","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 : 2021-09-17DOI: 10.1080/03605302.2022.2118608
Li Chen, Alexandra Holzinger, A. Jüngel, N. Zamponi
Abstract A mean-field-type limit from stochastic moderately interacting many-particle systems with singular Riesz potential is performed, leading to nonlocal porous-medium equations in the whole space. The nonlocality is given by the inverse of a fractional Laplacian, and the limit equation can be interpreted as a transport equation with a fractional pressure. The proof is based on Oelschläger’s approach and a priori estimates for the associated diffusion equations, coming from energy-type and entropy inequalities as well as parabolic regularity. An existence analysis of the fractional porous-medium equation is also provided, based on a careful regularization procedure, new variants of fractional Gagliardo–Nirenberg inequalities, and the div-curl lemma. A consequence of the mean-field limit estimates is the propagation of chaos property.
{"title":"Analysis and mean-field derivation of a porous-medium equation with fractional diffusion","authors":"Li Chen, Alexandra Holzinger, A. Jüngel, N. Zamponi","doi":"10.1080/03605302.2022.2118608","DOIUrl":"https://doi.org/10.1080/03605302.2022.2118608","url":null,"abstract":"Abstract A mean-field-type limit from stochastic moderately interacting many-particle systems with singular Riesz potential is performed, leading to nonlocal porous-medium equations in the whole space. The nonlocality is given by the inverse of a fractional Laplacian, and the limit equation can be interpreted as a transport equation with a fractional pressure. The proof is based on Oelschläger’s approach and a priori estimates for the associated diffusion equations, coming from energy-type and entropy inequalities as well as parabolic regularity. An existence analysis of the fractional porous-medium equation is also provided, based on a careful regularization procedure, new variants of fractional Gagliardo–Nirenberg inequalities, and the div-curl lemma. A consequence of the mean-field limit estimates is the propagation of chaos property.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"2217 - 2269"},"PeriodicalIF":1.9,"publicationDate":"2021-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46699864","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 : 2021-09-10DOI: 10.1080/03605302.2022.2056704
M. Akman, Johnny M. Lewis, A. Vogel
Abstract Let denote Euclidean n space and given k a positive integer let be a k-dimensional plane with If we first study the Martin boundary problem for solutions to the p-Laplace equation (called p-harmonic functions) in relative to We then use the results from our study to extend the work of Wolff on the failure of Fatou type theorems for p-harmonic functions in to p-harmonic functions in when Finally, we discuss generalizations of our work to solutions of p-Laplace type PDE (called -harmonic functions).
{"title":"Failure of Fatou type theorems for solutions to PDE of p-Laplace type in domains with flat boundaries","authors":"M. Akman, Johnny M. Lewis, A. Vogel","doi":"10.1080/03605302.2022.2056704","DOIUrl":"https://doi.org/10.1080/03605302.2022.2056704","url":null,"abstract":"Abstract Let denote Euclidean n space and given k a positive integer let be a k-dimensional plane with If we first study the Martin boundary problem for solutions to the p-Laplace equation (called p-harmonic functions) in relative to We then use the results from our study to extend the work of Wolff on the failure of Fatou type theorems for p-harmonic functions in to p-harmonic functions in when Finally, we discuss generalizations of our work to solutions of p-Laplace type PDE (called -harmonic functions).","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"1457 - 1503"},"PeriodicalIF":1.9,"publicationDate":"2021-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47887163","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 : 2021-09-08DOI: 10.1080/03605302.2021.1975132
J. Tello
Abstract We consider a Parabolic-Elliptic system of PDE’s with a chemotactic term in a N-dimensional unit ball describing the behavior of the density of a biological species “u” and a chemical stimulus “v.” The system includes a nonlinear chemotactic coefficient depending of “ ” i.e. the chemotactic term is given in the form for a positive constant χ when v satisfies the poisson equation We study the radially symmetric solutions under the assumption in the initial mass For χ large enough, we present conditions in the initial data, such that any regular solution of the problem blows up at finite time.
{"title":"Blow up of solutions for a Parabolic-Elliptic chemotaxis system with gradient dependent chemotactic coefficient","authors":"J. Tello","doi":"10.1080/03605302.2021.1975132","DOIUrl":"https://doi.org/10.1080/03605302.2021.1975132","url":null,"abstract":"Abstract We consider a Parabolic-Elliptic system of PDE’s with a chemotactic term in a N-dimensional unit ball describing the behavior of the density of a biological species “u” and a chemical stimulus “v.” The system includes a nonlinear chemotactic coefficient depending of “ ” i.e. the chemotactic term is given in the form for a positive constant χ when v satisfies the poisson equation We study the radially symmetric solutions under the assumption in the initial mass For χ large enough, we present conditions in the initial data, such that any regular solution of the problem blows up at finite time.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"307 - 345"},"PeriodicalIF":1.9,"publicationDate":"2021-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44804914","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 : 2021-09-02DOI: 10.1080/03605302.2022.2056703
E. Feireisl, Young-Sam Kwon
Abstract We consider global in time solutions of the Navier–Stokes–Fourier system describing the motion of a general compressible, viscous and heat conducting fluid far from equilibirum. Using a new concept of weak solution suitable to accommodate the inhomogeneous Dirichlet time dependent data we find sufficient conditions for the global in time weak solutions to be ultimately bounded.
{"title":"Asymptotic stability of solutions to the Navier–Stokes–Fourier system driven by inhomogeneous Dirichlet boundary conditions","authors":"E. Feireisl, Young-Sam Kwon","doi":"10.1080/03605302.2022.2056703","DOIUrl":"https://doi.org/10.1080/03605302.2022.2056703","url":null,"abstract":"Abstract We consider global in time solutions of the Navier–Stokes–Fourier system describing the motion of a general compressible, viscous and heat conducting fluid far from equilibirum. Using a new concept of weak solution suitable to accommodate the inhomogeneous Dirichlet time dependent data we find sufficient conditions for the global in time weak solutions to be ultimately bounded.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"1435 - 1456"},"PeriodicalIF":1.9,"publicationDate":"2021-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47720166","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 : 2021-09-02DOI: 10.1080/03605302.2021.1998909
B. Georgiev, Mayukh Mukherjee
Abstract We consider mass concentration properties of Laplace eigenfunctions that is, smooth functions satisfying the equation on a smooth closed Riemannian manifold. Using a heat diffusion technique, we first discuss mass concentration/localization properties of eigenfunctions around their nodal sets. Second, we discuss the problem of avoided crossings and (non)existence of nodal domains which continue to be thin over relatively long distances. Further, using the above techniques, we discuss the decay of Laplace eigenfunctions on Euclidean domains which have a central “thick” part and “thin” elongated branches representing tunnels of sub-wavelength opening. Finally, in an Appendix, we record some new observations regarding sub-level sets of the eigenfunctions and interactions of different level sets.
{"title":"Some applications of heat flow to Laplace eigenfunctions","authors":"B. Georgiev, Mayukh Mukherjee","doi":"10.1080/03605302.2021.1998909","DOIUrl":"https://doi.org/10.1080/03605302.2021.1998909","url":null,"abstract":"Abstract We consider mass concentration properties of Laplace eigenfunctions that is, smooth functions satisfying the equation on a smooth closed Riemannian manifold. Using a heat diffusion technique, we first discuss mass concentration/localization properties of eigenfunctions around their nodal sets. Second, we discuss the problem of avoided crossings and (non)existence of nodal domains which continue to be thin over relatively long distances. Further, using the above techniques, we discuss the decay of Laplace eigenfunctions on Euclidean domains which have a central “thick” part and “thin” elongated branches representing tunnels of sub-wavelength opening. Finally, in an Appendix, we record some new observations regarding sub-level sets of the eigenfunctions and interactions of different level sets.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"677 - 700"},"PeriodicalIF":1.9,"publicationDate":"2021-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42758979","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 : 2021-08-26DOI: 10.1080/03605302.2022.2047724
Y. Sire, C. Sogge, Chengbo Wang, Junyong Zhang
Abstract We provide reversed Strichartz estimates for the shifted wave equations on non-trapping asymptotically hyperbolic manifolds using cluster estimates for spectral projectors proved previously in such generality. As a consequence, we solve a problem left open in Sire et al [Trans. AMS 373(2020):7639-7668] about the endpoint case for global well-posedness of nonlinear wave equations. We also provide estimates in this context for the maximal wave operator.
{"title":"Reversed Strichartz estimates for wave on non-trapping asymptotically hyperbolic manifolds and applications","authors":"Y. Sire, C. Sogge, Chengbo Wang, Junyong Zhang","doi":"10.1080/03605302.2022.2047724","DOIUrl":"https://doi.org/10.1080/03605302.2022.2047724","url":null,"abstract":"Abstract We provide reversed Strichartz estimates for the shifted wave equations on non-trapping asymptotically hyperbolic manifolds using cluster estimates for spectral projectors proved previously in such generality. As a consequence, we solve a problem left open in Sire et al [Trans. AMS 373(2020):7639-7668] about the endpoint case for global well-posedness of nonlinear wave equations. We also provide estimates in this context for the maximal wave operator.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"1124 - 1132"},"PeriodicalIF":1.9,"publicationDate":"2021-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45866369","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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