{"title":"Global solvability to a two competing species chemotaxis Navier-Stokes system with singular sensitivity","authors":"Jiayi Han, Changchun Liu","doi":"10.3934/cpaa.2023090","DOIUrl":"https://doi.org/10.3934/cpaa.2023090","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70221550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The stochastic flocking model with far-field degenerate communication","authors":"Li Lv, Zibo Wang, Xiongtao Zhang","doi":"10.3934/cpaa.2023063","DOIUrl":"https://doi.org/10.3934/cpaa.2023063","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70221621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Hydrodynamic limits of Manton's Schrödinger system","authors":"Jeongho Kim, Bora Moon","doi":"10.3934/cpaa.2023066","DOIUrl":"https://doi.org/10.3934/cpaa.2023066","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70221643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Polynomial stability of stochastic heat equations","authors":"G. Lv, Hongjun Gao, Jinlong Wei","doi":"10.3934/cpaa.2023092","DOIUrl":"https://doi.org/10.3934/cpaa.2023092","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70221692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Elia Brué, Maria Colombo, Gianluca Crippa, Camillo De Lellis, M. Sorella
We answer positively to [BDL22, Question 2.4] by building new examples of solutions to the forced 3d-Navier-Stokes equations with vanishing viscosity, which exhibit anomalous dissipation and which enjoy uniform bounds in the space $L_t^3 C_x^{1/3 - varepsilon}$, for any fixed $varepsilon>0$. Our construction combines ideas of [BDL22] and [CCS22].
{"title":"Onsager critical solutions of the forced Navier-Stokes equations","authors":"Elia Brué, Maria Colombo, Gianluca Crippa, Camillo De Lellis, M. Sorella","doi":"10.3934/cpaa.2023071","DOIUrl":"https://doi.org/10.3934/cpaa.2023071","url":null,"abstract":"We answer positively to [BDL22, Question 2.4] by building new examples of solutions to the forced 3d-Navier-Stokes equations with vanishing viscosity, which exhibit anomalous dissipation and which enjoy uniform bounds in the space $L_t^3 C_x^{1/3 - varepsilon}$, for any fixed $varepsilon>0$. Our construction combines ideas of [BDL22] and [CCS22].","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46111723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the well-posedness for the inhomogeneous Hartree equation $ipartial_t u + Delta u = lambda(I_alpha ast |cdot|^{-b}|u|^p)|x|^{-b}|u|^{p-2}u$ in $H^s$, $sge0$. Until recently, its well-posedness theory has been intensively studied, focusing on solving the problem for the critical index $p=1+frac{2-2b+alpha}{n-2s}$ with $0le s le 1$, but the case $1/2leq s leq 1$ is still an open problem. In this paper, we develop the well-posedness theory in this case, especially including the energy-critical case. To this end, we approach to the matter based on the Sobolev-Lorentz space which can lead us to perform a finer analysis for this equation. This is because it makes it possible to control the nonlinearity involving the singularity $|x|^{-b}$ as well as the Riesz potential $I_alpha$ more effectively.
在$H^s$, $sge0$中研究了非齐次Hartree方程$ipartial_t u + Delta u = lambda(I_alpha ast |cdot|^{-b}|u|^p)|x|^{-b}|u|^{p-2}u$的适定性。直到最近，人们对其适定性理论进行了深入的研究，主要集中在用$0le s le 1$求解临界指标$p=1+frac{2-2b+alpha}{n-2s}$的问题上，但情况$1/2leq s leq 1$仍然是一个悬而未决的问题。在本文中，我们发展了这种情况下的适定性理论，特别是包括能量临界情况。为此，我们基于索博列夫-洛伦兹空间来处理这个问题，这可以使我们对这个方程进行更精细的分析。这是因为它可以更有效地控制涉及奇点$|x|^{-b}$以及Riesz势$I_alpha$的非线性。
{"title":"On well-posedness for inhomogeneous Hartree equations in the critical case","authors":"Seongyeon Kim","doi":"10.3934/cpaa.2023060","DOIUrl":"https://doi.org/10.3934/cpaa.2023060","url":null,"abstract":"We study the well-posedness for the inhomogeneous Hartree equation $ipartial_t u + Delta u = lambda(I_alpha ast |cdot|^{-b}|u|^p)|x|^{-b}|u|^{p-2}u$ in $H^s$, $sge0$. Until recently, its well-posedness theory has been intensively studied, focusing on solving the problem for the critical index $p=1+frac{2-2b+alpha}{n-2s}$ with $0le s le 1$, but the case $1/2leq s leq 1$ is still an open problem. In this paper, we develop the well-posedness theory in this case, especially including the energy-critical case. To this end, we approach to the matter based on the Sobolev-Lorentz space which can lead us to perform a finer analysis for this equation. This is because it makes it possible to control the nonlinearity involving the singularity $|x|^{-b}$ as well as the Riesz potential $I_alpha$ more effectively.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48352822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Existence of weak solutions for some elliptic systems","authors":"L. Boccardo, L. Orsina","doi":"10.2140/paa.2022.4.517","DOIUrl":"https://doi.org/10.2140/paa.2022.4.517","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"8 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79039601","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Existence of global strong solution for the\u0000Navier–Stokes–Korteweg system in one dimension for strongly degenerate viscosity\u0000coefficients","authors":"Cosmin Burtea, B. Haspot","doi":"10.2140/paa.2022.4.449","DOIUrl":"https://doi.org/10.2140/paa.2022.4.449","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"106 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76060094","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A nonlinear transmisson problem for an elastic full von Karman beam is considered here. We prove that the system possesses a compact global attractor.
本文讨论了弹性全冯-卡门梁的非线性传递问题。我们证明了系统具有一个紧致全局吸引子。
{"title":"Global attractors for a full von Karman beam transmission problem","authors":"T. Fastovska","doi":"10.3934/cpaa.2023022","DOIUrl":"https://doi.org/10.3934/cpaa.2023022","url":null,"abstract":"A nonlinear transmisson problem for an elastic full von Karman beam is considered here. We prove that the system possesses a compact global attractor.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49655490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 so-called large solutions of elliptic semilinear equations with non null sources term, thus solutions blowing up on the boundary of the domain for which reason they are greater than any other solution whenever Weak Maximum Principle holds. The main topic about large solutions is uniqueness results and their behavior near the boundary. It is much less than being simple. The structure of the semilinear equations considered includes the well known Keller-Osserman integral and an assumption on the ellipticity of the leading part of the differential operator. In our study an uniform ellipticity near the boundary is required. We consider source terms in the PDE whose boundary explosion is consistent with the Keller-Osserman condition. Extra Keller-Osserman explosions on the source are also studied, showing in particular that in some cases the PDE only admits large solutions.
{"title":"Large solutions of elliptic semilinear equations non-degenerate near the boundary","authors":"G. Díaz","doi":"10.3934/cpaa.2023006","DOIUrl":"https://doi.org/10.3934/cpaa.2023006","url":null,"abstract":"In this paper we study the so-called large solutions of elliptic semilinear equations with non null sources term, thus solutions blowing up on the boundary of the domain for which reason they are greater than any other solution whenever Weak Maximum Principle holds. The main topic about large solutions is uniqueness results and their behavior near the boundary. It is much less than being simple. The structure of the semilinear equations considered includes the well known Keller-Osserman integral and an assumption on the ellipticity of the leading part of the differential operator. In our study an uniform ellipticity near the boundary is required. We consider source terms in the PDE whose boundary explosion is consistent with the Keller-Osserman condition. Extra Keller-Osserman explosions on the source are also studied, showing in particular that in some cases the PDE only admits large solutions.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44136170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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