Isaac Neal, Calum Rickard, Steve Shkoller, Vlad Vicol
From an open set of initial data, we construct a family of classical solutions to the 1D nonisentropic compressible Euler equations which form $ C^{0,nu} $ cusps as a first singularity, for any $ nu in [frac{1}{2}, 1) $. For this range of $ nu $, this is the first result demonstrating the stable formation of such $ C^{0,nu} $ cusp-type singularities, also known as pre-shocks. The proof uses a new formulation of the differentiated Euler equations along the fast acoustic characteristic, and relies on a novel set of $ L^p $ energy estimates for all $ 1
从一个开放的初始数据集，我们构造了一维非等熵可压缩欧拉方程的经典解族，这些方程形成$ C^{0,nu} $顶点作为第一奇点，对于任何$ nu in [frac{1}{2}, 1) $。对于这个$ nu $范围，这是第一个证明这种$ C^{0,nu} $尖型奇点(也称为预冲击)稳定形成的结果。该证明使用了沿快速声学特性的微分欧拉方程的新公式，并依赖于所有$ 1<p<infty $的一组新的$ L^p $能量估计，这可能是独立的兴趣。
{"title":"A new type of stable shock formation in gas dynamics","authors":"Isaac Neal, Calum Rickard, Steve Shkoller, Vlad Vicol","doi":"10.3934/cpaa.2023118","DOIUrl":"https://doi.org/10.3934/cpaa.2023118","url":null,"abstract":"From an open set of initial data, we construct a family of classical solutions to the 1D nonisentropic compressible Euler equations which form $ C^{0,nu} $ cusps as a first singularity, for any $ nu in [frac{1}{2}, 1) $. For this range of $ nu $, this is the first result demonstrating the stable formation of such $ C^{0,nu} $ cusp-type singularities, also known as pre-shocks. The proof uses a new formulation of the differentiated Euler equations along the fast acoustic characteristic, and relies on a novel set of $ L^p $ energy estimates for all $ 1<p<infty $, which may be of independent interest.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"4 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":"135604154","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":"Hopf bifurcation in a two-species reaction-diffusion-advection competitive model with nonlocal delay","authors":"Tingting Wen, Xiaoli Wang, Guohong Zhang","doi":"10.3934/cpaa.2023036","DOIUrl":"https://doi.org/10.3934/cpaa.2023036","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":"70221177","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 positive solutions to critical Schrödinger system with mixed interactions in $ {mathbb R}^3 $","authors":"Song You, Houwang Li","doi":"10.3934/cpaa.2023097","DOIUrl":"https://doi.org/10.3934/cpaa.2023097","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"478 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70221766","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":"Infinitely many large energy solutions to a class of nonlocal discrete elliptic boundary value problems","authors":"Y. Long, Qinqin Zhang","doi":"10.3934/cpaa.2023037","DOIUrl":"https://doi.org/10.3934/cpaa.2023037","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":"70221217","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":"A remark on the local uniqueness by the monotonicity based method for the magnetic Schrödinger equation","authors":"Ryusei Yamashita","doi":"10.3934/cpaa.2023077","DOIUrl":"https://doi.org/10.3934/cpaa.2023077","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":"70221303","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":"Symmetry and monotonicity of positive solutions to systems involving general pseudo-relativistic operators","authors":"Xueying Chen, Guanfeng Li","doi":"10.3934/cpaa.2023045","DOIUrl":"https://doi.org/10.3934/cpaa.2023045","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":"70221345","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 emergent discrete-time dynamics of the Kuramoto model with frustration. For the discrete identical Kuramoto model with frustration, we present the exponential synchronization estimate for some admissible classes of initial configuration confined in a half circle. For the nonidentical oscillators, we exploit the second-order lifting of the original Kuramoto model with frustration and study the emergent behaviors of the corresponding discrete augmented model. More precisely, under a sufficient framework in terms of large coupling strength and small step size, we derive that the initial configuration distributed in a small set of a quarter circle will converge to frequency synchronization.
{"title":"On the synchronization of discrete-time Kuramoto model with frustration","authors":"Tingting Zhu","doi":"10.3934/cpaa.2023109","DOIUrl":"https://doi.org/10.3934/cpaa.2023109","url":null,"abstract":"In this paper, we study the emergent discrete-time dynamics of the Kuramoto model with frustration. For the discrete identical Kuramoto model with frustration, we present the exponential synchronization estimate for some admissible classes of initial configuration confined in a half circle. For the nonidentical oscillators, we exploit the second-order lifting of the original Kuramoto model with frustration and study the emergent behaviors of the corresponding discrete augmented model. More precisely, under a sufficient framework in terms of large coupling strength and small step size, we derive that the initial configuration distributed in a small set of a quarter circle will converge to frequency synchronization.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"36 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":"135700499","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}
Lyapunov exponent plays an important role in the dynamics of quasi-periodic cocycles, and admits an asymptotic formula for Schrödinger cocycle with large parameter (see [12] for analytic case, and see [15] for finitely smooth case). In this paper, we establish similar asymptotic formulas of Lyapunov exponent for two classes of important quasi-periodic cocycles: Szegő cocycles and Schrödinger Mosaic cocycles.
{"title":"On the Lyapunov exponent for some quasi-periodic cocycles with large parameter","authors":"Bowen Ding, Jinhao Liang","doi":"10.3934/cpaa.2023102","DOIUrl":"https://doi.org/10.3934/cpaa.2023102","url":null,"abstract":"Lyapunov exponent plays an important role in the dynamics of quasi-periodic cocycles, and admits an asymptotic formula for Schrödinger cocycle with large parameter (see [12] for analytic case, and see [15] for finitely smooth case). In this paper, we establish similar asymptotic formulas of Lyapunov exponent for two classes of important quasi-periodic cocycles: Szegő cocycles and Schrödinger Mosaic cocycles.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"254 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":"135496200","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":"Global dynamics of a two-species competition model in advective homogeneous environments","authors":"Jinyu Wei, Bin Liu","doi":"10.3934/cpaa.2023021","DOIUrl":"https://doi.org/10.3934/cpaa.2023021","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":"70221118","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":"Spatial Dynamics of a Nonlocal Reaction-Diffusion Epidemic Model in Time-Space Periodic Habitat","authors":"Ming-Zhen Xin, Bin-Guo Wang","doi":"10.3934/cpaa.2023072","DOIUrl":"https://doi.org/10.3934/cpaa.2023072","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"36 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70221252","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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