The Green–Naghdi type model in the Camassa–Holm regime derived in [Comm. Pure Appl. Anal. 14(6) (2015) 2203– 2230], describe the propagation of medium amplitude internal waves over medium amplitude topography variations. It is fully justified in the sense that it is well-posed, consistent with the full Euler system and converges to the latter with corresponding initial data. In this paper, we generalize this result by constructing a fully justified coupled asymptotic model in a more complex physical case of variable topography. More precisely, we are interested in specific bottoms wavelength of characteristic order λb = λ/α where λ is a characteristic horizontal length (wave-length of the interface). We assume a slowly varying topography with large amplitude (βα = O(√μ), where β characterizes the shape of the bottom). In addition, our system permits the full justification of any lower order, well-posed and consistent model. We apply the procedure to scalar models driven by simple unidirectional equations in the Camassa–Holm and long wave regimes and under some restrictions on the topography variations. We also show that wave breaking of solutions to such equations occurs in the Camassa–Holm regime with slow topography variations and for a specific set of parameters.
{"title":"Coupled and scalar asymptotic models for internal waves over variable topography","authors":"R. Lteif, Samer Israwi","doi":"10.3233/ASY-171440","DOIUrl":"https://doi.org/10.3233/ASY-171440","url":null,"abstract":"The Green–Naghdi type model in the Camassa–Holm regime derived in [Comm. Pure Appl. Anal. 14(6) (2015) 2203– 2230], describe the propagation of medium amplitude internal waves over medium amplitude topography variations. It is fully justified in the sense that it is well-posed, consistent with the full Euler system and converges to the latter with corresponding initial data. In this paper, we generalize this result by constructing a fully justified coupled asymptotic model in a more complex physical case of variable topography. More precisely, we are interested in specific bottoms wavelength of characteristic order λb = λ/α where λ is a characteristic horizontal length (wave-length of the interface). We assume a slowly varying topography with large amplitude (βα = O(√μ), where β characterizes the shape of the bottom). In addition, our system permits the full justification of any lower order, well-posed and consistent model. We apply the procedure to scalar models driven by simple unidirectional equations in the Camassa–Holm and long wave regimes and under some restrictions on the topography variations. We also show that wave breaking of solutions to such equations occurs in the Camassa–Holm regime with slow topography variations and for a specific set of parameters.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"487 1","pages":"61-98"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77052722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On a trigonometric inequality of Askey and Steinig","authors":"H. Alzer, M. Kwong","doi":"10.3233/ASY-171447","DOIUrl":"https://doi.org/10.3233/ASY-171447","url":null,"abstract":"","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"20 1","pages":"233-249"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80239876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A sharp regularity result for the Euler equation with a free interface","authors":"I. Kukavica, A. Tuffaha","doi":"10.3233/ASY-171459","DOIUrl":"https://doi.org/10.3233/ASY-171459","url":null,"abstract":"","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"114 1","pages":"121-145"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79243046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Approximate state-space controllability of linear singularly perturbed systems with two scales of state delays","authors":"V. Glizer","doi":"10.3233/ASY-171451","DOIUrl":"https://doi.org/10.3233/ASY-171451","url":null,"abstract":"","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"107 1","pages":"73-114"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77977890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Korteweg-de Vries equation on the quarter plane with asymptotically t-periodic data via the Fokas method","authors":"Byungsoo Moon, Guenbo Hwang","doi":"10.3233/ASY-171452","DOIUrl":"https://doi.org/10.3233/ASY-171452","url":null,"abstract":"","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"34 1","pages":"115-133"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85000602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the L p - L q decay estimate for the Stokes equations with free boundary conditions in an exterior domain","authors":"Y. Shibata","doi":"10.3233/ASY-171449","DOIUrl":"https://doi.org/10.3233/ASY-171449","url":null,"abstract":"","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"94 1","pages":"33-72"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79639804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Topology optimization method with respect to the insertion of small coated inclusion","authors":"Z. Belhachmi, A. B. Abda, B. Meftahi, H. Meftahi","doi":"10.3233/ASY-171441","DOIUrl":"https://doi.org/10.3233/ASY-171441","url":null,"abstract":"","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"40 1","pages":"99-119"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82296829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Asymptotic formulae for the risk of failure related to an elasto-plastic problem with noise","authors":"C. Feau, M. Laurière, L. Mertz","doi":"10.3233/ASY-171445","DOIUrl":"https://doi.org/10.3233/ASY-171445","url":null,"abstract":"","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"30 1","pages":"47-60"},"PeriodicalIF":0.0,"publicationDate":"2017-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80715245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We are concerned with the $3$D-Navier-Stokes equations with Coriolis force. Existence and uniqueness of global solutions in homogeneous Besov spaces are obtained for large speed of rotation. In the critical case of the regularity, we consider a suitable initial data class whose definition is based on the Stokes-Coriolis semigroup and Besov spaces. Moreover, we analyze the asymptotic behavior of solutions in that setting as the speed of rotation goes to infinity.
{"title":"Global well-posedness and asymptotic behavior for Navier-Stokes-Coriolis equations in homogeneous Besov spaces","authors":"L. Ferreira, V. Angulo-Castillo","doi":"10.3233/ASY-181496","DOIUrl":"https://doi.org/10.3233/ASY-181496","url":null,"abstract":"We are concerned with the $3$D-Navier-Stokes equations with Coriolis force. Existence and uniqueness of global solutions in homogeneous Besov spaces are obtained for large speed of rotation. In the critical case of the regularity, we consider a suitable initial data class whose definition is based on the Stokes-Coriolis semigroup and Besov spaces. Moreover, we analyze the asymptotic behavior of solutions in that setting as the speed of rotation goes to infinity.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"46 1","pages":"37-58"},"PeriodicalIF":0.0,"publicationDate":"2017-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73915340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Moderate deviations for systems of slow-fast diffusions","authors":"Matthew R. Morse, K. Spiliopoulos","doi":"10.3233/ASY-171434","DOIUrl":"https://doi.org/10.3233/ASY-171434","url":null,"abstract":"","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"9 1","pages":"97-135"},"PeriodicalIF":0.0,"publicationDate":"2017-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79912003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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