{"94d75063978007fd2183af88d8431c63elkepeter":{"DOI":"10.1007/s10231-020-00950-1","ISBN":"","ISSN":"0373-3114","URL":"https://doi.org/10.1007/s10231-020-00950-1","abstract":"The authors consider Euler's equations for free surface waves traveling on a body of density stratified water in the scenario when gravity and surface tension act as restoring forces. The flow is continuously stratified, and the water layer is bounded from below by an impermeable horizontal bed. Three equivalent classical formulations in a suitable setting of strong solutions which may describe nevertheless waves with singular density gradients are established. The availability of a weak formulation of the water wave problem, the regularity properties of the corresponding weak solutions, and methods from nonlinear functional analysis are used. The paper is organized as follows. In Section 2, the three formulations of the problem are introduced and their equivalence is established. In Section 3, the authors first introduce the notion of a weak solution to Dubreil-Jacotin's formulation and establish, by means of a shooting method, the existence of at least one laminar flow solution to this latter formulation. In Section 4, the equations are reformulated as an abstract bifurcation problem by using methods from nonlinear functional analysis.","annote":"","author":[{"family":"Escher","given":"Joachim"},{"family":"Knopf","given":"Patrik"},{"family":"Lienstromberg","given":"Christina"},{"family":"Matioc","given":"Bogdan-Vasile"}],"citation-label":"MR4142857","collection-editor":[],"collection-title":"","container-author":[],"container-title":"Ann. Mat. Pura Appl. (4)","documents":[],"edition":"","editor":[],"event-date":{"date-parts":[["2020"]],"literal":"2020"},"event-place":"","id":"94d75063978007fd2183af88d8431c63elkepeter","interhash":"f49cedf1153177fe651e0670050bf616","intrahash":"94d75063978007fd2183af88d8431c63","issue":"5","issued":{"date-parts":[["2020"]],"literal":"2020"},"keyword":"Lienstromberg gradients singular density","misc":{"mrclass":"76B15 (35B32 35C07 35Q35 76B70)","mrreviewer":"Gheorghe Procopiuc","fjournal":"Annali di Matematica Pura ed Applicata. Series IV","mrnumber":"4142857","issn":"0373-3114","doi":"10.1007/s10231-020-00950-1"},"note":"","number":"5","number-of-pages":"36","page":"1923--1959","page-first":"1923","publisher":"","publisher-place":"","status":"","title":"Stratified periodic water waves with singular density gradients","type":"article-journal","username":"elkepeter","version":"","volume":"199"},"c8514045542b75bf235e4b95dc4e664dmhartmann":{"DOI":"10.1007/s00162-012-0264-z","ISBN":"","ISSN":"0935-4964","URL":"http://dx.doi.org/10.1007/s00162-012-0264-z","abstract":"","annote":"","author":[{"family":"Brdar","given":"S."},{"family":"Baldauf","given":"M."},{"family":"Dedner","given":"A."},{"family":"Klöfkorn","given":"R."}],"citation-label":"brdar2012comparison","collection-editor":[],"collection-title":"","container-author":[],"container-title":"Theoretical and Computational Fluid Dynamics","documents":[],"edition":"","editor":[],"event-date":{"date-parts":[["2012"]],"literal":"2012"},"event-place":"","id":"c8514045542b75bf235e4b95dc4e664dmhartmann","interhash":"6904f2bea4a01c1e6a3c31769a8fccbf","intrahash":"c8514045542b75bf235e4b95dc4e664d","issue":"","issued":{"date-parts":[["2012"]],"literal":"2012"},"keyword":"Finite Density Discontinuous differences; Compressible current; Euler; gravity Navier???Stokes; Galerkin; Inertia flow; vorlaeufig","misc":{"issn":"0935-4964","owner":"kohlsk","language":"English","doi":"10.1007/s00162-012-0264-z"},"note":"","number":"","number-of-pages":"19","page":"1-20","page-first":"1","publisher":"Springer-Verlag","publisher-place":"","status":"","title":"Comparison of dynamical cores for NWP models: comparison of COSMO\n\tand Dune","type":"article-journal","username":"mhartmann","version":"","volume":""},"8fc4612e68ea6905b965ae8c7e92e656mhartmann":{"DOI":"","ISBN":"","ISSN":"","URL":"http://imajna.oxfordjournals.org/content/31/3/947.abstract","abstract":"We analyse the adaptive finite-element approximation to solutions\n\tof partial differential equations in variational formulation. Assuming\n\twell-posedness of the continuous problem and requiring only basic\n\tproperties of the adaptive algorithm, we prove convergence of the\n\tsequence of discrete solutions to the true one. The proof is based\n\ton the ideas by Morin, Siebert and Veeser but replaces local efficiency\n\tof the estimator by a local density property of the adaptively generated\n\tfinite-element spaces. As a result, estimators without a discrete\n\tlower bound are also included in our theory. The assumptions of the\n\tpresented framework are fulfilled by a large class of important applications,\n\testimators and adaptive strategies.","annote":"","author":[{"family":"Siebert","given":"Kunibert G."}],"citation-label":"siebert2011convergence","collection-editor":[],"collection-title":"","container-author":[],"container-title":"IMA Journal of Numerical Analysis","documents":[],"edition":"","editor":[],"event-date":{"date-parts":[["2011"]],"literal":"2011"},"event-place":"","id":"8fc4612e68ea6905b965ae8c7e92e656mhartmann","interhash":"704190075ecceaf1b26aac9eeed999c5","intrahash":"8fc4612e68ea6905b965ae8c7e92e656","issue":"3","issued":{"date-parts":[["2011"]],"literal":"2011"},"keyword":"adaptivity convergence elements finite density vorlaeufig","misc":{"owner":"kohlsk"},"note":"","number":"3","number-of-pages":"23","page":"947-970","page-first":"947","publisher":"","publisher-place":"","status":"","title":"A Convergence Proof for Adaptive Finite Elements without Lower Bound","type":"article-journal","username":"mhartmann","version":"","volume":"31"}}