{"title":"On the incompressible limit for a tumour growth model incorporating convective effects","authors":"Noemi David, Markus Schmidtchen","doi":"10.1002/cpa.22178","DOIUrl":"10.1002/cpa.22178","url":null,"abstract":"<p>In this work we study a tissue growth model with applications to tumour growth. The model is based on that of Perthame, Quirós, and Vázquez proposed in 2014 but incorporates the advective effects caused, for instance, by the presence of nutrients, oxygen, or, possibly, as a result of self-propulsion. The main result of this work is the incompressible limit of this model which builds a bridge between the density-based model and a geometry free-boundary problem by passing to a singular limit in the pressure law. The limiting objects are then proven to be unique.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 5","pages":"2613-2650"},"PeriodicalIF":3.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22178","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Log-Sobolev inequality for the \u0000 \u0000 \u0000 φ\u0000 2\u0000 4\u0000 \u0000 $varphi ^4_2$\u0000 and \u0000 \u0000 \u0000 φ\u0000 3\u0000 4\u0000 \u0000 $varphi ^4_3$\u0000 measures","authors":"Roland Bauerschmidt, Benoit Dagallier","doi":"10.1002/cpa.22173","DOIUrl":"10.1002/cpa.22173","url":null,"abstract":"<p>The continuum <math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>φ</mi>\u0000 <mn>2</mn>\u0000 <mn>4</mn>\u0000 </msubsup>\u0000 <annotation>$varphi ^4_2$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>φ</mi>\u0000 <mn>3</mn>\u0000 <mn>4</mn>\u0000 </msubsup>\u0000 <annotation>$varphi ^4_3$</annotation>\u0000 </semantics></math> measures are shown to satisfy a log-Sobolev inequality uniformly in the lattice regularisation under the optimal assumption that their susceptibility is bounded. In particular, this applies to all coupling constants in any finite volume, and uniformly in the volume in the entire high temperature phases of the <math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>φ</mi>\u0000 <mn>2</mn>\u0000 <mn>4</mn>\u0000 </msubsup>\u0000 <annotation>$varphi ^4_2$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>φ</mi>\u0000 <mn>3</mn>\u0000 <mn>4</mn>\u0000 </msubsup>\u0000 <annotation>$varphi ^4_3$</annotation>\u0000 </semantics></math> models.</p><p>The proof uses a general criterion for the log-Sobolev inequality in terms of the Polchinski (renormalisation group) equation, a recently proved remarkable correlation inequality for Ising models with general external fields, the Perron–Frobenius theorem, and bounds on the susceptibilities of the <math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>φ</mi>\u0000 <mn>2</mn>\u0000 <mn>4</mn>\u0000 </msubsup>\u0000 <annotation>$varphi ^4_2$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>φ</mi>\u0000 <mn>3</mn>\u0000 <mn>4</mn>\u0000 </msubsup>\u0000 <annotation>$varphi ^4_3$</annotation>\u0000 </semantics></math> measures obtained using skeleton inequalities.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 5","pages":"2579-2612"},"PeriodicalIF":3.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22173","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}