{"title":"广义无粘曲面拟地转方程不爆破的若干条件","authors":"Linrui Li, Mingli Hong, Lin Zheng","doi":"10.1155/2023/4420217","DOIUrl":null,"url":null,"abstract":"In this paper, we survey some non-blow-up results for the following generalized modified inviscid surface quasigeostrophic equation (GSQG) <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\"> <mfenced open=\"{\" close=\"\"> <mrow> <mtable class=\"smallmatrix\"> <mtr> <mtd columnalign=\"left\"> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>u</mi> <mo>·</mo> <mo>∇</mo> <mi>θ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd columnalign=\"left\"> <mi>u</mi> <mo>=</mo> <msup> <mrow> <mo>∇</mo> </mrow> <mrow> <mo>⊥</mo> </mrow> </msup> <mi>ψ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd columnalign=\"left\"> <mo>−</mo> <msup> <mrow> <mi>Λ</mi> </mrow> <mrow> <mi>β</mi> </mrow> </msup> <mi>ψ</mi> <mo>=</mo> <mi>θ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd columnalign=\"left\"> <mi>θ</mi> <mfenced open=\"(\" close=\")\"> <mrow> <mi>x</mi> <mo>,</mo> <mn>0</mn> </mrow> </mfenced> <mo>=</mo> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mfenced open=\"(\" close=\")\"> <mrow> <mi>x</mi> </mrow> </mfenced> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mfenced> </math> . This is a generalized surface quasigeostrophic equation (GSQG) with the velocity field <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\"> <mi>u</mi> </math> related to the scalar <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\"> <mi>θ</mi> </math> by <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\"> <mi>u</mi> <mo>=</mo> <mo>−</mo> <msup> <mrow> <mo>∇</mo> </mrow> <mrow> <mo>⊥</mo> </mrow> </msup> <msup> <mrow> <mi>Λ</mi> </mrow> <mrow> <mo>−</mo> <mi>β</mi> </mrow> </msup> <mi>θ</mi> </math> , where <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M5\"> <mn>1</mn> <mo>≤</mo> <mi>β</mi> <mo>≤</mo> <mn>2</mn> </math> . We prove that there is no finite-time singularity if the level set of generalized quasigeostrophic equation does not have a hyperbolic saddle, and the angle of opening of the saddle can go to zero at most as an exponential decay. Moreover, we give some conditions that rule out the formation of sharp fronts for generalized inviscid surface quasigeostrophic equation, and we obtain some estimates on the formation of semiuniform fronts. These results greatly weaken the geometrical assumptions which rule out the collapse of a simple hyperbolic saddle in finite time.","PeriodicalId":49111,"journal":{"name":"Advances in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Some Conditions of Non-Blow-Up of Generalized Inviscid Surface Quasigeostrophic Equation\",\"authors\":\"Linrui Li, Mingli Hong, Lin Zheng\",\"doi\":\"10.1155/2023/4420217\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we survey some non-blow-up results for the following generalized modified inviscid surface quasigeostrophic equation (GSQG) <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M1\\\"> <mfenced open=\\\"{\\\" close=\\\"\\\"> <mrow> <mtable class=\\\"smallmatrix\\\"> <mtr> <mtd columnalign=\\\"left\\\"> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>u</mi> <mo>·</mo> <mo>∇</mo> <mi>θ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd columnalign=\\\"left\\\"> <mi>u</mi> <mo>=</mo> <msup> <mrow> <mo>∇</mo> </mrow> <mrow> <mo>⊥</mo> </mrow> </msup> <mi>ψ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd columnalign=\\\"left\\\"> <mo>−</mo> <msup> <mrow> <mi>Λ</mi> </mrow> <mrow> <mi>β</mi> </mrow> </msup> <mi>ψ</mi> <mo>=</mo> <mi>θ</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd columnalign=\\\"left\\\"> <mi>θ</mi> <mfenced open=\\\"(\\\" close=\\\")\\\"> <mrow> <mi>x</mi> <mo>,</mo> <mn>0</mn> </mrow> </mfenced> <mo>=</mo> <msub> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mfenced open=\\\"(\\\" close=\\\")\\\"> <mrow> <mi>x</mi> </mrow> </mfenced> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mfenced> </math> . This is a generalized surface quasigeostrophic equation (GSQG) with the velocity field <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M2\\\"> <mi>u</mi> </math> related to the scalar <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M3\\\"> <mi>θ</mi> </math> by <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M4\\\"> <mi>u</mi> <mo>=</mo> <mo>−</mo> <msup> <mrow> <mo>∇</mo> </mrow> <mrow> <mo>⊥</mo> </mrow> </msup> <msup> <mrow> <mi>Λ</mi> </mrow> <mrow> <mo>−</mo> <mi>β</mi> </mrow> </msup> <mi>θ</mi> </math> , where <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M5\\\"> <mn>1</mn> <mo>≤</mo> <mi>β</mi> <mo>≤</mo> <mn>2</mn> </math> . We prove that there is no finite-time singularity if the level set of generalized quasigeostrophic equation does not have a hyperbolic saddle, and the angle of opening of the saddle can go to zero at most as an exponential decay. Moreover, we give some conditions that rule out the formation of sharp fronts for generalized inviscid surface quasigeostrophic equation, and we obtain some estimates on the formation of semiuniform fronts. These results greatly weaken the geometrical assumptions which rule out the collapse of a simple hyperbolic saddle in finite time.\",\"PeriodicalId\":49111,\"journal\":{\"name\":\"Advances in Mathematical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-10-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Mathematical Physics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1155/2023/4420217\",\"RegionNum\":4,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Mathematical Physics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1155/2023/4420217","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
Some Conditions of Non-Blow-Up of Generalized Inviscid Surface Quasigeostrophic Equation
In this paper, we survey some non-blow-up results for the following generalized modified inviscid surface quasigeostrophic equation (GSQG) . This is a generalized surface quasigeostrophic equation (GSQG) with the velocity field related to the scalar by , where . We prove that there is no finite-time singularity if the level set of generalized quasigeostrophic equation does not have a hyperbolic saddle, and the angle of opening of the saddle can go to zero at most as an exponential decay. Moreover, we give some conditions that rule out the formation of sharp fronts for generalized inviscid surface quasigeostrophic equation, and we obtain some estimates on the formation of semiuniform fronts. These results greatly weaken the geometrical assumptions which rule out the collapse of a simple hyperbolic saddle in finite time.
期刊介绍:
Advances in Mathematical Physics publishes papers that seek to understand mathematical basis of physical phenomena, and solve problems in physics via mathematical approaches. The journal welcomes submissions from mathematical physicists, theoretical physicists, and mathematicians alike.
As well as original research, Advances in Mathematical Physics also publishes focused review articles that examine the state of the art, identify emerging trends, and suggest future directions for developing fields.