Nonuniqueness of Weak Solutions to the SQG Equation

被引：76

作者：

Buckmaster, Tristan ^{[1
,4
]}

Shkoller, Steve ^{[2
]}

Vicol, Vlad ^{[3
,5
]}

机构：

[1] Courant Inst, New York, NY 10012 USA

[2] Univ Calif Davis, Dept Math, One Shields Ave, Davis, CA 95616 USA

[3] Princeton Univ, Princeton, NJ 08544 USA

[4] Princeton Univ, Dept Math, Fine Hall,304 Washington Rd, Princeton, NJ 08544 USA

[5] NYU, Courant Inst, 251 Mercer St, New York, NY 10012 USA

来源：

COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS | 2019年 / 72卷 / 09期

基金：

美国国家科学基金会;

关键词：

QUASI-GEOSTROPHIC EQUATION; SHARP FRONTS; ANOMALOUS DISSIPATION; ENERGY-CONSERVATION; ONSAGERS CONJECTURE; MAXIMUM-PRINCIPLES; WELL-POSEDNESS; EXISTENCE; SINGULARITIES; REGULARITY;

D O I：

10.1002/cpa.21851

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering Open Problem 11 of De Lellis and Szekelyhidi in 2012. Moreover, we also show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian. In view of the results of Marchand in 2008, we establish that for the dissipative SQG equation, weak solutions may be constructed in the same function space both via classical weak compactness arguments and via convex integration. (c) 2019 Wiley Periodicals, Inc.

引用

页码：1809 / 1874

页数：66

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