Continuous properties of the solution map for the Euler equations

Journal Article · Sat Mar 15 00:00:00 EDT 2014 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4867622· OSTI ID:22251070

Tang, Hao; Liu, Zhengrong ^[1]

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1. Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640 (China)

In this paper, we study the dependence on initial data of solutions to the incompressible Euler equations in Besov spaces. We show that for s > n/p + 1, p ∈ (1, ∞), and r ∈ [1, ∞], the solution map u{sub 0}↦u is Hölder continuous in Besov space B{sub p,r}{sup s} equipped with weaker topology. When the space variable x is taken to be periodic, we obtain a family of explicit periodic solutions. Furthermore, we prove that for any s∈R and 1 ⩽ r ⩽ ∞, the solution map is not globally uniformly continuous in B{sub 2,r}{sup s}(T{sup n}), which extends some results of Himonas and Misiołek [Commun. Math. Phys. 296(1), 285–301 (2010)].

• OSTI ID: 22251070
• Journal Information: Journal of Mathematical Physics, Vol. 55, Issue 3; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
EQUATIONS
MAPS
MATHEMATICAL SOLUTIONS
MATHEMATICAL SPACE
PERIODICITY
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