Acta Mathematica Scientia ›› 2018, Vol. 38 ›› Issue (5): 1591-1616.

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HÖLDER CONTINUOUS SOLUTIONS OF BOUSSINESQ EQUATIONS

Tao TAO1, Liqun ZHANG2   

  1. 1. School of Mathematics Sciences, Peking University, Beijing 100871, China;School of Mathematics, Shandong University, Jinan 250100, China;
    2. Academy of Mathematic and System Science, CAS, Beijing 100190, China;School of Mathematical Sciences, UCAS, Beijing 100049, China
  • Received:2017-04-01 Online:2018-11-09 Published:2018-11-09
  • Supported by:
    The research was partially supported by the NSFC (11471320 and 11631008).

Abstract: We show the existence of dissipative Hölder continuous solutions of the Boussinesq equations. More precise, for any β ∈ (0, 1/5), a time interval[0, T] and any given smooth energy profile e:[0, T] → (0, ∞), there exist a weak solution (v, θ) of the 3d Boussinesq equations such that (v, θ) ∈ Cβ(T3×[0, T]) with e(t)=∫T3|v(x, t)|2dx for all t ∈[0, T]. This extend the result of [2] about Onsager's conjecture into Boussinesq equation and improve our previous result in [30].

Key words: Boussinesq equations, Hölder continuous solutions, prescribed kinetic energy

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