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AIMS Mathematics

2017, Volume 2, Issue 2: 336-347. doi: 10.3934/Math.2017.2.336
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Research article

Logarithmically improved regularity criteria for the Boussinesq equations

  • 1.
    Department of Mathematics, University of Mostaganem, Box 227, Mostaganem, 27000, Algeria
  • 2.
    Dipartimento di Mathematicae Informatica, Universit à di Catania Viale Andrea Doria, 6,95125 Catania, Italy
  • Received: 24 April 2017 Accepted: 22 May 2017 Published: 06 June 2017

  • In this paper, logarithmically improved regularity criteria for the Boussinesq equations are established under the framework of Besov space $\overset{.}{B}_{\infty, \infty }.{-r}$. We prove the solution $(u, \theta)$ is smooth up to time $T>0$ provided that $ \int_0 . T\frac{{{{\left\| {u( \cdot ,t)} \right\|}_{{{\mathop B\limits^. }_{\infty ,\infty }}. - r}}\;.\frac{2}{{1 - r}}}}{{\log (e + {{\left\| {u(t,.)} \right\|}_{{{\mathop B\limits^. }_{\infty ,\infty }}. - r\;}})}}dt < \infty $ for some $0\leq r < 1$ or $||u( \cdot ,t){||_{{L^\infty }(0,T;{\mathop B\limits^{·}}{_{\infty ,\infty }^{ - 1}}\;\;({{\mathbb{R} }^3}))}}<<1.$ This result improves some previous works.

    Citation: Sadek Gala, Mohamed Mechdene, Maria Alessandra Ragusa. Logarithmically improved regularity criteria for the Boussinesq equations[J]. AIMS Mathematics, 2017, 2(2): 336-347. doi: 10.3934/Math.2017.2.336

    Related Papers:

  • In this paper, logarithmically improved regularity criteria for the Boussinesq equations are established under the framework of Besov space $\overset{.}{B}_{\infty, \infty }.{-r}$. We prove the solution $(u, \theta)$ is smooth up to time $T>0$ provided that $ \int_0 . T\frac{{{{\left\| {u( \cdot ,t)} \right\|}_{{{\mathop B\limits^. }_{\infty ,\infty }}. - r}}\;.\frac{2}{{1 - r}}}}{{\log (e + {{\left\| {u(t,.)} \right\|}_{{{\mathop B\limits^. }_{\infty ,\infty }}. - r\;}})}}dt < \infty $ for some $0\leq r < 1$ or $||u( \cdot ,t){||_{{L^\infty }(0,T;{\mathop B\limits^{·}}{_{\infty ,\infty }^{ - 1}}\;\;({{\mathbb{R} }^3}))}}<<1.$ This result improves some previous works.


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