- Open Access
Blow-up criteria for a two-fluid model of the truncated Euler equations
© Xu et al.; licensee Springer. 2013
- Received: 4 March 2013
- Accepted: 18 September 2013
- Published: 7 November 2013
In this paper, we study a two-fluid model of the truncated Euler equations with partial viscosity. We obtain new blow-up criteria for a solution of the system in terms of vorticity in the homogeneous Besov space .
- blow-up criterion
- truncated Euler equations
- Besov space
where denotes the fluid velocity vector field, is the scalar pressure, , while and are given initial velocity and initial temperature, respectively, with . For more detailed background of the system, we refer the readers to  and the references therein. When , the system reduces to the well-known Navier-Stokes equations.
Our purpose in this paper is to establish blow-up criteria of a strong solution for the two-fluid model of the truncated Euler equations in terms of vorticity in the homogeneous Besov space . The main difficulty is without viscosity in the second equation for system (1.1).
Now we state our main results as follows.
then the solution can be extended smoothly beyond .
Throughout this paper, we introduce some function spaces, notations and important inequalities.
for and , where ∗ denotes the convolution of functions defined on .
and when , we just let .
Lemma 2.1 
holds for all .
Lemma 2.2 
and C is a constant independent of f.
In this section, we prove Theorem 1.1.
where we use the Hölder inequality, the Young inequality and (2.2).
This completes the proof. □
Research supported by the National Natural Science Foundation of China (11161041), the Fundamental Research Funds for the Central Universities (No. 31920130006) and Middle-Younger Scientific Research Fund (No. 12XB39).
- Krstulovic G, Brachet M: Two-fluid model of the truncated Euler equations. Physica D 2008, 237: 2015–2019. 10.1016/j.physd.2007.11.008MATHMathSciNetView ArticleGoogle Scholar
- Sun J, Fan J: Regularity criterion for a two-fluid model of the truncated Euler equations. Acta Math. Sci. 2010, 30: 1693–1698. (in Chinese)MATHMathSciNetGoogle Scholar
- Triebel H North-Holland Mathematical Library 18. In Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam; 1978.Google Scholar
- Meyer, Y, Gerard, P, Oru, F: Inégalités de Sobolev précisées. In: Séminaire Équations aux dérivées partielles (Polytechnique) (1996–1997), Exp. No. 4, 8pGoogle Scholar
- Ladyzhenskaya O, Solonnikov V, Ural’tseva N Translations of Mathematical Monographs. In Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc., Providence; 1968. Translated from the Russian by S. SmithGoogle Scholar
- Kato T, Ponce G: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 1988, 41(7):891–907. 10.1002/cpa.3160410704MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.