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 .
1 Introduction and main results
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 .
2 Preliminaries and lemmas
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.
3 Proof of the main results
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).
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