© W. Chen and J. Fan. 2011
Received: 3 October 2010
Accepted: 16 January 2011
Published: 13 February 2011
where is the fluid density, is the fluid velocity field, is the "filtered" fluid velocity, and is the pressure, which are unknowns. is the lengthscale parameter that represents the width of the filter, and for simplicity, we will take . is a bounded domain with smooth boundary .
When , the above system reduces to the well-known Leray- model and has been studied in [1, 2]. When , the above system reduces to the classical density-dependent Navier-Stokes equation, which has received many studies [3–6]. Specifically, it is proved in [3, 4] that the density-dependent Navier-Stokes equations has a unique locally smooth solution if the following two hypotheses (H1) and (H2) are satisfied:
where and represent the unknown magnetic field and the "filtered" magnetic field, respectively. is the lengthscale parameter representing the width of the filter and we will take for simplicity. is the unit outward vector to . When and , the above system (1.3)–(1.9) reduces to the well-known density-dependent MHD equations, which have been studied by many authors (see [7–9] and referees therein). When and , the above system has been studied in  recently, and also modified models were analyzed in . In this paper, we will prove the following theorem.
Since the proof of Theorem 1.1 is similar to and simpler than that of Theorem 1.2, we only prove Theorem 1.2 for concision.
2. Proof of Theorem 1.2
By similar argument as that in [3, 4], it is easy to prove that there are and a unique smooth solution to the problem (1.3)–(1.9) in , and we only need to establish some a priori estimates for any time. Therefore, in the following estimates, we assume that the solution is sufficiently smooth.
This completes the proof.
This work is partially supported by ZJNSF (Grant no. R6090109) and NSFC (Grant no. 10971197).
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