The existence, uniqueness, and regularity for an incompressible Newtonian flow with intrinsic degree of freedom

Abstract

We consider the regularity and uniqueness of solution to the Cauchy problem of a mathematical model for an incompressible, homogeneous, Newtonian fluid, taking into account internal degree of freedom. We first show there exist uniquely a local strong solution. Then we show this solution can be extend to the whole interval [0,T] if the velocity u, or its gradient ∇ u, or the pressure p belongs to some function class, which are similar with that of incompressible Navier–Stokes equations. Our result shows that the solution is unique in these classes, and that velocity field plays a more prominent role in the existence theory of strong solution than the angular velocity field. Finally, if the L3 ∕ 2‐norm of the initial angular velocity vector and some homogeneous Besov norm of initial velocity field are small, then there exists uniquely a global strong solution. Copyright © 2012 John Wiley & Sons, Ltd.

Existence and asymptotic behavior for an incompressible Newtonian flow with intrinsic degree of freedom

“…In , it was shown that there uniquely exists a local strong solution ( u , p , ω ) to under some suitable assumptions on initial data. In this paper, we can further show the existence of following small strong solution in : Theorem Let , and for some p > n .…”

“…But it is worth to point out that, for incompressible Navier–Stokes equations, if one solution is weak, and another one is strong with same initial data, then the two solutions agree with each other (for the details, see the summarization on uniqueness in . Whereas for , there is no similar result, and unique result shown in holds only provided two solutions are strong, because of the strong nonlinear term F ( p ) with respect to pressure.…”

“…Recent, the authors studied the Cauchy problem of in . It was shown in that there locally exists a unique strong solution to for some m ≥ 1, when initial velocity u 0 belongs to some Sobolev space with α ∈ (0,1] and q > max{3(2 m − 1) (1 − α ) ∕ (1 + α ),3}and is divergence free in the sense of distribution, and the initial angular velocity field with .…”

“…Recent, the authors studied the Cauchy problem of in . It was shown in that there locally exists a unique strong solution to for some m ≥ 1, when initial velocity u 0 belongs to some Sobolev space with α ∈ (0,1] and q > max{3(2 m − 1) (1 − α ) ∕ (1 + α ),3}and is divergence free in the sense of distribution, and the initial angular velocity field with . And this local strong solution can be extended to be global, provided any one of following five conditions hold: for 1 ∕ p + 3 ∕ 2 q ≤ 1 ∕ 2 and q > 3, , for 1 ∕ p + 3 ∕ 2 r = 1 with 1 < p ≤ 2 and 3 ∕ 2 < r < ∞ , …”

“…Further, a global strong solution was obtained in under the assumption that the L 3 ∕ 2 ‐norm of the initial angular velocity vector and the homogeneous Besov norm of initial velocity field are small with 1 ∕ p + 3 ∕ 2 q = 1 ∕ 2 for some p > 2 and some q > 3.…”