Singular initial data and uniform global bounds for the hyper-viscous Navier-Stokes equation with periodic boundary conditions

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:190

Article

Avrin, J

关键词: hyper-viscosity;    local existence;    global existence;    regularity;   

DOI  :  10.1016/S0022-0396(02)00155-9

来源: Elsevier

【 摘 要 】

In the hyper-viscous Navier-Stokes equations of incompressible flow, the operator A = -Delta is replaced by A(alpha,a,b) drop aA(alpha) + bA for real numbers alpha,a,b with alpha greater than or equal to I and b greater than or equal to 0. We treat here the case a>0 and equip A (and hence A,,,b) with periodic boundary conditions over a rectangular solid Omegasubset ofR(n). For initial data in L-p(Omega) with alphagreater than or equal ton/(2p) + 1/2 we establish local existence and uniqueness of strong solutions, generalizing a result of Giga/Miyakawa for alpha = I and b = 0. Specializing to the case p 2, which holds a particular physical relevance in terms of the total energy of the system, it is somewhat interesting to note that the condition alphagreater than or equal ton/4 + 1/2 is sufficient also to establish global existence of these unique regular solutions and uniform higher-order bounds. For the borderline case alpha = n/4 + 1/2 we generalize standard existing (for n = 3) folklore results and use energy techniques and Gronwall's inequality to obtain first a time-dependent H-alpha-bound, and then convert to a time-independent global exponential H-alpha-bound. This is to be expected, given that uniform bounds already exist for n = 2, alpha = 1 ([6, pp. 78-79]), and the folklore bounds already suggest that the alphagreater than or equal ton/4 + 1/2 cases for ngreater than or equal to3 should behave as well as the n = 2 case. What is slightly less expected is that the ngreater than or equal to3 cases are easier to prove and give better bounds, e.g. the uniform bound for ngreater than or equal to3 depends on the square of the data in the exponential rather than the fourth power for n = 2. More significantly, for alpha > n/4 + 1/2 we use our own entirely semigroup techniques to obtain uniform global bounds which bootstrap directly from the uniform L-2-estimate and are algebraic in terms of the uniform L-2-bounds on the initial and forcing data. The integer powers on the square of the data increase without bound as alphadown arrown/4 + 1/2, thus anticipating the exponential bound in the borderline case alpha = n/4 + 1/2. We prove our results for the case a = I and b = 0; the general case with a > 0 and b greater than or equal to 0 can be recovered by using orm-equivalence. We note that the hyperviscous Navier-Stokes equations have both physical and numerical application. (C) 2002 Elsevier Science (USA). All rights reserved.

【 授权许可】

Free