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

Abstract

In the hyper-viscous Navier–Stokes equations of incompressible flow, the operator A=− Δ is replaced by A α, a, b ≡ aA α + bA for real numbers α, a, b with α⩾1 and b⩾0. We treat here the case a>0 and equip A (and hence A α, a, b ) with periodic boundary conditions over a rectangular solid Ω⊂R n . For initial data in L p(Ω) with α⩾ n/(2 p)+1/2 we establish local existence and uniqueness of strong solutions, generalizing a result of Giga/Miyakawa for α=1 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 α⩾ n/4+1/2 is sufficient also to establish global existence of these unique regular solutions and uniform higher-order bounds. For the borderline case α= 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 α -bound, and then convert to a time-independent global exponential H α -bound. This is to be expected, given that uniform bounds already exist for n=2, α=1 ([6, pp. 78–79]), and the folklore bounds already suggest that the α⩾ n/4+1/2 cases for n⩾3 should behave as well as the n=2 case. What is slightly less expected is that the n⩾3 cases are easier to prove and give better bounds, e.g. the uniform bound for n⩾3 depends on the square of the data in the exponential rather than the fourth power for n=2. More significantly, for α> 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 α↓ n/4+1/2, thus “anticipating” the exponential bound in the borderline case α= n/4+1/2. We prove our results for the case a=1 and b=0; the general case with a>0 and b⩾0 can be recovered by using norm-equivalence. We note that the hyperviscous Navier–Stokes equations have both physical and numerical application.