Navier–Stokes equation in super-critical spaces \( E_{p , q}^{s} \)

Abstract

In this paper we develop a new way to study the global existence and uniqueness for the Navier–Stokes equation (NS) and consider the initial data in a class of modulation spaces E_{p,q}^{s} with exponentially decaying weights (s < 0,\:1 < p,q < \infty ) for which the norms are defined by \|f\|_{E_{p,q}^{s}} = \left(\:\sum \limits_{k \in \mathbb{Z}^{d}}2^{s|k|q}\|\mathscr{F}^{−1}\chi _{k + [0,1]^{d}}\mathscr{F}f\|_{p}^{q}\right)^{1/ q}. The space E_{p,q}^{s} is a rather rough function space and cannot be treated as a subspace of tempered distributions. For example, we have the embedding H^{\sigma } \subset E_{2,1}^{s} for any \sigma < 0 and s < 0 . It is known that H^{\sigma } ( \sigma < d/ 2−1 ) is a super-critical space of NS, it follows that E_{2,1}^{s} ( s < 0 ) is also super-critical for NS. We show that NS has a unique global mild solution if the initial data belong to E_{2,1}^{s} ( s < 0 ) and their Fourier transforms are supported in \mathbb{R}_{I}^{d}: = \{\xi \in \mathbb{R}^{d}:\:\xi _{i}⩾0,\:i = 1,...,d\} . Similar results hold for the initial data in E_{r,1}^{s} with 2 < r⩽d . Our results imply that NS has a unique global solution if the initial value u_{0} is in L^{2} with \mathrm{\sup p}\:\hat{u}_{0}\: \subset \mathbb{R}_{I}^{d} .

Complex valued semi-linear heat equations in super-critical spaces $E^s_σ$

“…(ii) Let s = 0. Assume that u 0 E 0 σ is sufficiently small, then (13) has a unique solution u satisfying (14)…”

“…The Fourier transforms on S ′ 1 can be defined by duality (cf. [14]), namely, for any f ∈ S ′ 1 , its Fourier transform F f = f satisfies…”

“…Recently, Navier-Stokes equation in supercritical modulation spaces E s p,1 was studied in [14], where a general idea to handle a class of supercritical initial data was presented. In this paper, we will consider the semi-linear heat equation (SLH) ∂ t u − ∆u − f (u) = 0, u(0, x) = u 0 (x) (3) in the supercritical spaces E s σ with s < 0, where f (u) = u m (or f (u) = e u − 1), u is a complex-valued distribution function of (t, x) ∈ R + × R d , m ≥ 2 is a natural number.…”

Characterisation of the Weyl-Hörmander classes by time-frequency shifts

“…Sjöstrand later generalised the theory in [38] to broader classes of symbols with limited regularity. These articles have sprung a line of research concerning ΨDOs and FIOs with symbols in modulation and Wiener amalgam spaces and their application to PDEs; see [1,5,10,11,12,13,20,25,26,27,40,41] and the references therein. In [37], Sjöstrand proved the following characterisation of M ∞,1 (R 2n ) (see the proof of [37,Proposition 3.1]).…”