Navier-Stokes equation in super-critical spaces $ {E}_{p,q}^{s}$

Feichtinger, Hans G.; Gröchenig, Karlheinz; Li, Kuijie; Wang, Baoxiang

doi:10.1016/j.anihpc.2020.06.002

Navier-Stokes equation in super-critical spaces

E_{p, q}^{s}

Feichtinger, Hans G.¹ ; Gröchenig, Karlheinz ¹ ; Li, Kuijie ² ; Wang, Baoxiang ³

¹ a Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria
² b School of Mathematical Sciences, Fudan University, Shanghai, 200433, China
³ c LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China

Annales de l'I.H.P. Analyse non linéaire, janvier – février 2021, Tome 38 (2021) no. 1, pp. 139-173.

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Résumé

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}} = {(\sum_{k \in Z^{d}} 2^{s | k | q} {‖ F^{- 1} χ_{k + {[0, 1]}^{d}} F f ‖}_{p}^{q})}^{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^{σ} \subset E_{2, 1}^{s}

for any

σ < 0

and

s < 0

. It is known that

H^{σ}

(

σ < 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

R_{I}^{d} : = {ξ \in R^{d} : ξ_{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

supp {\hat{u}}_{0} \subset R_{I}^{d}

DOI : 10.1016/j.anihpc.2020.06.002

Classification : 35Q55, 42B35, 42B37
Mots-clés : Navier-Stokes equation, Modulation spaces, Negative exponential weight, Global well-posedness

Affiliations des auteurs :

Feichtinger, Hans G. ¹ ; Gröchenig, Karlheinz ¹ ; Li, Kuijie ² ; Wang, Baoxiang ³

@article{AIHPC_2021__38_1_139_0,
     author = {Feichtinger, Hans G. and Gr\"ochenig, Karlheinz and Li, Kuijie and Wang, Baoxiang},
     title = {Navier-Stokes equation in super-critical spaces $ {E}_{p,q}^{s}$},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {139--173},
     publisher = {Elsevier},
     volume = {38},
     number = {1},
     year = {2021},
     doi = {10.1016/j.anihpc.2020.06.002},
     mrnumber = {4200480},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2020.06.002/}
}

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Feichtinger, Hans G.; Gröchenig, Karlheinz; Li, Kuijie; Wang, Baoxiang. Navier-Stokes equation in super-critical spaces $ {E}_{p,q}^{s}$. Annales de l'I.H.P. Analyse non linéaire, janvier – février 2021, Tome 38 (2021) no. 1, pp. 139-173. doi : 10.1016/j.anihpc.2020.06.002. http://www.numdam.org/articles/10.1016/j.anihpc.2020.06.002/

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