Premium
On pointwise decay rates of time‐periodic solutions to the Navier–Stokes equation
Author(s) -
Nakatsuka Tomoyuki
Publication year - 2021
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800377
Subject(s) - pointwise , mathematics , mathematical analysis , navier–stokes equations , order (exchange) , space (punctuation) , stokes' law , stokes flow , physics , geometry , mechanics , compressibility , flow (mathematics) , linguistics , philosophy , finance , economics
We study the existence of a time‐periodic solution with pointwise decay properties to the Navier–Stokes equation in the whole space. We show that if the time‐periodic external force is sufficiently small in an appropriate sense, then there exists a time‐periodic solution { u , p } of the Navier–Stokes equation such that| ∇ ju ( t , x ) | = O ( | x | 1 − n − j ) and| ∇ jp ( t , x ) | = O ( | x | − n − j )( j = 0 , 1 , … )uniformly in t ∈ R as | x | → ∞ . Our solution decays faster than the time‐periodic Stokes fundamental solution and the faster decay of its spatial derivatives of higher order is also described.