Decay Results of the Nonstationary Navier–Stokes Flows in Half-Spaces

Decay Results of the Nonstationary Navier–Stokes Flows in Half-Spaces There is a long standing problem on the nonstationary Navier–Stokes equations which pertains to how to characterize the L −time asymptotic expansion of the Navier–Stokes flows in the half space. Beyond a few partial results, new progress has not yet to be made on this open question. In this article, we give a confirmed answer to this problem; namely, a thorough characterization on L −summability is revealed. In order to prove this result, we need to avoid the unboundedness of the projection operator, which is overcome by treating an elliptic Neumann problem. Finally, using the weighted estimates on the heat kernel’s convolution, we obtain 1 n the exact profile structure of the asymptotic expansion in L (R ). In addition, some crucial estimates on the fractional spatial derivatives of the non-stationary Stokes and Navier–Stokes flows are established for the first time, which allows for a better understanding of the L −decay problem. 1. Introduction and Main Results In this paper, we consider the decay properties, as t −→ ∞, of solutions to the nonstationary Navier–Stokes equations in the half space ∂ u − u + (u ·∇)u +∇ p = 0in R × (0, ∞), ⎪ t ∇· u http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Rational Mechanics and Analysis Springer Journals

Decay Results of the Nonstationary Navier–Stokes Flows in Half-Spaces

Loading next page...
 
/lp/springer_journal/decay-results-of-the-nonstationary-navier-stokes-flows-in-half-spaces-SqsKyKfydl
Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2018 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Physics; Classical Mechanics; Physics, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Fluid- and Aerodynamics
ISSN
0003-9527
eISSN
1432-0673
D.O.I.
10.1007/s00205-018-1263-z
Publisher site
See Article on Publisher Site

Abstract

There is a long standing problem on the nonstationary Navier–Stokes equations which pertains to how to characterize the L −time asymptotic expansion of the Navier–Stokes flows in the half space. Beyond a few partial results, new progress has not yet to be made on this open question. In this article, we give a confirmed answer to this problem; namely, a thorough characterization on L −summability is revealed. In order to prove this result, we need to avoid the unboundedness of the projection operator, which is overcome by treating an elliptic Neumann problem. Finally, using the weighted estimates on the heat kernel’s convolution, we obtain 1 n the exact profile structure of the asymptotic expansion in L (R ). In addition, some crucial estimates on the fractional spatial derivatives of the non-stationary Stokes and Navier–Stokes flows are established for the first time, which allows for a better understanding of the L −decay problem. 1. Introduction and Main Results In this paper, we consider the decay properties, as t −→ ∞, of solutions to the nonstationary Navier–Stokes equations in the half space ∂ u − u + (u ·∇)u +∇ p = 0in R × (0, ∞), ⎪ t ∇· u

Journal

Archive for Rational Mechanics and AnalysisSpringer Journals

Published: Jun 1, 2018

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off