scholarly journals Regularity of Solutions to the Navier-Stokes Equations Evolving from Small Data in BMO−1

2007 ◽  
Vol 2007 ◽  
Author(s):  
Pierre Germain ◽  
Nataša Pavlović ◽  
Gigliola Staffilani
Keyword(s):  
Stokes Equations ◽  
Regularity Of Solutions ◽  
Navier Stokes ◽  
Small Data ◽  
Navier Stokes Equations

scholarly journals In-Flow and Out-Flow Problem for the Stokes System

2020 ◽  
Vol 22 (4) ◽  
Author(s):  
Bernard Nowakowski ◽  
Gerhard Ströhmer
Keyword(s):  
Stokes System ◽  
Flow Problem ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Tangential Component ◽  
Regularity Of Solutions ◽  
Navier Stokes ◽  
Lateral Part ◽  
Navier Stokes Equations ◽  
Stationary Navier Stokes Equations

AbstractWe investigate the existence and regularity of solutions to the stationary Stokes system and non-stationary Navier–Stokes equations in three dimensional bounded domains with in- and out-lets. We assume that on the in- and out-flow parts of the boundary the pressure is prescribed and the tangential component of the velocity field is zero, whereas on the lateral part of the boundary the fluid is at rest.

scholarly journals The Navier–Stokes regularity problem

2020 ◽  
Vol 378 (2174) ◽  
pp. 20190526
Author(s):  
James C. Robinson
Keyword(s):  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Regularity Of Solutions ◽  
Navier Stokes ◽  
Initial Condition ◽  
Theme Issue ◽  
Uniqueness Of Solutions ◽  
Rigorous Results ◽  
Navier Stokes Equations

There is currently no proof guaranteeing that, given a smooth initial condition, the three-dimensional Navier–Stokes equations have a unique solution that exists for all positive times. This paper reviews the key rigorous results concerning the existence and uniqueness of solutions for this model. In particular, the link between the regularity of solutions and their uniqueness is highlighted. This article is part of the theme issue ‘Stokes at 200 (Part 1)’.

scholarly journals ON APPROXIMATE SOLUTIONS OF SEMILINEAR EVOLUTION EQUATIONS II: GENERALIZATIONS, AND APPLICATIONS TO NAVIER–STOKES EQUATIONS

2008 ◽  
Vol 20 (06) ◽  
pp. 625-706 ◽  
Author(s):  
CARLO MOROSI ◽  
LIVIO PIZZOCCHERO
Keyword(s):  
Exact Solution ◽  
Global Existence ◽  
Exponential Decay ◽  
Evolution Equations ◽  
Stokes Equations ◽  
Approximate Solutions ◽  
Navier Stokes ◽  
Small Data ◽  
Navier Stokes Equations ◽  
Semilinear Evolution Equations

In our previous paper [12], a general framework was outlined to treat the approximate solutions of semilinear evolution equations; more precisely, a scheme was presented to infer from an approximate solution the existence (local or global in time) of an exact solution, and to estimate their distance. In the first half of the present work, the abstract framework of [12] is extended, so as to be applicable to evolutionary PDEs whose nonlinearities contain derivatives in the space variables. In the second half of the paper, this extended framework is applied to the incompressible Navier–Stokes equations, on a torus Td of any dimension. In this way, a number of results are obtained in the setting of the Sobolev spaces ℍn(Td), choosing the approximate solutions in a number of different ways. With the simplest choices we recover local existence of the exact solution for arbitrary data and external forces, as well as global existence for small data and forces. With the supplementary assumption of exponential decay in time for the forces, the same decay law is derived for the exact solution with small (zero mean) data and forces. The interval of existence for arbitrary data, the upper bounds on data and forces for global existence, and all estimates on the exponential decay of the exact solution are derived in a fully quantitative way (i.e. giving the values of all the necessary constants; this makes a difference with most of the previous literature). Next, the Galerkin approximate solutions are considered and precise, still quantitative estimates are derived for their ℍn distance from the exact solution; these are global in time for small data and forces (with exponential time decay of the above distance, if the forces decay similarly).

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