Electron. J. Diff. Equ.,
Vol. 2015 (2015), No. 170, pp. 124.
Local and global lowregularity solutions to generalized
Lerayalpha equations
Nathan Pennington
Abstract:
It has recently become common to study approximating equations
for the NavierStokes equation. One of these is the Lerayalpha equation,
which regularizes the NavierStokes equation by replacing (in most locations)
the solution u with
. Another is the generalized
NavierStokes equation, which replaces the Laplacian with a Fourier multiplier
with symbol of the form
(
is the standard NavierStokes equation),
and recently in [16] Tao also considered multipliers of
the form
, where g is (essentially) a logarithm.
The generalized Lerayalpha equation combines these two modifications by
incorporating the regularizing term and replacing the Laplacians with more
general Fourier multipliers, including allowing for g terms similar to those
used in [16]. Our goal in this paper is to obtain existence and
uniqueness results with low regularity and/or nonL^2 initial data.
We will also use energy estimates to extend some of these local existence
results to global existence results.
Submitted May 15, 2015. Published June 18, 2015.
Math Subject Classifications: 76D05, 35A02, 35K58.
Key Words: Lerayalpha model; Besov space; fractional Laplacian.
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Nathan Pennington
Department of Mathematics, Creighton University
2500 California Plaza
Omaha, NE 68178, USA
email: nathanpennington@creighton.edu

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