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Erschienen in:
2007 | OriginalPaper | Buchkapitel
Local-in-time existence of strong solutions of the n-dimensional Burgers equation via discretizations
verfasst von : João Paulo Teixeira
Erschienen in: The Strength of Nonstandard Analysis
Verlag: Springer Vienna
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Consider the equation:
$$ u_t = \nu \Delta u - (u \cdot \nabla )u + f{\mathbf{ }}for{\mathbf{ }}x \in [0,1]^n {\mathbf{ }}and{\mathbf{ }}t \in (0,\infty ), $$
together with periodic boundary conditions and initial condition
u
(
t
, 0) =
g
(
x
). This corresponds a Navier-Stokes problem where the incompressibility condition has been dropped. The major difficulty in existence proofs for this simplified problem is the unbounded advection term, (
u
· ∇)
u
.
We present a proof of local-in-time existence of a smooth solution based on a discretization by a suitable Euler scheme. It will be shown that this solution exists in an interval [0,
T
), where
T
≤ 1/
C
, with
C
depending only on
n
and the values of the Lipschitz constants of
f
and
u
at time 0. The argument given is based directly on local estimates of the solutions of the discretized problem.
