Attracting Fixed Points for the Kuramoto--Sivashinsky Equation: A Computer Assisted Proof
نویسندگان
چکیده
منابع مشابه
Attracting Fixed Points for the Kuramoto-Sivashinsky Equation: A Computer Assisted Proof
We present a computer assisted proof of the existence of several attracting fixed points for the Kuramoto–Sivashinsky equation ut = (u )x − uxx − νuxxxx, u(x, t) = u(x+ 2π, t), u(x, t) = −u(−x, t), where ν > 0. The method is general and can be applied to other dissipative PDEs.
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We consider the family of destabilized Kuramoto-Sivashinsky equations in one spatial dimension ut + νuxxxx + βuxx + γuux = αu for α,ν ≥ 0 and β ,γ ∈ R. For certain parameter values, shock-like stationary solutions have been numerically observed. In this work we verify the existence of several such solutions using the framework of self-consistent bounds and validated numerics.
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We apply the method of self-consistent bounds to prove the existence of multiple steady state bifurcations for Kuramoto-Sivashinski PDE on the line with odd and periodic boundary conditions.
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A new theorem is applied to the Kuramoto-Sivashinsky equation with L-periodic boundary conditions, proving the existence of an asymptotically complete inertial manifold attracting all initial data. Combining this result with a new estimate of the size of the globally absorbing set yields an improved estimate of the dimension, N ∼ L.
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ژورنال
عنوان ژورنال: SIAM Journal on Applied Dynamical Systems
سال: 2002
ISSN: 1536-0040
DOI: 10.1137/s111111110240176x