516 research outputs found
Fast growth of the vorticity gradient in symmetric smooth domains for 2D incompressible ideal flow
We construct an initial data for the two-dimensional Euler equation in a
bounded smooth symmetric domain such that the gradient of vorticity in
grows as a double exponential in time for all time. Our
construction is based on the recent result by Kiselev and \v{S}ver\'{a}k.Comment: 18 page
Dissipation enhancement for a degenerated parabolic equation
In this paper, we quantitatively consider the enhanced-dissipation effect of
the advection term to the parabolic -Laplacian equations. More precisely, we
show the mixing property of flow for the passive scalar enhances the
dissipation process of the -Laplacian in the sense of decay, that is,
the decay can be arbitrarily fast. The main ingredient of our argument is
to understand the underlying iteration structure inherited from the parabolic
-Laplacian equations. This extends the dissipation enhancement result of the
advection diffusion equation by Yuanyuan Feng and Gautam Iyer into a non-linear
setting.Comment: 22 page
Suppression of epitaxial thin film growth by mixing
We consider following fourth-order parabolic equation with gradient
nonlinearity on the two-dimensional torus with and without advection of an
incompressible vector field in the case : \begin{equation*}
\partial_t u + (-\Delta)^2 u = -\nabla\cdot(|\nabla u|^{p-2}\nabla u).
\end{equation*} The study of this form of equations arises from mathematical
models that simulate the epitaxial growth of the thin film. We prove the local
existence of mild solutions for any initial data lies in in both cases.
Our main result is: in the advective case, if the imposed advection is
sufficiently mixing, then the global existence of solution can be proved, and
the solution will converge exponentially to a homogeneous mixed state. While in
the absence of advection, there exist initial data in
such that the solution will blow up in finite time.Comment: 33 page
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