1,236 research outputs found
Spectral stability of Prandtl boundary layers: an overview
In this paper we show how the stability of Prandtl boundary layers is linked
to the stability of shear flows in the incompressible Navier Stokes equations.
We then recall classical physical instability results, and give a short
educational presentation of the construction of unstable modes for Orr
Sommerfeld equations. We end the paper with a conjecture concerning the
validity of Prandtl boundary layer asymptotic expansions.Comment: 17 page
Large deviations for velocity-jump processes and non-local Hamilton-Jacobi equations
We establish a large deviation theory for a velocity jump process, where new
random velocities are picked at a constant rate from a Gaussian distribution.
The Kolmogorov forward equation associated with this process is a linear
kinetic transport equation in which the BGK operator accounts for the changes
in velocity. We analyse its asymptotic limit after a suitable rescaling
compatible with the WKB expansion. This yields a new type of Hamilton Jacobi
equation which is non local with respect to velocity variable. We introduce a
dedicated notion of viscosity solution for the limit problem, and we prove
well-posedness in the viscosity sense. The fundamental solution is explicitly
computed, yielding quantitative estimates for the large deviations of the
underlying velocity-jump process {\em \`a la Freidlin-Wentzell}. As an
application of this theory, we conjecture exact rates of acceleration in some
nonlinear kinetic reaction-transport equations
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