Course highlight: APMA2210: Topics in Differential Equations

APMA2210: Topics in Differential Equations
Stability of Noncharacteristic Viscous Boundary Layers
Instructor: Toan Nguyen

Description:
My plan for this course is to discuss asymptotic stability of noncharacteristic boundary layers in gas dynamics and magnetohydrodynamics equations. A main difficulty in the stability analysis is that there is no spectral gap between the imaginary axis and the essential spectrum of the linearized operator about the layer. Standard semi-group methods therefore do not seem to apply and, at best, algebraic temporal decay can be expected in case of stability. Pointwise estimates for the Green function are then useful and sufficient for analysis of the (linear and) nonlinear stability. The general mathematical approach to be covered in the course is the so-called pointwise semi-group or Evans function approach developed by Zumbrun-Howard and Mascia-Zumbrun in their studies of (orbital) asymptotic stability of viscous shock waves. We discuss its application in the context of boundary layers. In particular, I shall discuss in detail the gap/conjugation lemma, the tracking/reduction lemma, construction of the resolvent kernel, the spectral and Evans function theory, and the construction of the Green function with sharp pointwise estimates.

Time permitting, I shall also discuss a few recent developments on stability of boundary layers for a more general class of hyperbolic-parabolic conservation laws in one or multi-dimensional spaces. Certain numerical evidences for stability might as well be demonstrated in the end of the course.

Course Information:
Meeting time: MWF 2 - 2:50pm, B&H 163
Office hours: by appointment