APMA2210: Topics in Differential Equations

Stability of Noncharacteristic Viscous Boundary Layers

Course Information:

Instructor: Toan T. Nguyen

Class meeting time: Wednesdays 12:30-3:00pm 

Meeting place: Room 104, 37 Manning.

Office hours: by appointment

Email: Toan_Nguyen@Brown.edu

Course Description:

This course is devoted to study asymptotic stability of boundary layers in compressible gas dynamics equations. Boundary layers to be considered are assumed to be noncharacteristic, which include those so called inflow/outflow or suction/blowing boundary layers.  In studies of the stability analysis for such a boundary layer, one of main difficulties 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 applications 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 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 course.

Main References:

1. Matsumura, A. and Nishihara, K., Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas, Comm. Math. Phys., 222 (2001), no. 3, 449--474.

2. K.~Zumbrun and P.~Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J., 47(3):741--871, 1998.

3. K, Zumbrun, Instantaneous shock location and one-dimensional nonlinear stability of viscous shock wave, arXiv:0909.2422v1.

4.  Yarahmadian, Shantia ; Zumbrun, Kevin . Pointwise green function bounds and long-time stability of
large-amplitude noncharacteristic boundary layers. SIAM J. Math. Anal. 40 (2009), no. 6, 2328--2350

5. J. Humpherys and K. Zumbrun, Spectral stability of small amplitude shock profiles for dissipative symmetric hyperbolic–parabolic systems. Z. Angew. Math. Phys. 53 (2002) 20–34.

6. To be updated...