Research Interests
While the theory for one dimensional stability of traveling waves is by now fairly well developed, the case of multidimensional stability of one dimensional structures, such as planar shocks or profiles is far less understood. Nevertheless, starting with the inviscid results of Majda [Maj1, Maj2] for planar shocks of multidimensional concervation laws and the more recent results by Gues, Metivier, Williams and Zumbrun for planar viscous shocks [GMWZ1] and other scenarios [GMWZ2,GMWZ3] a theory of multidimensional stability of one dimensional structures is begining to emerge. However, there are still many important open questions in the stability of planar profiles. For instance, the multidimensional stability of Boltzmann profiles has not yet been resolved (c.f. [Y][LY] for recent one dimensional results). Even less understood is the case of multidimensional stability of multidimensional structures such as vortices or spherical shocks. For instance, the short-time existence result in [Maj2], valid even when the shock is a compact manifold rather than simply planar, does not describe how the stability depends on the curvature of the manifold. The simplest structure in which to investigate the effect of curvature on stability (and hence on the existence time of a nearby perturbed structure) is a spherical shock.
Global Solutions of Multidimensional Conservation Laws
A long standing problem in the field of hyperbolic conservation laws is to obtain a long-time existence result for the Cauchy problem for a system of conservation laws in more than one spatial dimension. For one dimension, there is a global BV solution provided the BV norm of the initial data is small enough [Gl]. In more than one spatial dimension however, it is well known that BV is not the correct space for which to prove a long-time existence result [B,R,Del], and as of yet the correct space has not been unequivocably identified. To understand some of the obstructions to global existence one could first attempt to understand the case of gloabl radial solutions of multidimensional conservation laws. Here one much understand the behavior of spherical waves converging to the origin. As argued by Glimm, this is the most likely worst-case scenario for regularity so that any long-time existence theory must include this case.
Other Interests
Geometric singular perturbation theory and the Evans bundle
Hyperbolic initial boundary value problems
Stochastic evolution equations
Any hard, interesting problem which I can make some progress solving!!!!
References
[BB] S. Bianchini & A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Annals of Mathematics, 161 (2005), pp 223-342.
[B] A. Bressan, An ill posed Cauchy problem for a hyperbolic system in two space dimensions, Rend. Sem. Mat. Univ. Padova, 110, (2003).
[Del] C. DeLellis, Blowup of the BV norm in the multidimensional Keyfitz and Kranzer system, Duke Math. J. 127, no. 2 (2005), 313–339.
[J] C.K.R.T Jones
[Gl] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm.
Pure Appl. Math. 18, (1965), 697-715.
[GMWZ1] O. Gues, G. Metivier, M. Williams, and K. Zumbrun, Multidimensional viscous
shocks I: degenerate symmetrizers and long time stability, J. Amer. Math. Soc., 18, (2005),
no.1, 61-120.
[GMWZ2] O. Gues, G. Metivier, M. Williams, and K. Zumbrun, Existence and stability of multidimensional shock fronts in the vanishing viscosity limit, Arch. Rat. Mech. Anal. 175. pp 151-244 (2004).
[GMWZ3] O. Gues, G. Metivier, M. Williams, and K. Zumbrun, Navier-Stokes regularization of multidimensional Euler
shocks, to appear in Ann. Scient. ENS.
[Ma1] A Majda, The stability of multi-dimensional shock fronts -- a new
problem for linear hyperbolic equations, Mem. Amer. Math. Soc., No.
275, AMS, Providence, 1983.
[Ma2] A. Majda, The
existence of multidimensional shock fronts, Mem. Amer. Math. Soc.
No. 281, AMS, Providence, 1983.
[R] J.Rauch, BV estimates fail for most quasilinear hyperbolic systems
in dimensions greater than one, Comm. Math. Phys., 106 (1986),
pp 481–484.
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