Piotr Bizoń

Piotr Bizoń

Dissipation through dispersion

Dissipation of energy by radiation is a fundamental phenomenon which is responsible for the asymptotic stability of solutions of extended Hamiltonian systems [1] . In particular, for dispersive wave equations defined on spatially unbounded domains, the relaxation to a stationary equilibrium is due the radiation of excess energy to infinity. This occurs in a variety of physical situations, ranging from dynamics of gas bubbles in a compressible fluid to the formation of black holes in gravitational collapse. The radiation emitted during the approach to equilibrium encodes information about the final equilibrium state and thus can be used to determine its properties (in fact, future applications of this inverse problem in the gravitational wave astronomy are one of our motivations). Despite physical importance, mathematical understanding of dissipation-by-dispersion phenomena is still in its infancy, especially in the non-perturbative regime where initial data are not close to the final equilibrium. This prompted me to study this phenomenon is toy models (Skyrme model, equivariant wave maps, and spherically-symmetric Yang-Mills fields). Together with Michał Kahl we have employed the hyperboloidal formulation of the initial value problem (introduced by Helmut Friedrich [2] and developed by Anil Zenginoglu [3] ) to study this problem [4,5] . The hyperboloidal approach is ideally suited for studying relaxation to static solutions because it inherently incorporates the dissipation of energy by dispersion. Another big advantage is that the convergence to the attractor occurs pointwise on the entire spatial hypersurfaces (the leaves of the hyperboloidal foliation), including the null infinity. Moreover, in this formulation the quasinormal modes (aka scattering resonances) can be defined as genuine eigenmodes of a certain non self-adjoint linear operator - this has both conceptual and computational advantages over the standard definitions involving outgoing-wave boundary conditions (this was first pointed out by Bernd Schmidt [6] and independently developed rigorously by Claude Warnick for asymptotically AdS black holes [7] ). Recently, together with Patryk Mach we combined the hyperboloidal approach with the Galerkin method to analyze dynamics of the Yang-Mills field on the hyperbolic space [8] .

  1. A. Soffer, Dissipation through dispersion, in Nonlinear dynamics and renormalization group, CRM Proc. Lecture Notes 27, 175, Amer. Math. Soc., Providence, RI, 2001. ↩
  2. H. Friedrich, Cauchy problems for the conformal vacuum field equations in general relativity, Comm. Math. Phys. 91 , 445 (1983) ↩
  3. A. Zenginoglu, Hyperboloidal foliations and scri-fixing, Classical Quantum Gravity 25, 145002 (2008) ↩
  4. P. Bizoń, M. Kahl, Wave maps on a wormhole, Phys. Rev. D 91, 065003 (2015) ↩
  5. P. Bizoń, M. Kahl, A Yang-Mills field on the extremal Reissner-Nordstrm black hole, Class. Quantum Grav. 33, 175013 (2016)↩
  6. B.G. Schmidt, On relativistic stellar oscillations, Gravity Research Foundation essay (1993). ↩
  7. C.M. Warnick, On quasinormal modes of asymptotically anti-de Sitter black holes, Comm. Math. Phys. 333, 959 (2015)↩
  8. P. Bizoń, P. Mach, Global dynamics of a Yang-Mills field on an asymptotically hyperbolic space, Trans. Amer. Math. Soc. 369, 2029 (2017) ↩