Department of Mathematics

Analysis and PDE

  •  The Formation of Shock Singularities in Solutions to Wave Equation Systems with Multiple Speeds
  •  04/10/2017
  •  4:10 PM - 5:00 PM
  •  C517 Wells Hall
  •  Jared Speck, Massachusetts Institute of Technology

In this talk, I will describe my recent work on the formation of shock singularities in solutions to quasilinear wave equation systems in 2D with more than one speed, that is, systems with at least two distinct wave operators. In the systems under study, the fast wave forms a shock singularity while the slow wave remains regular, even though the two waves interact all the way up to the singularity. This work represents an extension of the remarkable proofs of shock formation for scalar quasilinear wave equations provided by S. Alinhac, as well as the breakthrough sharpening of Alinhac's results by D. Christodoulou for the scalar wave equations of irrotational fluid mechanics. Both results crucially relied on the construction of geometric vectorfields that are adapted to the wave characteristics, whose intersection drives the singularity formation. THe key new difficulty for systems with multiple speeds is that the geometric vectorfields are, by necessity, precisely adapted to the characteristics of the shock-forming (fast) wave. Thus, there is no freedom left to adapt the vectorfields to the characteristics of the slow wave, and for this reason, they exhibit very poor commutation properties with the slow wave operator. To overcome this difficulty, we rely in part on some ideas from our recent joint work with J. Luk, in which we proved a shock formation result for the compressible Euler equations with vorticity, which we formulated as a wave-transport system featuring precisely one wave operator.

 

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