2017/09/13 by Jared Speck
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Boundary (topology) #Cartesian coordinate system #Curvilinear coordinates #Eikonal equation #Geometrical optics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Nonlinear system #Partial differential equation #Singularity #Variable (mathematics) #math-ph #math.AP #math.MP #msc:35L45 #msc:35L67
paper · pdf · doi:10.2140/paa.2019.1.447
published as Pure Appl. Analysis 1 (2019) 447-514 · 68 pages, 2 figures
arxiv created 2017/09/13 · openalex created_date 2017/09/25 · openalex publication_date 2019/07/17 · arxiv updated 2019/07/31 · openalex updated_date 2026/08/05
In [math] spatial dimensions, we study the Cauchy problem for a genuinely nonlinear quasilinear transport equation coupled to a quasilinear symmetric hyperbolic subsystem of a rather general type. For an open set (relative to a suitable Sobolev topology) of regular initial data that are close to the data of a simple plane wave, we give a sharp, constructive proof of shock formation in which the transport variable remains bounded but its first-order Cartesian coordinate partial derivatives blow up in finite time. Moreover, we prove that, at least at the low derivative levels, the singularity does not propagate into the symmetric hyperbolic variables: they and their first-order Cartesian coordinate partial derivatives remain bounded, even though they interact with the transport variable all the way up to its singularity. The formation of the singularity is tied to the finite-time degeneration, relative to the Cartesian coordinates, of a system of geometric coordinates adapted to the characteristics of the transport operator. Two crucial features of the proof are that relative to the geometric coordinates, all solution variables remain smooth, and that the finite-time degeneration coincides with the intersection of the transport characteristics. Compared to prior shock formation results in more than one spatial dimension, in which the blowup occurred in solutions to quasilinear wave equations, the main new features of the present work are: (i) we develop a theory of nonlinear geometric optics for transport operators, which is compatible with the coupling and which allows us to implement a quasilinear geometric vector field method, even though the regularity properties of the corresponding eikonal function are less favorable compared to the wave equation case and (ii) we allow for a full quasilinear coupling; i.e., the principal coefficients in all equations are allowed to depend on all solution variables.