Multidimensional nonlinear geometric optics for transport operators with applications to stable shock formation

Abstract

In n ≥ 1 spatial dimensions, we study the Cauchy problem for a 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 the singularity does not propagate into the symmetric hyperbolic variables: they and their firstorder 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 characteristics. Compared to prior shock formation results in more than one spatial dimension, in which the blowup occurred in solutions to 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 vectorfield 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.

On the stability of multi-dimensional rarefaction waves I: the energy estimates

“…The most recent work [1] [2] of Abbrescia and Speck further studies the structure of the singular boundary of the maximal developments of the data. For the new developments on the shock formation in other hyperbolic equations under the geometric frame work of Christodoulou, we refer the readers to [30], [31], [49], [50], [55], [56] and [57]. The work of Christodoulou also inspired research on the low regularity theory on Euler equations, see the series of work [29], [28] and [58] and also a sharper result [60] of Q. Wang.…”

Stable ODE-type blowup for some quasilinear wave equations with derivative-quadratic nonlinearities

“…This approach is more robust and is capable of accommodating solutions such that the blowup occurs along a hypersurface, which, in the problem of shock formation, is what typically occurs along a portion of the boundary of the maximal development. Christodoulou's results have since been extended in many directions, including to apply to other wave equations [9,68], different sets of initial data [50,51,69], the compressible Euler equations with non-zero vorticity [45,46,63], systems of wave equations with multiple speeds [64], and quasilinear systems in which a solution to a transport equation forms a shock [65]. Some of the earlier extensions are explained in detail in the survey article [26].…”