In
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.
