Vol. 1, No. 3, 2019

Download this article
Download this article For screen
For printing
Recent Issues
Volume 2, Issue 4
Volume 2, Issue 3
Volume 2, Issue 2
Volume 2, Issue 1
Volume 1, Issue 4
Volume 1, Issue 3
Volume 1, Issue 2
Volume 1, Issue 1
The Journal
About the Journal
Editorial Board
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN (electronic): 2578-5885
ISSN (print): 2578-5893
Author Index
To Appear
Other MSP Journals
This article is available for purchase or by subscription. See below.
Multidimensional nonlinear geometric optics for transport operators with applications to stable shock formation

Jared Speck

Vol. 1 (2019), No. 3, 447–514

In n 1 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.

PDF Access Denied

However, your active subscription may be available on Project Euclid at

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

blowup, characteristics, eikonal equation, eikonal function, simple wave, singularity, stable blowup, vector field method, wave breaking
Mathematical Subject Classification 2010
Primary: 35L67
Secondary: 35L45
Received: 17 February 2019
Revised: 14 April 2019
Accepted: 22 April 2019
Published: 17 July 2019
Jared Speck
Department of Mathematics
Vanderbilt University
Nashville, TN
United States