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2019 Multidimensional nonlinear geometric optics for transport operators with applications to stable shock formation
Jared Speck
Pure Appl. Anal. 1(3): 447-514 (2019). DOI: 10.2140/paa.2019.1.447

Abstract

In n1 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.

Citation

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Jared Speck. "Multidimensional nonlinear geometric optics for transport operators with applications to stable shock formation." Pure Appl. Anal. 1 (3) 447 - 514, 2019. https://doi.org/10.2140/paa.2019.1.447

Information

Received: 17 February 2019; Revised: 14 April 2019; Accepted: 22 April 2019; Published: 2019
First available in Project Euclid: 31 July 2019

zbMATH: 07114664
MathSciNet: MR3985091
Digital Object Identifier: 10.2140/paa.2019.1.447

Subjects:
Primary: 35L67
Secondary: 35L45

Keywords: blowup , characteristics , eikonal equation , eikonal function , simple wave , singularity , stable blowup , vector field method , wave breaking

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.1 • No. 3 • 2019
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