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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Conormal and piecewise smooth solutions to quasilinear wave equations
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by Seong Joo Kang PDF
Trans. Amer. Math. Soc. 347 (1995), 1-35 Request permission

Abstract:

In this paper, we show first that if a solution $u$ of the equation ${P_2}(t,x,u,Du,D)u = f(t,x,u,Du)$, where ${P_2}(t,x,u,Du,D)$ is a second order strictly hyperbolic quasilinear operator, is conormal with respect to a single characteristic hypersurface $\Sigma$ of ${P_2}$ in the past and $\Sigma$ is smooth in the past, then $\Sigma$ is smooth and $u$ is conormal with respect to $\Sigma$ for all time. Second, let ${\Sigma _0}$ and ${\Sigma _1}$ be characteristic hypersurfaces of ${P_2}$ which intersect transversally and let $\Gamma = {\Sigma _0} \cap {\Sigma _1}$. If ${\Sigma _0}$ and ${\Sigma _1}$ are smooth in the past and $u$ is conormal with repect to $\{ {\Sigma _0},{\Sigma _1}\}$ in the past, then $\Gamma$ is smooth, and $u$ is conormal with respect to $\{ {\Sigma _0},{\Sigma _1}\}$ locally in time outside of $\Gamma$, even though ${\Sigma _0}$ and ${\Sigma _1}$ are no longer necessarily smooth across $\Gamma$. Finally, we show that if $u(0,x)$ and ${\partial _t}u(0,x)$ are in an appropriate Sobolev space and are piecewise smooth outside of $\Gamma$, then $u$ is piecewise smooth locally in time outside of ${\Sigma _0} \cup {\Sigma _1}$.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 347 (1995), 1-35
  • MSC: Primary 35L70
  • DOI: https://doi.org/10.1090/S0002-9947-1995-1282889-2
  • MathSciNet review: 1282889