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Conormal and piecewise smooth solutions to quasilinear wave equations


Author: Seong Joo Kang
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
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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

DOI: https://doi.org/10.1090/S0002-9947-1995-1282889-2
Keywords: Strictly hyperbolic quasilinear equation, conormal, piecewise smooth
Article copyright: © Copyright 1995 American Mathematical Society

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