## Conormal and piecewise smooth solutions to quasilinear wave equations

HTML articles powered by AMS MathViewer

- 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 ${\Sigma _0} \cup {\Sigma _1}$.## References

- S. Alinhac,
*Paracomposition et application aux équations non-linéaires*, Bony-Sjöstrand-Meyer seminar, 1984–1985, École Polytech., Palaiseau, 1985, pp. Exp. No. 11, 11 (French). MR**819777**
—, - Michael Beals,
*Self-spreading and strength of singularities for solutions to semilinear wave equations*, Ann. of Math. (2)**118**(1983), no. 1, 187–214. MR**707166**, DOI 10.2307/2006959 - Michael Beals,
*Propagation and interaction of singularities in nonlinear hyperbolic problems*, Progress in Nonlinear Differential Equations and their Applications, vol. 3, Birkhäuser Boston, Inc., Boston, MA, 1989. MR**1033737**, DOI 10.1007/978-1-4612-4554-4 - Michael Beals and Michael Reed,
*Propagation of singularities for hyperbolic pseudodifferential operators with nonsmooth coefficients*, Comm. Pure Appl. Math.**35**(1982), no. 2, 169–184. MR**644021**, DOI 10.1002/cpa.3160350203 - Michael Beals and Michael Reed,
*Microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems*, Trans. Amer. Math. Soc.**285**(1984), no. 1, 159–184. MR**748836**, DOI 10.1090/S0002-9947-1984-0748836-7
J. M. Bony, - Thomas Messer,
*Propagation of singularities of hyperbolic systems*, Indiana Univ. Math. J.**36**(1987), no. 1, 45–77. MR**876991**, DOI 10.1512/iumj.1987.36.36003 - Yves Meyer,
*Remarques sur un théorème de J.-M. Bony*, Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), 1981, pp. 1–20 (French). MR**639462** - L. Nirenberg,
*On elliptic partial differential equations*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)**13**(1959), 115–162. MR**109940**
—, - Jeffrey Rauch,
*Singularities of solutions to semilinear wave equations*, J. Math. Pures Appl. (9)**58**(1979), no. 3, 299–308. MR**544255** - Jeffrey Rauch and Michael C. Reed,
*Propagation of singularities for semilinear hyperbolic equations in one space variable*, Ann. of Math. (2)**111**(1980), no. 3, 531–552. MR**577136**, DOI 10.2307/1971108 - Jeffrey Rauch and Michael Reed,
*Striated solutions of semilinear, two-speed wave equations*, Indiana Univ. Math. J.**34**(1985), no. 2, 337–353. MR**783919**, DOI 10.1512/iumj.1985.34.34020

*Interaction d’ondes simple pour des équations complétement nonlineaires*, Sém. d’E.D.P. no. 8 (1985-1986), École Polytechnique, Paris.

*Interaction des singularités pour les équations aux dérivées partielles non-linéaires*, Sem. Goulaouic-Meyer-Schwartz

**22**(1979-80). —,

*Interaction des singularités pour les équations aux dérivées partielles non-linéaires*, Sem. Goulaouic-Meyer-Schwartz

**2**(1981-82).

*Lecture on linear partial differential equation*, C.B.M.S. Regional Conf. Ser. in Math., no. 17, Amer. Math. Soc., Providence, RI, 1973.

## 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