Nonlinear Hyperbolic Smoothing at a Focal Point

Jean-Luc Joly[1]; Guy Métivier[2]; Jeffrey Rauch[3]

  • [1] MAB, Université de Bordeaux I, 33405 Talence, FRANCE
  • [2] IRMAR, Université de Rennes I,35042 Rennes, FRANCE
  • [3] Department of Mathematics, University of Michigan, Ann Arbor 48109 MI, USA

Séminaire Équations aux dérivées partielles (1998-1999)

  • Volume: 1998-1999, page 1-11

Abstract

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The nonlinear dissipative wave equation u t t - Δ u + | u t | h - 1 u t = 0 in dimension d > 1 has strong solutions with the following structure. In 0 ≤ t < 1 the solutions have a focusing wave of singularity on the incoming light cone | x | = 1 - t . In { t ≥ 1 } that is after the focusing time, they are smoother than they were in { 0 ≤ t < 1 } . The examples are radial and piecewise smooth in { 0 ≤ t < 1 }

How to cite

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Joly, Jean-Luc, Métivier, Guy, and Rauch, Jeffrey. "Nonlinear Hyperbolic Smoothing at a Focal Point." Séminaire Équations aux dérivées partielles 1998-1999 (1998-1999): 1-11. <http://eudml.org/doc/10979>.

@article{Joly1998-1999,
abstract = {The nonlinear dissipative wave equation $u_\{tt\}-\Delta u + |u_t|^\{h-1\}u_t=0$ in dimension $d&gt;1$ has strong solutions with the following structure. In $0\le t &lt;1$ the solutions have a focusing wave of singularity on the incoming light cone $|x|=1-t$. In $\lbrace t\ge 1\rbrace $ that is after the focusing time, they are smoother than they were in $\lbrace 0\le t&lt;1\rbrace $. The examples are radial and piecewise smooth in $\lbrace 0\le t&lt;1 \rbrace $},
affiliation = {MAB, Université de Bordeaux I, 33405 Talence, FRANCE; IRMAR, Université de Rennes I,35042 Rennes, FRANCE; Department of Mathematics, University of Michigan, Ann Arbor 48109 MI, USA},
author = {Joly, Jean-Luc, Métivier, Guy, Rauch, Jeffrey},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {dissipative wave equation; strong solutions; incoming light cone; focusing time},
language = {eng},
pages = {1-11},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Nonlinear Hyperbolic Smoothing at a Focal Point},
url = {http://eudml.org/doc/10979},
volume = {1998-1999},
year = {1998-1999},
}

TY - JOUR
AU - Joly, Jean-Luc
AU - Métivier, Guy
AU - Rauch, Jeffrey
TI - Nonlinear Hyperbolic Smoothing at a Focal Point
JO - Séminaire Équations aux dérivées partielles
PY - 1998-1999
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 1998-1999
SP - 1
EP - 11
AB - The nonlinear dissipative wave equation $u_{tt}-\Delta u + |u_t|^{h-1}u_t=0$ in dimension $d&gt;1$ has strong solutions with the following structure. In $0\le t &lt;1$ the solutions have a focusing wave of singularity on the incoming light cone $|x|=1-t$. In $\lbrace t\ge 1\rbrace $ that is after the focusing time, they are smoother than they were in $\lbrace 0\le t&lt;1\rbrace $. The examples are radial and piecewise smooth in $\lbrace 0\le t&lt;1 \rbrace $
LA - eng
KW - dissipative wave equation; strong solutions; incoming light cone; focusing time
UR - http://eudml.org/doc/10979
ER -

References

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  1. J.-M. Bony, Intéraction des singularités pour les équations aux dérivées partielles non linéaires, Sém. Goulaouic-Meyer-Schwartz, 1979-80 #22, and 1981-82 #2. Zbl0498.35017MR544505
  2. R. Courant and D. Hilbert, Methods of Mathematical Physics Vol. I, Interscience, N.Y, 1953. Zbl0051.28802MR65391
  3. P. Gerard and J. Rauch, Propagation de la régularité locale de solutions d’équations hyperboliques nonlinéaires, Ann. Inst. Fourier, 37(1987), 65-85. Zbl0617.35079
  4. J.-L. Joly, G. Métivier, and J. Rauch, Focussing and Absorbtion of Nonlinear Oscillations, Journées aux Dérivées Partielles, St. Jean de Monts, École Polytéchnique Publ.,(1993) III-1 to III-11. Zbl0815.35071MR1240527
  5. J.-L. Joly, G. Métivier, and J. Rauch, Focusing at a point and absorbtion of nonlinear oscillations, Trans. AMS. (347)1995, 3921-3969. Zbl0857.35087MR1297533
  6. J.-L. Joly, G. Métivier, and J. Rauch, Caustics for dissipative semilinear oscillations, in Geometric Optics and Related Topics, F. Colombini and N. Lerner eds, Birkhaüser, Boston, 1997, 245-266. Zbl0891.35098MR2033498
  7. J.-L. Joly, G. Métivier, and J. Rauch, L p estimates for oscillatory integrals and caustics for dissipative semilinear oscillations, preprint. Zbl0891.35098MR2033498
  8. J.-L. Lions, and W. Strauss, Some nonlinear evolution equations, Bull. Soc. Math. France 93(1965), 43-96 Zbl0132.10501MR199519
  9. J. Rauch and M. Reed, Jump discontinuities of semilinear, strictly hyperbolic sytems in two variables: creation and propagation, Comm. Math. Phys. 81(1981) 203-227. Zbl0468.35064MR632757
  10. J. Rauch and M. Reed, Striated solutions of semilinear, two-speed wave equations, Indiana U. Math. J. 34(1985) 337-353. Zbl0559.35053MR783919
  11. J. Rauch and M. Reed, Nonlinear superposition and absorbtion of delta waves in one space dimension, J. Funct. anal. 73(1987), 152-178. Zbl0661.35058MR890661
  12. J. Rauch and M. Reed, Bounded stratified and striated solutions of hyperbolic systems, in Nonlinear Partial Differential Equations and Their Applications Vol. IX, H. Brezis and J. L. Lions, eds., Pitman Research Notes in Math., 181(1989), 334-351. Zbl0695.35124MR992654
  13. E. Stein and G. Weiss, Fractional integrals in n dimensional Euclidean space, J. Math. and Mech., (1958), 503-514. Zbl0082.27201MR98285

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