Nonlinear Hyperbolic Smoothing at a Focal Point

Joly, Jean-Luc; Métivier, Guy; Rauch, Jeffrey

Joly, Jean-Luc ¹ ; Métivier, Guy ² ; Rauch, Jeffrey 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 (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1998-1999), Exposé no. 5, 11 p.

Résumé

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 $0 \leq t < 1$ the solutions have a focusing wave of singularity on the incoming light cone $| x | = 1 - t$ . In ${t \geq 1}$ that is after the focusing time, they are smoother than they were in ${0 \leq t < 1}$ . The examples are radial and piecewise smooth in ${0 \leq t < 1}$

MR Zbl

Affiliations des auteurs :

Joly, Jean-Luc ¹ ; Métivier, Guy ² ; Rauch, Jeffrey 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

@article{SEDP_1998-1999____A5_0,
     author = {Joly, Jean-Luc and M\'etivier, Guy and Rauch, Jeffrey},
     title = {Nonlinear {Hyperbolic} {Smoothing} at a {Focal} {Point}},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
     note = {talk:5},
     pages = {1--11},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {1998-1999},
     zbl = {1059.35516},
     mrnumber = {1721323},
     language = {en},
     url = {http://www.numdam.org/item/SEDP_1998-1999____A5_0/}
}

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 (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
N1  - talk:5
PY  - 1998-1999
SP  - 1
EP  - 11
PB  - Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://www.numdam.org/item/SEDP_1998-1999____A5_0/
LA  - en
ID  - SEDP_1998-1999____A5_0
ER  -

%0 Journal Article
%A Joly, Jean-Luc
%A Métivier, Guy
%A Rauch, Jeffrey
%T Nonlinear Hyperbolic Smoothing at a Focal Point
%J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
%Z talk:5
%D 1998-1999
%P 1-11
%I Centre de mathématiques Laurent Schwartz, École polytechnique
%U http://www.numdam.org/item/SEDP_1998-1999____A5_0/
%G en
%F SEDP_1998-1999____A5_0

Joly, Jean-Luc; Métivier, Guy; Rauch, Jeffrey. Nonlinear Hyperbolic Smoothing at a Focal Point. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1998-1999), Exposé no. 5, 11 p. http://www.numdam.org/item/SEDP_1998-1999____A5_0/

Bibliographie
Cité par

[B] 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. | EuDML | Numdam | MR | Zbl

[CH] R. Courant and D. Hilbert, Methods of Mathematical Physics Vol. I, Interscience, N.Y, 1953. | MR | Zbl

[GR] P. Gerard and J. Rauch, Propagation de la régularité locale de solutions d’équations hyperboliques nonlinéaires, Ann. Inst. Fourier, 37(1987), 65-85. | EuDML | Numdam | Zbl

[JMR1] 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. | EuDML | Numdam | MR | Zbl

[JMR2] J.-L. Joly, G. Métivier, and J. Rauch, Focusing at a point and absorbtion of nonlinear oscillations, Trans. AMS. (347)1995, 3921-3969. | MR | Zbl

[JMR3] 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. | MR | Zbl

[JMR4] J.-L. Joly, G. Métivier, and J. Rauch, $L^{p}$ estimates for oscillatory integrals and caustics for dissipative semilinear oscillations, preprint. | MR

[LS] J.-L. Lions, and W. Strauss, Some nonlinear evolution equations, Bull. Soc. Math. France 93(1965), 43-96 | EuDML | Numdam | MR | Zbl

[RR1] 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. | MR | Zbl

[RR2] J. Rauch and M. Reed, Striated solutions of semilinear, two-speed wave equations, Indiana U. Math. J. 34(1985) 337-353. | MR | Zbl

[RR3] J. Rauch and M. Reed, Nonlinear superposition and absorbtion of delta waves in one space dimension, J. Funct. anal. 73(1987), 152-178. | MR | Zbl

[RR4] 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. | MR | Zbl

[SW] E. Stein and G. Weiss, Fractional integrals in $n$ dimensional Euclidean space, J. Math. and Mech., (1958), 503-514. | MR | Zbl