On the wave equation with multiplicities and space-dependent irregular coefficients

Garetto, Claudia

doi:10.1090/tran/8319

On the wave equation with multiplicities and space-dependent irregular coefficients
HTML articles powered by AMS MathViewer

by Claudia Garetto PDF

Trans. Amer. Math. Soc. 374 (2021), 3131-3176 Request permission

Abstract:

In this paper we study the well-posedness of the Cauchy problem for a wave equation with multiplicities and space-dependent irregular coefficients. As in Garetto and Ruzhansky [Arch. Ration. Mech. Anal. 217 (2015), pp. 113–154], in order to give a meaningful notion of solution, we employ the notion of very weak solution, which construction is based on a parameter dependent regularisation of the coefficients via mollifiers. We prove that, even with distributional coefficients, a very weak solution exists for our Cauchy problem and it converges to the classical one when the coefficients are smooth. The dependence on the mollifiers of very weak solutions is investigated at the end of the paper in some instructive examples.

References

Arshyn Altybay, Michael Ruzhansky, and Niyaz Tokmagambetov, Wave equation with distributional propagation speed and mass term: numerical simulations, Appl. Math. E-Notes 19 (2019), 552–562. MR 4027900
A. Altybay, M. Ruzhansky, M. E. Sebih, and N. Tokmagambetov, The heat equation with singular potential, arXiv:2004.11255, 2020.
A. Altybay, M. Ruzhansky, M. E. Sebih, and N. Tokmagambetov, Fractional Schrödinger equations with potential of higher order singularities, arXiv:2004.10182, 2020.
A. Altybay, M. Ruzhansky, M. E. Sebih, and N. Tokmagambetov, Fractional Klein-Gordon equation with singular mass term, arXiv:2004.10145, 2020.
Fredrik Andersson, Maarten V. de Hoop, Hart F. Smith, and Gunther Uhlmann, A multi-scale approach to hyperbolic evolution equations with limited smoothness, Comm. Partial Differential Equations 33 (2008), no. 4-6, 988–1017. MR 2424386, DOI 10.1080/03605300701629393
H. A. Biagioni and M. Oberguggenberger, Generalized solutions to the Korteweg-de Vries and the regularized long-wave equations, SIAM J. Math. Anal. 23 (1992), no. 4, 923–940. MR 1166565, DOI 10.1137/0523049
M. D. Bronšteĭn, The Cauchy problem for hyperbolic operators with characteristics of variable multiplicity, Trudy Moskov. Mat. Obshch. 41 (1980), 83–99 (Russian). MR 611140
Valeriy Brytik, Maarten V. de Hoop, Hart F. Smith, and Gunther Uhlmann, Decoupling of modes for the elastic wave equation in media of limited smoothness, Comm. Partial Differential Equations 36 (2011), no. 10, 1683–1693. MR 2832159, DOI 10.1080/03605302.2011.558554
Peter Caday, Maarten V. de Hoop, Vitaly Katsnelson, and Gunther Uhlmann, Scattering control for the wave equation with unknown wave speed, Arch. Ration. Mech. Anal. 231 (2019), no. 1, 409–464. MR 3894555, DOI 10.1007/s00205-018-1283-8
Ferruccio Colombini and Tamotu Kinoshita, On the Gevrey well posedness of the Cauchy problem for weakly hyperbolic equations of higher order, J. Differential Equations 186 (2002), no. 2, 394–419. MR 1942215, DOI 10.1016/S0022-0396(02)00009-8
Ferruccio Colombini and Tamotu Kinoshita, On the Gevrey wellposedness of the Cauchy problem for weakly hyperbolic equations of 4th order, Hokkaido Math. J. 31 (2002), no. 1, 39–60. MR 1888269, DOI 10.14492/hokmj/1350911769
Ferruccio Colombini and Sergio Spagnolo, An example of a weakly hyperbolic Cauchy problem not well posed in $C^{\infty }$, Acta Math. 148 (1982), 243–253. MR 666112, DOI 10.1007/BF02392730
Ferruccio Colombini, Ennio De Giorgi, and Sergio Spagnolo, Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6 (1979), no. 3, 511–559 (French). MR 553796
Ferruccio Colombini, Enrico Jannelli, and Sergio Spagnolo, Nonuniqueness in hyperbolic Cauchy problems, Ann. of Math. (2) 126 (1987), no. 3, 495–524. MR 916717, DOI 10.2307/1971359
Piero D’Ancona and Tamotu Kinoshita, On the wellposedness of the Cauchy problem for weakly hyperbolic equations of higher order, Math. Nachr. 278 (2005), no. 10, 1147–1162. MR 2155966, DOI 10.1002/mana.200310299
Piero D’Ancona and Sergio Spagnolo, Quasi-symmetrization of hyperbolic systems and propagation of the analytic regularity, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), no. 1, 169–185 (English, with Italian summary). MR 1618976
Maarten de Hoop, Günther Hörmann, and Michael Oberguggenberger, Evolution systems for paraxial wave equations of Schrödinger-type with non-smooth coefficients, J. Differential Equations 245 (2008), no. 6, 1413–1432. MR 2436448, DOI 10.1016/j.jde.2008.06.022
Maarten V. de Hoop, Sean F. Holman, Hart F. Smith, and Gunther Uhlmann, Regularity and multi-scale discretization of the solution construction of hyperbolic evolution equations with limited smoothness, Appl. Comput. Harmon. Anal. 33 (2012), no. 3, 330–353. MR 2950133, DOI 10.1016/j.acha.2012.01.001
Nils Dencker, On the propagation of singularities for pseudo-differential operators with characteristics of variable multiplicity, Comm. Partial Differential Equations 17 (1992), no. 9-10, 1709–1736. MR 1187627, DOI 10.1080/03605309208820901
Claudia Garetto and Christian Jäh, Well-posedness of hyperbolic systems with multiplicities and smooth coefficients, Math. Ann. 369 (2017), no. 1-2, 441–485. MR 3694652, DOI 10.1007/s00208-016-1436-8
Claudia Garetto and Michael Ruzhansky, On the well-posedness of weakly hyperbolic equations with time-dependent coefficients, J. Differential Equations 253 (2012), no. 5, 1317–1340. MR 2927383, DOI 10.1016/j.jde.2012.05.001
Claudia Garetto and Michael Ruzhansky, Weakly hyperbolic equations with non-analytic coefficients and lower order terms, Math. Ann. 357 (2013), no. 2, 401–440. MR 3096513, DOI 10.1007/s00208-013-0910-9
Claudia Garetto and Michael Ruzhansky, Hyperbolic second order equations with non-regular time dependent coefficients, Arch. Ration. Mech. Anal. 217 (2015), no. 1, 113–154. MR 3338443, DOI 10.1007/s00205-014-0830-1
Claudia Garetto and Michael Ruzhansky, A note on weakly hyperbolic equations with analytic principal part, J. Math. Anal. Appl. 412 (2014), no. 1, 1–14. MR 3145777, DOI 10.1016/j.jmaa.2013.09.011
Claudia Garetto and Michael Ruzhansky, On $C^\infty$ well-posedness of hyperbolic systems with multiplicities, Ann. Mat. Pura Appl. (4) 196 (2017), no. 5, 1819–1834. MR 3694745, DOI 10.1007/s10231-017-0639-2
Claudia Garetto, Christian Jäh, and Michael Ruzhansky, Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I: well-posedness, Math. Ann. 372 (2018), no. 3-4, 1597–1629. MR 3880309, DOI 10.1007/s00208-018-1672-1
Claudia Garetto, Christian Jäh, and Michael Ruzhansky, Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, II: microlocal analysis, J. Differential Equations 269 (2020), no. 10, 7881–7905. MR 4113191, DOI 10.1016/j.jde.2020.05.038
Michael Grosser, Michael Kunzinger, Michael Oberguggenberger, and Roland Steinbauer, Geometric theory of generalized functions with applications to general relativity, Mathematics and its Applications, vol. 537, Kluwer Academic Publishers, Dordrecht, 2001. MR 1883263, DOI 10.1007/978-94-015-9845-3
Lars Hörmander, Hyperbolic systems with double characteristics, Comm. Pure Appl. Math. 46 (1993), no. 2, 261–301. MR 1199200, DOI 10.1002/cpa.3160460207
Lars Hörmander, Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26, Springer-Verlag, Berlin, 1997. MR 1466700
G. Hörmann and Maarten V. de Hoop, Microlocal analysis and global solutions of some hyperbolic equations with discontinuous coefficients, Acta Appl. Math. 67 (2001), no. 2, 173–224. MR 1848743, DOI 10.1023/A:1010614332739
Günther Hörmann and Maarten V. De Hoop, Detection of wave front set perturbations via correlation: foundation for wave-equation tomography, Appl. Anal. 81 (2002), no. 6, 1443–1465. MR 1956070, DOI 10.1080/0003681021000035489
V. Ja. Ivriĭ and V. M. Petkov, Necessary conditions for the correctness of the Cauchy problem for non-strictly hyperbolic equations, Uspehi Mat. Nauk 29 (1974), no. 5(179), 3–70 (Russian). Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), III. MR 0427843
Enrico Jannelli, The hyperbolic symmetrizer: theory and applications, Advances in phase space analysis of partial differential equations, Progr. Nonlinear Differential Equations Appl., vol. 78, Birkhäuser Boston, Boston, MA, 2009, pp. 113–139. MR 2664607, DOI 10.1007/978-0-8176-4861-9_{7}
Enrico Jannelli and Giovanni Taglialatela, Homogeneous weakly hyperbolic equations with time dependent analytic coefficients, J. Differential Equations 251 (2011), no. 4-5, 995–1029. MR 2812580, DOI 10.1016/j.jde.2011.04.009
Kunihiko Kajitani and Yasuo Yuzawa, The Cauchy problem for hyperbolic systems with Hölder continuous coefficients with respect to the time variable, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2006), no. 4, 465–482. MR 2297720
Ilia Kamotski and Michael Ruzhansky, Regularity properties, representation of solutions, and spectral asymptotics of systems with multiplicities, Comm. Partial Differential Equations 32 (2007), no. 1-3, 1–35. MR 2304141, DOI 10.1080/03605300600856816
Tamotu Kinoshita and Sergio Spagnolo, Hyperbolic equations with non-analytic coefficients, Math. Ann. 336 (2006), no. 3, 551–569. MR 2249759, DOI 10.1007/s00208-006-0009-7
V. V. Kucherenko and Yu. V. Osipov, The Cauchy problem for nonstrictly hyperbolic equations, Mat. Sb. (N.S.) 120(162) (1983), no. 1, 84–111, 144 (Russian). MR 687338
F. Lafon and M. Oberguggenberger, Generalized solutions to symmetric hyperbolic systems with discontinuous coefficients: the multidimensional case, J. Math. Anal. Appl. 160 (1991), no. 1, 93–106. MR 1124079, DOI 10.1016/0022-247X(91)90292-8
Maria Mascarello and Luigi Rodino, Partial differential equations with multiple characteristics, Mathematical Topics, vol. 13, Akademie Verlag, Berlin, 1997. MR 1608649
R. B. Melrose and G. A. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math. 32 (1979), no. 4, 483–519. MR 528633, DOI 10.1002/cpa.3160320403
R. B. Melrose and G. A. Uhlmann, Microlocal structure of involutive conical refraction, Duke Math. J. 46 (1979), no. 3, 571–582. MR 544247, DOI 10.1215/S0012-7094-79-04630-1
Juan Carlos Muñoz, Michael Ruzhansky, and Niyaz Tokmagambetov, Wave propagation with irregular dissipation and applications to acoustic problems and shallow waters, J. Math. Pures Appl. (9) 123 (2019), 127–147. MR 3912648, DOI 10.1016/j.matpur.2019.01.012
M. Oberguggenberger, Multiplication of distributions and applications to partial differential equations, Pitman Research Notes in Mathematics Series, vol. 259, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. MR 1187755
O. A. Oleĭnik, On the Cauchy problem for weakly hyperbolic equations, Comm. Pure Appl. Math. 23 (1970), 569–586. MR 264227, DOI 10.1002/cpa.3160230403
Yujiro Ohya and Shigeo Tarama, Le problème de Cauchy à caractéristiques multiples dans la classe de Gevrey. I. Coefficients hölderiens en $t$, Hyperbolic equations and related topics (Katata/Kyoto, 1984) Academic Press, Boston, MA, 1986, pp. 273–306 (French, with English summary). MR 925253
Cesare Parenti and Alberto Parmeggiani, On the Cauchy problem for hyperbolic operators with double characteristics, Comm. Partial Differential Equations 34 (2009), no. 7-9, 837–888. MR 2560303, DOI 10.1080/03605300902892360
Michael Ruzhansky and Niyaz Tokmagambetov, Wave equation for operators with discrete spectrum and irregular propagation speed, Arch. Ration. Mech. Anal. 226 (2017), no. 3, 1161–1207. MR 3712280, DOI 10.1007/s00205-017-1152-x
Michael Ruzhansky and Nurgissa Yessirkegenov, Very weak solutions to hypoelliptic wave equations, J. Differential Equations 268 (2020), no. 5, 2063–2088. MR 4046183, DOI 10.1016/j.jde.2019.09.020
M. E. Sebih and J. Wirth, On a wave equation with singular dissipation, arXiv:2002.00825, 2020.
Sergio Spagnolo and Giovanni Taglialatela, Some inequalities of Glaeser-Bronšteĭn type, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 17 (2006), no. 4, 367–375. MR 2287708, DOI 10.4171/RLM/474
Sergio Spagnolo and Giovanni Taglialatela, Homogeneous hyperbolic equations with coefficients depending on one space variable, J. Hyperbolic Differ. Equ. 4 (2007), no. 3, 533–553. MR 2339807, DOI 10.1142/S0219891607001240
Karen Yagdjian, The Cauchy problem for hyperbolic operators, Mathematical Topics, vol. 12, Akademie Verlag, Berlin, 1997. Multiple characteristics; Micro-local approach. MR 1469977