Nonlinear waves and dispersive equations. Abstracts from the workshop held June 26 – July 2, 2022. (English) Zbl 1519.00023
Summary: Nonlinear dispersive equations are models for nonlinear waves in a wide range of physical contexts. Mathematically they display an interplay between linear dispersion and nonlinear interactions, which can result in a wide range of outcomes from finite time blow-up to solitons and scattering. They are linked to many areas of mathematics and physics, ranging from integrable systems and harmonic analysis to fluid dynamics, geometry and general relativity and probability.
MSC:
| 00B05 | Collections of abstracts of lectures |
| 00B25 | Proceedings of conferences of miscellaneous specific interest |
| 35-06 | Proceedings, conferences, collections, etc. pertaining to partial differential equations |
| 35Qxx | Partial differential equations of mathematical physics and other areas of application |
| 37-06 | Proceedings, conferences, collections, etc. pertaining to dynamical systems and ergodic theory |
| 37Kxx | Dynamical system aspects of infinite-dimensional Hamiltonian and Lagrangian systems |
| 76-06 | Proceedings, conferences, collections, etc. pertaining to fluid mechanics |
| 76-XX | Fluid mechanics |
Software:
FresnelSurfaceReferences:
| [1] | A. Abanov, E. Bettelheim, P. Wiegmann, Integrable hydrodynamics of Calogero-Sutherland model : bidirectional Benjamin-Ono equation, J. Phys. A : Math. Theor. 42 (2009), 135201-135224. · Zbl 1168.37022 |
| [2] | R. Badreddine, The Calogero-Sutherland DNLS equation, PhD thesis, Orsay, in preparation. |
| [3] | P. Gérard, T. Kappeler, On the integrability of the Benjamin-Ono equation on the torus, Comm. Pure Appl. Math 74 (2021), 1674-1747. · Zbl 1471.35354 |
| [4] | P. Gérard, E. Lenzmann, The Calogero-Moser derivative NLS equation, in preparation. |
| [5] | D. Pelinovsky, R. Grimshaw, A spectral transform for the intermediate nonlinear Schrödinger equation, J. Math. Physics 36 (1995), 4203-4219. · Zbl 0845.35119 |
| [6] | R. Sun, Complete integrability of the Benjamin-Ono equation on the multi-soliton manifold, Comm. Math. Phys. 383 (2021), 1051-1092. · Zbl 1465.35355 |
| [7] | L. Farah, J. Holmer, S. Roudenko, and K. Yang, Asymptotic stability of solitary waves of the 3D quadratic Zakharov-Kuznetsov equation, arxiv.2006.00193 [math.AP]. |
| [8] | S. Kinoshita, Global well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equa-tion in 2D, Ann. Inst. H. Poincaré C Anal. Non Linéaire 38 (2021), 451-505. · Zbl 1458.35373 |
| [9] | S. Herr and S. Kinoshita, The Zakharov-Kuznetsov equation in high dimensions: small initial data of critical regularity, J. Evol. Equ. 21 (2021), 2105-2121. · Zbl 1476.35223 |
| [10] | S. Herr and S. Kinoshita, Subcritical well-posedness results for the Zakharov-Kuznetsov equation in dimension three and higher, Ann. Inst. Fourier (accepted), arXiv:2001.09047 [math.AP]. · Zbl 1515.35249 |
| [11] | D. Lannes, F. Linares, J. -C. Saut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, Prog. Nonlinear Differ. Equ. Appl., 84 (2013), 181-213. · Zbl 1273.35263 |
| [12] | V. E. Zakharov, E. A. Kuznetsov, Three-dimensional solitons, Sov. Phys. JETP, 39 (1974), 285-286. |
| [13] | M. Aizenman. Proof of the triviality of ϕ 4 d field theory and some mean-field features of Ising models for d > 4. Phys. Rev. Lett., 47(1):1-4, 1981. |
| [14] | M. Aizenman and H. Duminil-Copin. Marginal triviality of the scaling limits of critical 4D Ising and φ 4 4 models. Ann. of Math. (2), 194(1):163-235, 2021. · Zbl 1489.60150 |
| [15] | J. Bourgain. Periodic nonlinear Schrödinger equation and invariant measures. Comm. Math. Phys., 166(1):1-26, 1994. · Zbl 0822.35126 |
| [16] | J. Bourgain. Invariant measures for the 2D-defocusing nonlinear Schrödinger equation. Comm. Math. Phys., 176(2):421-445, 1996. · Zbl 0852.35131 |
| [17] | J. Bourgain. Nonlinear Schrödinger equations. In Hyperbolic equations and frequency interactions (Park City, UT, 1995), volume 5 of IAS/Park City Math. Ser., pages 3-157. Amer. Math. Soc., Providence, RI, 1999. · Zbl 0952.35127 |
| [18] | G. Da Prato and A. Debussche. Strong solutions to the stochastic quantization equa-tions. Ann. Probab., 31(4):1900-1916, 2003. · Zbl 1071.81070 |
| [19] | J. Froehlich. On the triviality of λϕ 4 d and the approach to the critical point in d (−) > 4 dimensions. Nucl. Phys. B, 200:281-296, 1982. |
| [20] | J. Glimm and A. Jaffe. Positivity of the φ 4 · Zbl 0177.28203 |
| [21] | Hamiltonian. Fortschr. Physik, 21:327-376, 1973. |
| [22] | M. Hairer. A theory of regularity structures. Invent. Math., 198(2):269-504, 2014. · Zbl 1332.60093 |
| [23] | K. Iwata. An infinite-dimensional stochastic differential equation with state space C(R). Probab. Theory Related Fields, 74(1):141-159, 1987. · Zbl 0587.60044 |
| [24] | E. Nelson. A quartic interaction in two dimensions. In Mathematical Theory of Ele-mentary Particles (Proc. Conf., Dedham, Mass., 1965), pages 69-73. M.I.T. Press, Cambridge, Mass., 1966. |
| [25] | T. Tao. Geometric renormalization of large energy wave maps. In Journées “Équations aux Dérivées Partielles”, pages Exp. No. XI, 32.École Polytech., Palaiseau, 2004. · Zbl 1087.58019 |
| [26] | P. E. Zhidkov. An invariant measure for a nonlinear wave equation. Nonlinear Anal., 22(3):319-325, 1994. · Zbl 0803.35098 |
| [27] | G. Chen and F. Pusateri, The nonlinear Schrödinger equation with an L 1 potential, to appear in Analysis & PDE. arXiv:1912.10949. |
| [28] | G. Chen, Long-time dynamics of small solutions to 1d cubic nonlinear Schrödinger equa-tions with a trapping potential, Preprint arXiv:2106.10106. |
| [29] | P. Germain, F. Pusateri and F. Rousset, The nonlinear Schrödinger equation with a poten-tial., Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018),no. 6, 1477-1530. · Zbl 1406.35355 |
| [30] | H. Lindblad, J. Lührmann and A. Soffer, Decay and asymptotics for the 1D Klein-Gordon equation with variable coefficient cubic nonlinearities. Preprint arXiv:1907.09922. · Zbl 1455.35021 |
| [31] | H. Lindblad,, J. Lührmann, W. Schlag, and A. Soffer, On modified scattering for 1D qua-dratic Klein-Gordon equation with non-generic potentials. To appear in IMRN. · Zbl 1517.35196 |
| [32] | J. Lührmann and W. Schlag, Asymptotic stability of the sine-Gordon kink under odd perturbations. Preprint arXiv:2106.09605 . |
| [33] | Bjoern Bringmann, Rowan Killip, and Monica Vişan, Global well-posedness for the fifth-order KdV equation in H −1 (R), Ann. PDE 7 (2021), no. 2, Paper No. 21, 46. · Zbl 1493.35089 |
| [34] | Luc Molinet, A note on ill posedness for the KdV equation, Differential Integral Equations 24 (2011), no. 7-8, 759-765. · Zbl 1249.35292 |
| [35] | , Sharp ill-posedness results for the KdV and mKdV equations on the torus, Adv. Math. 230 (2012), no. 4-6, 1895-1930. · Zbl 1263.35194 |
| [36] | Rowan Killip and Monica Vişan, KdV is well-posed in H −1 , Ann. of Math. (2) 190 (2019), no. 1, 249-305. · Zbl 1426.35203 |
| [37] | H. Bahouri and G. Perelman, Global well-posedness for the derivative nonlinear Schrödinger equation. Preprint, 2020. arXiv:2012.01923. |
| [38] | H. A. Biagioni and F. Linares. Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc., 353(9):3649-3659, 2001. · Zbl 0970.35154 |
| [39] | Z. Guo and Y. Wu. Global well-posedness for the derivative nonlinear Schrödinger equation in H 1 2 (R). Discrete Contin. Dyn. Syst., 37(1):257-264, 2017. · Zbl 1359.35181 |
| [40] | B. Harrop-Griffiths, R. Killip, M. Ntekoume, and M. Visan. Global well-posedness for the derivative nonlinear Schrödinger equation in L 2 (R). Preprint, 2022. arXiv:2204.12548. References |
| [41] | E. Giorgi, S. Klainerman and J. Szeftel, Wave equations estimates and the nonlinear stability of slowly rotating Kerr black holes, arXiv:2205.14808. |
| [42] | S. Klainerman and J. Szeftel, Global Non-Linear Stability of Schwarzschild Spacetime under Polarized Perturbations, Annals of Math Studies, 210. Princeton University Press, Princeton NJ, 2020, xviii+856 pp. · Zbl 1469.83002 |
| [43] | S. Klainerman and J. Szeftel, Construction of GCM spheres in perturbations of Kerr, to appear in Annals of PDE. · Zbl 1501.35394 |
| [44] | S. Klainerman and J. Szeftel, Effective results in uniformization and intrinsic GCM spheres in perturbations of Kerr, to appear in Annals of PDE. · Zbl 1501.35395 |
| [45] | S. Klainerman and J. Szeftel, Kerr stability for small angular momentum, arXiv:2104.11857. · Zbl 07715571 |
| [46] | D. Shen, Construction of GCM hypersurfaces in perturbations of Kerr, arXiv:2205.12336. References · Zbl 1524.35642 |
| [47] | Hajer Bahouri and Patrick Gérard, High frequency approximation of solutions to critical nonlinear wave equations, American Journal of Mathematics 121 (1999), no. 1, 131-175. · Zbl 0919.35089 |
| [48] | Timothy Candy, Minimal non-scattering solutions for the Zakharov system, preprint avail-able at arXiv:2205.08867. · Zbl 1284.35356 |
| [49] | Timothy Candy, Sebastian Herr, and Kenji Nakanishi, Global wellposedness for the energy-critical Zakharov system below the ground state, Advances in Mathematics 384 (2021), 107746. · Zbl 1479.35766 |
| [50] | Timothy Candy, Sebastian Herr, and Kenji Nakanishi, The Zakharov system in dimension d 4, to appear in Journal of the European Mathematical Society (2021). · Zbl 1479.35766 |
| [51] | Zihua Guo and Kenji Nakanishi, The Zakharov system in 4d radial energy space below the ground state, American Journal of Mathematics 143 (2021), no. 5. · Zbl 1512.35420 |
| [52] | Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), no. 3, 645-675. MR 2257393 · Zbl 1115.35125 |
| [53] | Nader Masmoudi and Kenji Nakanishi, Energy convergence for singular limits of Zakharov type systems, Inventiones mathematicae 172 (2008), no. 3, 535-583. · Zbl 1143.35090 |
| [54] | Vladimir E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP 35 (1972), no. 5, 908-914. |
| [55] | Soliton resolution for the radial quadratic wave equation in six space dimensions Charles Collot (joint work with Thomas Duyckaerts, Carlos Kenig and Frank Merle) References |
| [56] | Bahouri, H., & Gérard, P. (1999). High frequency approximation of solutions to critical nonlinear wave equations. American Journal of Mathematics, 121(1), 131-175. · Zbl 0919.35089 |
| [57] | Collot, C., Duyckaerts, T., Kenig, C., & Merle, F. (2022). Soliton resolution for the radial quadratic wave equation in six space dimensions. arXiv preprint arXiv:2201.01848. |
| [58] | Duyckaerts, T., Kenig, C., & Merle, F. (2013). Classification of the radial solutions of the focusing, energy-critical wave equation. Cambridge Journal of Mathematics, 1(1), 75-144. · Zbl 1308.35143 |
| [59] | Duyckaerts, T., Kenig, C. E., & Merle, F. (2019). Soliton resolution for the radial critical wave equation in all odd space dimensions. To appear in Acta Mathematica. |
| [60] | Duyckaerts, T., Kenig, C., Martel, Y., & Merle, F. (2022). Soliton resolution for critical co-rotational wave maps and radial cubic wave equation. Communications in Mathematical Physics, 391(2), 779-871. · Zbl 1491.35292 |
| [61] | Jendrej, J., & Lawrie, A. (2021). Soliton resolution for equivariant wave maps. arXiv preprint arXiv:2106.10738. |
| [62] | Jendrej, J., & Lawrie, A. (2022). Soliton resolution for the energy-critical nonlinear wave equation in the radial case. arXiv preprint arXiv:2203.09614. |
| [63] | Jia, H., & Kenig, C. (2017). Asymptotic decomposition for semilinear wave and equivariant wave map equations. American Journal of Mathematics, 139(6), 1521-1603. · Zbl 1391.35278 |
| [64] | Soffer, A. (2006). Soliton dynamics and scattering. In International congress of mathemati-cians (Vol. 3, pp. 459-471) · Zbl 1111.35054 |
| [65] | Tao, T. (2009). Why are solitons stable?. Bulletin of the American Mathematical Society, 46(1), 1-33. · Zbl 1155.35082 |
| [66] | D. Christodoulou and S. Klainerman, The global nonlinear stability of the Minkowski space, volume 41 of Priceton Mathematical Series, Princeton University Press (1993). · Zbl 0733.35105 |
| [67] | A. Ionescu and B. Pausader, The Einstein-Klein-Gordon Coupled System: Global Stability of the Minkowski Solution, Annals of Mathematics Studies, Princeton Press University (2022). · Zbl 1524.83010 |
| [68] | P. LeFloch and Y. Ma, The global nonlinear stability of Minkowski space for self-gravitating massive fields, Commun. Math. Phys. 346, 603-665 (2016). · Zbl 1359.83003 |
| [69] | H. Lindblad and I. Rodnianski, The global stability of Minkowski space-time in harmonic guage, Ann. of Math. (2), 171(3):1401-1477, (2010) · Zbl 1192.53066 |
| [70] | E. Witten, Instability of the Kaluza-Klein vacuum, Nuclear Physics B 195 (1982) 481-492. · Zbl 0900.53036 |
| [71] | Z. Wyatt, The weak null condition and Kaluza-Klein spacetimes, J. Hyperbolic Diff. Equ. 15(2):210-258 (2018) · Zbl 1394.35500 |
| [72] | R. Schippa, R. Schnaubelt Quasilinear Maxwell equations in two dimensions, accepted to Pure and Applied Analysis (2022+). |
| [73] | R. Mandel, R. Schippa Time-harmonic solutions for Maxwell’s equations in anisotropic media and Bochner-Riesz estimates with negative index for non-elliptic surfaces., Ann. Henri Poincar’e 23 (2022), no. 5, 1831-1880. · Zbl 1489.42015 |
| [74] | R. Schippa Resolvent estimates for time-harmonic Maxwell’s equations in the partially anisotropic case., J. Fourier Anal. Appl. 28 (2022), no. 2, Paper 16, 31 pp. · Zbl 07488199 |
| [75] | R. Mandel, R. Schippa Maple Worksheet for Fresnel surface., https://arxiv.org/src/2103.17176v1/anc/FresnelSurface.mw (2021) |
| [76] | D. Tataru Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation., Amer. J. Math. 122 (2000), no. 2, 349-376. · Zbl 0959.35125 |
| [77] | D. Tataru Strichartz estimates for second order hyperbolic operators with nonsmooth coef-ficients. II. Amer. J. Math. 123 (2001), no. 3, 385-423. · Zbl 0988.35037 |
| [78] | D. Tataru Strichartz estimates for second order hyperbolic operators with nonsmooth coef-ficients. III. J. Amer. Math. Soc. 15 (2002), no. 2, 419-442. · Zbl 0990.35027 |
| [79] | T. Elgindi and I. Jeong Symmetries and critical phenomena in fluidt. Communications on Pure and Applied Mathematics. 73.2 (2020), 257-316. · Zbl 1442.76031 |
| [80] | Boris Khesin, Gerard Misiolek and Alexander Shnirelman, Geometric Hydrodynamics in Open Problems, ArXiv Preprint 2022. · Zbl 1509.35198 |
| [81] | Y. Angelopoulos, S. Aretakis, and D. Gajic. Late-time asymptotics for the wave equation on spherically symmetric, stationary spacetimes. Adv. Math., 323:529-621, 2018. · Zbl 1381.83051 |
| [82] | P. Bizon and A. Rostworowski. A Note about late-time wave tails on a dynamical back-ground. Phys. Rev. D, 81:084047, 2010. |
| [83] | M. Dafermos and I. Rodnianski. A new physical-space approach to decay for the wave equation with applications to black hole spacetimes. In XVIth International Congress on Mathematical Physics, pages 421-432. World Sci. Publ., Hackensack, NJ, 2010. · Zbl 1211.83019 |
| [84] | R. Donninger, W. Schlag, and A. Soffer. A proof of Price’s law on Schwarzschild black hole manifolds for all angular momenta. Adv. Math., 226(1):484-540, 2011. · Zbl 1205.83041 |
| [85] | P. Hintz. A sharp version of price’s law for wave decay on asymptotically flat spacetimes. 04 2020. · Zbl 1484.83012 |
| [86] | S. Klainerman. Uniform decay estimates and the Lorentz invariance of the classical wave equation. Comm. Pure Appl. Math., 38(3):321-332, 1985. · Zbl 0635.35059 |
| [87] | J. Luk and S.-J. Oh. Quantitative decay rates for dispersive solutions to the Einstein-scalar field system in spherical symmetry. Anal. PDE, 8(7):1603-1674, 2015. · Zbl 1326.35385 |
| [88] | J. Luk and S.-J. Oh. Proof of linear instability of the Reissner-Nordström Cauchy horizon under scalar perturbations. Duke Math. J., 166(3):437-493, 2017. · Zbl 1373.35306 |
| [89] | J. Metcalfe, D. Tataru, and M. Tohaneanu. Price’s law on nonstationary space-times. Adv. Math., 230(3):995-1028, 2012. · Zbl 1246.83070 |
| [90] | G. Moschidis. The r p -weighted energy method of Dafermos and Rodnianski in general asymptotically flat spacetimes and applications. Ann. PDE, 2(1):Art. 6, 194, 2016. · Zbl 1397.35031 |
| [91] | J. Oliver and J. Sterbenz. A vector field method for radiating black hole spacetimes. Anal. PDE, 13(1):29-92, 2020. · Zbl 1437.35088 |
| [92] | R. H. Price. Nonspherical perturbations of relativistic gravitational collapse. 1. Scalar and gravitational perturbations. Phys. Rev. D, 5:2419-2438, 1972. |
| [93] | D. Tataru. Local decay of waves on asymptotically flat stationary space-times. Amer. J. Math., 135(2):361-401, 2013. · Zbl 1266.83033 |
| [94] | Mihaela Ifrim and Daniel Tataru Global solutions for 1D cubic defocusing dispersive equa-tions: Part I, arXiv:2205.12212 |
| [95] | Fabrice Planchon and Luis Vega, Bilinear virial identities and applications, Ann. Sci.Éc. Norm. Supér. (4), 42(2):261-290, 2009. · Zbl 1192.35166 |
| [96] | G. Chen, Long-time dynamics of small solutions to 1d cubic nonlinear Schrödinger equa-tions with a trapping potential, Preprint arXiv:2106.10106. |
| [97] | G. Chen, J. Liu, B. Lu, Long-time asymptotics and stability for the sine-Gordon equation, Preprint arXiv:2009.04260. |
| [98] | G. Chen, F. Pusateri, On the 1d cubic NLS with a non-generic potential, Preprint arXiv:2205.01487. |
| [99] | S. Cuccagna, M. Maeda, Asymptotic stability of kink with internal modes under odd per-turbation, Preprint arXiv:2203.13468. · Zbl 1500.35038 |
| [100] | J.-M. Delort, N. Masmoudi, On the stability of kink solutions of the Φ 4 model in 1 + 1 space time dimensions, Preprint hal-02862414. |
| [101] | P. Germain, F. Pusateri, Quadratic Klein-Gordon equations with a potential in one dimen-sion, Preprint arXiv:2006.15688. · Zbl 1495.35126 |
| [102] | P. Germain, F. Pusateri, Z. Zhang, On 1d quadratic Klein-Gordon equations with a potential and symmetries, Preprint arXiv:2202.13273. · Zbl 1508.35120 |
| [103] | M. Kowalczyk, Y. Martel, C. Muñoz, H. Van Den Bosch, A sufficient condition for asymp-totic stability of kinks in general (1 + 1)-scalar field models, Ann. PDE 7 (2021), no. 1, Paper No. 10, 98. · Zbl 1469.35150 |
| [104] | M. Kowalczyk, Y. Martel, C. Muñoz, Kink dynamics in the φ 4 model: asymptotic stability for odd perturbations in the energy space, J. Amer. Math. Soc. 30 (2017), no. 3, 769-798. · Zbl 1387.35419 |
| [105] | M. Kowalczyk, Y. Martel, Kink dynamics under odd perturbations for (1 + 1)-scalar field models with one internal mode, Preprint arXiv:2203.04143. |
| [106] | M. Kowalczyk, Y. Martel, C. Muñoz, Nonexistence of small, odd breathers for a class of nonlinear wave equations, Lett. Math. Phys. 107 (2017), no. 5, 921-931. · Zbl 1384.35109 |
| [107] | M. Kowalczyk, Y. Martel, C. Muñoz, Soliton dynamics for the 1D NLKG equation with symmetry and in the absence of internal modes, to appear in J. Eur. Math. Soc. · Zbl 1486.35122 |
| [108] | T. Léger, F. Pusateri, Internal modes and radiation damping for quadratic Klein-Gordon in 3D, Preprint arXiv:2112.13163. |
| [109] | H. Lindblad, J. Lührmann, W. Schlag, A. Soffer, On modified scattering for 1D quadratic Klein-Gordon equations with non-generic potentials, to appear in Int. Math. Res. Not. · Zbl 1517.35196 |
| [110] | H. Lindblad, J. Lührmann, A. Soffer, Decay and asymptotics for the 1D Klein-Gordon equation with variable coefficient cubic nonlinearities, SIAM J. Math. Anal. 52 (2020), no. 6, 6379-6411. · Zbl 1455.35021 |
| [111] | H. Lindblad, J. Lührmann, A. Soffer, Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities, Arch. Ration. Mech. Anal. 241 (2021), no. 3, 1459-1527. · Zbl 1475.35305 |
| [112] | J. Lührmann, Y. Li, Soliton dynamics for the 1D quadratic Klein-Gordon equation with symmetry, Preprint arXiv:2203.11371. · Zbl 1503.35201 |
| [113] | J. Lührmann, W. Schlag, Asymptotic stability of the sine-Gordon kink under odd perturba-tions, Preprint arXiv:2106.09605. |
| [114] | Y. Martel, Asymptotic stability of solitary waves for the 1D cubic-quintic Schrödinger equa-tion with no internal mode, Preprint arXiv:2110.01492. · Zbl 1510.35310 |
| [115] | G. Pöschl, E. Teller, Bemerkungen zur Quantenmechanik des anharmonischen Oszillators, Zeitschrift für Physik 83 (1933), no. 3-4, 143-151. · JFM 59.1555.02 |
| [116] | C. Collot, T.-E. Ghoul, N. Masmoudi, V.T. Nguyen, Collapsing-ring blowup solutions for the Keller-Segel system in three dimensions and higher, arXiv:2112.15518 · Zbl 1518.35141 |
| [117] | M. A. Herrero, E. Medina, E., J. J. L. Velázquez, Self-similar blow-up for a reaction-diffusion system, Journal of Computational and Applied Mathematics, 97 (1998), 99-119. · Zbl 0934.35066 |
| [118] | T. Senba, Blowup behavior of radial solutions to Jäger-Luckhaus system in high dimensional domains, Funkcialaj Ekvacioj Serio Internacia, 48 (2005), 247-271. · Zbl 1116.35065 |
| [119] | Y. Giga, N. Mizoguchi, T. Senba, Asymptotic behavior of type I blowup solutions to a parabolic-elliptic system of drift-diffusion type, Archive for Rational Mechanics and Analysis, 201 (2011), 549-573. · Zbl 1270.35131 |
| [120] | M.P. Brenner, P. Constantin, L.P. Kadanoff, A. Schenkel, S.C. Venkataramani, Diffusion, attraction and collapse, Nonlinearity, 12 (1999), 549-573. |
| [121] | E.F. Keller and L.A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. · Zbl 1170.92306 |
| [122] | W. Jäger, S. Luckhaus. On explosions of solutions to a system of partial differential equations modelling chemotaxis, Transactions of the American Mathematical Society, 329 (1992), 819-824. · Zbl 0746.35002 |
| [123] | M.A. Herrero, E. Medina, E., J.J.L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), 1739-1754. · Zbl 0909.35071 |
| [124] | Y. Ascasibar, R. Granero-Belinchón, J.M. Moreno, An approximate treatment of gravita-tional collapse, Physica D: Nonlinear Phenomena, 262 (2013), 71-82. · Zbl 1434.35241 |
| [125] | Boris Ettinger. Well-posedness of the three-form field equation and the minimal surface equation in Minkowski space. ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)-University of California, Berkeley. |
| [126] | J. Ginibre and G. Velo. Generalized Strichartz inequalities for the wave equation. J. Funct. Anal., 133(1):50-68, 1995. · Zbl 0849.35064 |
| [127] | Lars Hörmander. Lectures on nonlinear hyperbolic differential equations, volume 26 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. · Zbl 0881.35001 |
| [128] | Springer-Verlag, Berlin, 1997. |
| [129] | Thomas J. R. Hughes, Tosio Kato, and Jerrold E. Marsden. Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativ-ity. Arch. Rational Mech. Anal., 63(3):273-294 (1977), 1976. · Zbl 0361.35046 |
| [130] | L. V. Kapitanskiȋ. Estimates for norms in Besov and Lizorkin-Triebel spaces for solutions of second-order linear hyperbolic equations. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 171(Kraev. Zadachi Mat. Fiz. i Smezh. Voprosy Teor. Funktsiȋ. 20):106-162, 185-186, 1989. · Zbl 0725.35022 |
| [131] | Markus Keel and Terence Tao. Endpoint Strichartz estimates. Amer. J. Math., 120(5):955-980, 1998. · Zbl 0922.35028 |
| [132] | Sergiu Klainerman, Igor Rodnianski, and Jeremie Szeftel. The bounded L 2 curvature con-jecture. Invent. Math., 202(1):91-216, 2015. · Zbl 1330.53089 |
| [133] | Hans Lindblad. Counterexamples to local existence for quasilinear wave equations. Math. Res. Lett., 5(5):605-622, 1998. · Zbl 0932.35149 |
| [134] | Gerd Mockenhaupt, Andreas Seeger, and Christopher D. Sogge. Local smoothing of Fourier integral operators and Carleson-Sjölin estimates. J. Amer. Math. Soc., 6(1):65-130, 1993. · Zbl 0776.58037 |
| [135] | S. Klainerman. The null condition and global existence to nonlinear wave equations. In Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984), volume 23 of Lectures in Appl. Math., pages 293-326. Amer. Math. Soc., Providence, RI, 1986. · Zbl 0599.35105 |
| [136] | Gaspard Ohlmann. Ill-posedness of a quasilinear wave equation in two dimensions for data in H 7/4 . arXiv e-prints, page arXiv:2107.03732, July 2021. · Zbl 1467.05009 |
| [137] | Hart F. Smith. A parametrix construction for wave equations with C 1,1 coefficients. Ann. Inst. Fourier (Grenoble), 48(3):797-835, 1998. · Zbl 0974.35068 |
| [138] | Hart F. Smith and Christopher D. Sogge. On Strichartz and eigenfunction estimates for low regularity metrics. Math. Res. Lett., 1(6):729-737, 1994. · Zbl 0832.35018 |
| [139] | Hart F. Smith and Daniel Tataru. Sharp counterexamples for Strichartz estimates for low regularity metrics. Math. Res. Lett., 9(2-3):199-204, 2002. · Zbl 1003.35075 |
| [140] | Hart F. Smith and Daniel Tataru. Sharp local well-posedness results for the nonlinear wave equation. Ann. of Math. (2), 162(1):291-366, 2005. · Zbl 1098.35113 |
| [141] | Christopher D. Sogge. Lectures on non-linear wave equations. International Press, Boston, MA, second edition, 2008. · Zbl 1165.35001 |
| [142] | Daniel Tataru. Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. II. Amer. J. Math., 123(3):385-423, 2001. · Zbl 0988.35037 |
| [143] | Daniel Tataru. Nonlinear wave equations. In Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), pages 209-220. Higher Ed. Press, Beijing, 2002. References · Zbl 1136.35417 |
| [144] | B. Dodson, A determination of blowup solutions to the focusing, quintic NLS with mass equal to the mass of the soliton, Arxiv 2104.11690 (2021). |
| [145] | B. Dodson, A determination of blowup solutions to the focusing NLS with mass equal to the mass of the soliton, Arxiv 2106.02723 (2021). |
| [146] | R. Killip, D. Li, M. Visan, X. Zhang, Characterization of minimal-mass blowup solutions to the focusing, mass-critical NLS, SIAM Jpurnal on mathematical analysis 41.1 (2009) 219-236. · Zbl 1184.35293 |
| [147] | F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Mathematical Journal 69.2 (1993) 427-454. · Zbl 0808.35141 |
| [148] | Lars Hörmander. The lifespan of classical solutions of non-linear hyperbolic equations. In Pseudo-Differential Operators, pages 214-280. Springer, 1987. · Zbl 0632.35045 |
| [149] | Lars Hörmander. On the fully nonlinear Cauchy problem with small data. II. In Microlocal analysis and nonlinear waves (Minneapolis, MN, 1988-1989), volume 30 of IMA Vol. Math. Appl., pages 51-81. Springer, New York, 1991. · Zbl 0783.35036 |
| [150] | Lars Hörmander. Lectures on nonlinear hyperbolic differential equations, volume 26. Springer Science & Business Media, 1997. · Zbl 0881.35001 |
| [151] | Fritz John. Blow-up for quasi-linear wave equations in three space dimensions. Communi-cations on Pure and Applied Mathematics, 34(1):29-51, 1981. · Zbl 0453.35060 |
| [152] | Fritz John. Blow-up of radial solutions of utt = c 2 (ut)∆u in three space dimensions. Matemática Aplicada e Computacional, 4(1):3-18, 1985. · Zbl 0597.35082 |
| [153] | Jared Speck. Shock formation in small-data solutions to 3D quasilinear wave equations, volume 214 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2016. · Zbl 1373.35005 |
| [154] | Dongxiao Yu. Modified scattering for a scalar quasilinear wave equation satisfying the weak null condition. ProQuest LLC, Ann Arbor, MI, 2021. Thesis (Ph.D.)-University of Califor-nia, Berkeley. · Zbl 1461.35159 |
| [155] | Dongxiao Yu. Modified wave operators for a scalar quasilinear wave equation satisfying the weak null condition. Communications in Mathematical Physics, 382(3):1961-2013, 2021. · Zbl 1461.35159 |
| [156] | Dongxiao Yu. Nontrivial global solutions to some quasilinear wave equations in three space dimensions, 2022. |
| [157] | I. Egorova, Z. Gladka, V. Kotlyarov, and G. Teschl,, Long-time asymptotics for the Korteweg-de Vries equation with step-like initial data, Nonlinearity 26 (2013), no. 7, 1839-1864. · Zbl 1320.35308 |
| [158] | R. Killip and M. Vişan, KdV is well-posed in H −1 , Ann. of Math. (2) 190 (2019), no. 1, 249-305. · Zbl 1426.35203 |
| [159] | T. Laurens, Global well-posedness for H −1 (R) perturbations of KdV with exotic spatial asymptotics, Preprint arXiv:2104.11346, to appear in Comm. Math. Phys. · Zbl 1509.35266 |
| [160] | T. Laurens, KdV on an incoming tide, Nonlinearity 35 (2022), no. 1, 343-387. References · Zbl 1479.35737 |
| [161] | A. Sá Barreto, G. Uhlmann, and Y. Wang, Inverse Scattering for Critical Semilinear Wave Equations. Preprint arXiv:2003.03822. · Zbl 1500.35220 |
| [162] | R. Carles and I. Gallagher, Analyticity of the scattering operator for semilinear dispersive equations. Comm. Math. Phys. 286 (2009), no. 3, 1181-1209. · Zbl 1173.35677 |
| [163] | R. Killip, J. Murphy, and M. Visan, The scattering map determines the nonlinearity. Preprint arXiv:2207.02414. · Zbl 1512.35542 |
| [164] | P. D. Lax, Translation invariant spaces. Acta Math. 101 (1959), 163-178. · Zbl 0085.09102 |
| [165] | C. S. Morawetz and W. A. Strauss, On a nonlinear scattering operator. Comm. Pure Appl. Math. 26 (1973), 47-54. · Zbl 0265.35057 |
| [166] | H. Sasaki, Inverse scattering problems for the Hartree equation whose interaction potential decays rapidly. J. Differential Equations 252 (2012), no. 2, 2004-2023. · Zbl 1230.35131 |
| [167] | M. Watanabe, Time-dependent method for non-linear Schrödinger equations in inverse scat-tering problems. J. Math. Anal. Appl. 459 (2018), no. 2, 932-944. · Zbl 1382.35280 |
| [168] | R. Weder, Inverse scattering for the nonlinear Schrödinger equation. II. Reconstruction of the potential and the nonlinearity in the multidimensional case. Proc. Amer. Math. Soc. 129 (2001), no. 12, 3637-3645. · Zbl 0986.35125 |
| [169] | C. Bardos and P. Degond. Global existence for the Vlasov-Poisson equation in 3 space vari-ables with small initial data. Annales de l’Institut Henri Poincaré. Analyse Non Linéaire, 2(2):101-118, 1985. · Zbl 0593.35076 |
| [170] | S. Caprino and C. Marchioro. On the plasma-charge model. Kinetic and Related Models, 3(2):241-254, 2010. · Zbl 1193.82045 |
| [171] | G. Crippa, S. Ligabue, and C. Saffirio. Lagrangian solutions to the Vlasov-Poisson system with a point charge. Kinetic and Related Models, 11(6):1277-1299, 2018. · Zbl 1405.35215 |
| [172] | L. Desvillettes, E. Miot, and C. Saffirio. Polynomial propagation of moments and global existence for a Vlasov-Poisson system with a point charge. Annales de l’Institut Henri Poincaré. Analyse Non Linéaire, 32(2):373-400, 2015. · Zbl 1323.35178 |
| [173] | P. Flynn, Z. Ouyang, B. Pausader, and K. Widmayer. Scattering map for the Vlasov-Poisson system. Peking Mathematical Journal, to appear (arXiv preprint 2101.01390), 2021. |
| [174] | R. M. Höfer and R. Winter. A fast point charge interacting with the screened Vlasov-Poisson system. arXiv preprint 2205.00035, Apr. 2022. |
| [175] | A. Ionescu, B. Pausader, X. Wang, and K. Widmayer. Nonlinear Landau damping for the Vlasov-Poisson system in R 3 : the Poisson equilibrium. arXiv preprint 2205.04540, May 2022. |
| [176] | A. D. Ionescu, B. Pausader, X. Wang, and K. Widmayer. On the Asymptotic Behavior of Solutions to the Vlasov-Poisson System. International Mathematics Research Notices. IMRN, (12):8865-8889, 2022. · Zbl 1491.35082 |
| [177] | D. Li and X. Zhang. Asymptotic growth bounds for the 3-D Vlasov-Poisson system with point charges. Mathematical Methods in the Applied Sciences, 41(9):3294-3306, 2018. · Zbl 1393.35246 |
| [178] | P.-L. Lions and B. Perthame. Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system. Inventiones Mathematicae, 105(2):415-430, 1991. · Zbl 0741.35061 |
| [179] | C. Marchioro, E. Miot, and M. Pulvirenti. The Cauchy problem for the 3-D Vlasov-Poisson system with point charges. Archive for Rational Mechanics and Analysis, 201(1):1-26, 2011. · Zbl 1321.76081 |
| [180] | S. Pankavich. Exact large time behavior of spherically symmetric plasmas. SIAM Journal on Mathematical Analysis, 53(4):4474-4512, 2021. · Zbl 1476.35275 |
| [181] | B. Pausader and K. Widmayer. Stability of a point charge for the Vlasov-Poisson system: the radial case. Communications in Mathematical Physics, 385(3):1741-1769, 2021. · Zbl 1475.35343 |
| [182] | B. Pausader, K. Widmayer, and J. Yang. Stability of a point charge for the repulsive Vlasov-Poisson system. arXiv preprint 2207.05644, July 2022. |
| [183] | H. Koch and X. Liao, Conserved energies for the one dimensional Gross-Pitaevskii equation, Adv. Math., 377 (2021), Paper No. 107467, 83. · Zbl 1455.35236 |
| [184] | H. Koch and X. Liao, Conserved energies for the one dimensional Gross-Pitaevskii equation: low regularity case, https://arxiv.org/abs/2204.06293, (2022). |
| [185] | S. Agrawal and T. Alazard. Refined Rellich boundary inequalities for the derivatives of a harmonic function. arXiv:2205.04756 · Zbl 1526.35014 |
| [186] | A. R. Brodsky. On the asymptotic behavior of solutions of the wave equations. Proc. Amer. Math. Soc., 18:207-208, 1967. · Zbl 0149.06704 |
| [187] | Robert T. Glassey and Walter A. Strauss. Decay of a Yang-Mills field coupled to a scalar field. Comm. Math. Phys., 67(1):51-67, 1979. · Zbl 0425.35085 |
| [188] | Peter D. Lax and Ralph S. Phillips. Scattering theory, volume 26 of Pure and Applied Mathematics. Academic Press, Inc., Boston, MA, second edition, 1989. With appendices by Cathleen S. Morawetz and Georg Schmidt. · Zbl 0697.35004 |
| [189] | Lord Rayleigh. Hydrodynamical notes. The London, Edinburgh, and Dublin Philosophi-cal Magazine and Journal of Science, 21(122):177-195, 1911. (Also in Rayleigh’s Scientific Papers, 6, 1920: 11-14). |
| [190] | Robert S. Strichartz. Asymptotic behavior of waves. J. Functional Analysis, 40(3):341-357, 1981. · Zbl 0484.35070 |
| [191] | Vladimir E. Zakharov. Stability of periodic waves of finite amplitude on the surface of a deep fluid. Journal of Applied Mechanics and Technical Physics, 9(2):190-194, 1968. Reporter: Dongxiao Yu |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
