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The nonlinear fractional relativistic Schrödinger equation: Existence, multiplicity, decay and concentration results
Dispersive estimates for the wave and the Klein-Gordon equations in large time inside the Friedlander domain
Sorbonne Université, CNRS, LJLL, F-75005 Paris, France |
We prove global in time dispersion for the wave and the Klein-Gordon equation inside the Friedlander domain by taking full advantage of the space-time localization of caustics and a precise estimate of the number of waves that may cross at a given, large time. Moreover, we uncover a significant difference between Klein-Gordon and the wave equation in the low frequency, large time regime, where Klein-Gordon exhibits a worse decay than the wave, unlike in the flat space.
References:
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Strichartz estimates for the wave equation on manifolds with boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1817-1829.
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Parametrix and propagation of singularities for the interior mixed hyperbolic problem, J. Analyse Math., 32 (1977), 17-62.
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O. Ivanovici, G. Lebeau and F. Planchon, Strichartz estimates for the wave equation inside strictly convex 2d model domain, preprint, https://arXiv.org/abs/2008.03598. |
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O. Ivanovici, G. Lebeau and F. Planchon,
Dispersion for the wave equation inside strictly convex domains I: the Friedlander model case, Ann. of Math. (2), 180 (2014), 323-380.
doi: 10.4007/annals.2014.180.1.7. |
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O. Ivanovici, G. Lebeau and F. Planchon, New counterexamples to Strichartz estimates for the wave equation on a 2D model convex domain, to appear in J. Ec. polytech. Math., https://arXiv.org/abs/2008.02716. |
| [7] |
J. Kato and T. Ozawa,
Endpoint strichartz estimates for the Klein-Gordon equation in two space dimensions and some applications, J. Math. Pures Appl., 95 (2011), 48-71.
doi: 10.1016/j.matpur.2010.10.001. |
| [8] |
S. Machihara, K. Nakanishi and T. Ozawa,
Small global solutions and the nonrelativistic limit for the nonlinear dirac equation, Rev. Mat. Iberoamericana, 19 (2003), 179-194.
doi: 10.4171/RMI/342. |
| [9] |
R. B. Melrose,
Equivalence of glancing hypersurfaces, Invent. Math., 37 (1976), 165-191.
doi: 10.1007/BF01390317. |
| [10] |
R. B. Melrose and J. Sjöstrand,
Singularities of boundary value problems. I, Comm. Pure Appl. Math., 31 (1978), 593-617.
doi: 10.1002/cpa.3160310504. |
| [11] |
R. B. Melrose and J. Sjöstrand,
Singularities of boundary value problems. II, Comm. Pure Appl. Math., 35 (1982), 129-168.
doi: 10.1002/cpa.3160350202. |
| [12] |
R. B. Melrose and M. E. Taylor, Boundary Problems for Wave Equations With Grazing and Gliding Rays, Available at https://www.unc.edu/math/Faculty/met/wavep.html. |
| [13] |
R. B. Melrose and M. E. Taylor,
The radiation pattern of a diffracted wave near the shadow boundary, Comm. Partial Differential Equations, 11 (1986), 599-672.
doi: 10.1080/03605308608820439. |
| [14] |
O. Vallée and M. Soares, Airy Functions and Applications to Physics, Imperial College Press, London; Distributed by World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2004.
doi: 10.1142/p345. |
show all references
References:
| [1] |
M. D. Blair, H. F. Smith and Ch. D. Sogge,
Strichartz estimates for the wave equation on manifolds with boundary, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1817-1829.
doi: 10.1016/j.anihpc.2008.12.004. |
| [2] |
G. Eskin,
Parametrix and propagation of singularities for the interior mixed hyperbolic problem, J. Analyse Math., 32 (1977), 17-62.
doi: 10.1007/BF02803574. |
| [3] |
O. Ivanovici, R. Lascar, G. Lebeau and F. Planchon., Dispersion for the wave equation inside strictly convex domains II: The general case, preprint, https://arXiv.org/abs/1605.08800. |
| [4] |
O. Ivanovici, G. Lebeau and F. Planchon, Strichartz estimates for the wave equation inside strictly convex 2d model domain, preprint, https://arXiv.org/abs/2008.03598. |
| [5] |
O. Ivanovici, G. Lebeau and F. Planchon,
Dispersion for the wave equation inside strictly convex domains I: the Friedlander model case, Ann. of Math. (2), 180 (2014), 323-380.
doi: 10.4007/annals.2014.180.1.7. |
| [6] |
O. Ivanovici, G. Lebeau and F. Planchon, New counterexamples to Strichartz estimates for the wave equation on a 2D model convex domain, to appear in J. Ec. polytech. Math., https://arXiv.org/abs/2008.02716. |
| [7] |
J. Kato and T. Ozawa,
Endpoint strichartz estimates for the Klein-Gordon equation in two space dimensions and some applications, J. Math. Pures Appl., 95 (2011), 48-71.
doi: 10.1016/j.matpur.2010.10.001. |
| [8] |
S. Machihara, K. Nakanishi and T. Ozawa,
Small global solutions and the nonrelativistic limit for the nonlinear dirac equation, Rev. Mat. Iberoamericana, 19 (2003), 179-194.
doi: 10.4171/RMI/342. |
| [9] |
R. B. Melrose,
Equivalence of glancing hypersurfaces, Invent. Math., 37 (1976), 165-191.
doi: 10.1007/BF01390317. |
| [10] |
R. B. Melrose and J. Sjöstrand,
Singularities of boundary value problems. I, Comm. Pure Appl. Math., 31 (1978), 593-617.
doi: 10.1002/cpa.3160310504. |
| [11] |
R. B. Melrose and J. Sjöstrand,
Singularities of boundary value problems. II, Comm. Pure Appl. Math., 35 (1982), 129-168.
doi: 10.1002/cpa.3160350202. |
| [12] |
R. B. Melrose and M. E. Taylor, Boundary Problems for Wave Equations With Grazing and Gliding Rays, Available at https://www.unc.edu/math/Faculty/met/wavep.html. |
| [13] |
R. B. Melrose and M. E. Taylor,
The radiation pattern of a diffracted wave near the shadow boundary, Comm. Partial Differential Equations, 11 (1986), 599-672.
doi: 10.1080/03605308608820439. |
| [14] |
O. Vallée and M. Soares, Airy Functions and Applications to Physics, Imperial College Press, London; Distributed by World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2004.
doi: 10.1142/p345. |
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