Uniqueness of bounded solutions for the homogeneous relativistic Landau equation with Coulomb interactions
Authors:
Robert M. Strain and Zhenfu Wang
Journal:
Quart. Appl. Math. 78 (2020), 107-145
MSC (2010):
Primary 82D10, 35Q70, 35Q75, 35B45, 35A02.
DOI:
https://doi.org/10.1090/qam/1545
Published electronically:
July 8, 2019
MathSciNet review:
4042221
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Abstract: We prove the uniqueness of weak solutions to the spatially homogeneous special relativistic Landau equation under the conditional assumption that the solution satisfies $(p^0)^7 F(t,p) \in L^1 ([0,T]; L^\infty )$. The existence of standard weak solutions to the relativistic Landau equation has been shown recently in [J. Funct. Anal. 277 (2019), pp. 1139–1201].
References
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- S. T. Belyaev and G. I. Budker, Relativistic kinetic equation, Dokl. Akad. Nauk SSSR (N.S.) 107 (1956), 807–810 (Russian). MR 0083886
- S. T. Belyaev and G. I. Budker, Boltzmann’s equation for an electron gas in which collisions are infrequent, Plasma Physics and the problem of controlled thermonuclear reactions (M.A. Leontovich, ed.), Pergamon Press, New York, 1961, p. 431.
- Abhay G. Bhatt and Rajeeva L. Karandikar, Invariant measures and evolution equations for Markov processes characterized via martingale problems, Ann. Probab. 21 (1993), no. 4, 2246–2268. MR 1245309
- Alexander Bobylev, Irene M. Gamba, and Chenglong Zhang, On the rate of relaxation for the Landau kinetic equation and related models, J. Stat. Phys. 168 (2017), no. 3, 535–548. MR 3670754, DOI https://doi.org/10.1007/s10955-017-1814-y
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- Kleber Carrapatoso, Laurent Desvillettes, and Lingbing He, Estimates for the large time behavior of the Landau equation in the Coulomb case, Arch. Ration. Mech. Anal. 224 (2017), no. 2, 381–420. MR 3614751, DOI https://doi.org/10.1007/s00205-017-1078-3
- L. Desvillettes, Entropy dissipation estimates for the Landau equation in the Coulomb case and applications, J. Funct. Anal. 269 (2015), no. 5, 1359–1403. MR 3369941, DOI https://doi.org/10.1016/j.jfa.2015.05.009
- Laurent Desvillettes, Entropy dissipation estimates for the Landau equation: general cross sections, From particle systems to partial differential equations. III, Springer Proc. Math. Stat., vol. 162, Springer, [Cham], 2016, pp. 121–143. MR 3557719, DOI https://doi.org/10.1007/978-3-319-32144-8_6
- Laurent Desvillettes and Cédric Villani, On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness, Comm. Partial Differential Equations 25 (2000), no. 1-2, 179–259. MR 1737547, DOI https://doi.org/10.1080/03605300008821512
- Laurent Desvillettes and Cédric Villani, On the spatially homogeneous Landau equation for hard potentials. II. $H$-theorem and applications, Comm. Partial Differential Equations 25 (2000), no. 1-2, 261–298. MR 1737548, DOI https://doi.org/10.1080/03605300008821513
- François Golse, Cyril Imbert, Clément Mouhot, and Alexis F. Vasseur, Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19 (2019), no. 1, 253–295. MR 3923847
- Nicolas Fournier, Uniqueness of bounded solutions for the homogeneous Landau equation with a Coulomb potential, Comm. Math. Phys. 299 (2010), no. 3, 765–782. MR 2718931, DOI https://doi.org/10.1007/s00220-010-1113-9
- Nicolas Fournier and Hélène Guérin, On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity, J. Stat. Phys. 131 (2008), no. 4, 749–781. MR 2398952, DOI https://doi.org/10.1007/s10955-008-9511-5
- Nicolas Fournier and Hélène Guérin, Well-posedness of the spatially homogeneous Landau equation for soft potentials, J. Funct. Anal. 256 (2009), no. 8, 2542–2560. MR 2502525, DOI https://doi.org/10.1016/j.jfa.2008.11.008
- Tadahisa Funaki, The diffusion approximation of the spatially homogeneous Boltzmann equation, Duke Math. J. 52 (1985), no. 1, 1–23. MR 791288, DOI https://doi.org/10.1215/S0012-7094-85-05201-9
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- Robert T. Glassey and Walter A. Strauss, On the derivatives of the collision map of relativistic particles, Transport Theory Statist. Phys. 20 (1991), no. 1, 55–68. MR 1105532, DOI https://doi.org/10.1080/00411459108204708
- Robert T. Glassey and Walter A. Strauss, Asymptotic stability of the relativistic Maxwellian, Publ. Res. Inst. Math. Sci. 29 (1993), no. 2, 301–347. MR 1211782, DOI https://doi.org/10.2977/prims/1195167275
- Robert T. Glassey and Walter A. Strauss, The relativistic Boltzmann equation near equilibrium, Spectral and scattering theory and applications, Adv. Stud. Pure Math., vol. 23, Math. Soc. Japan, Tokyo, 1994, pp. 105–111. MR 1275397, DOI https://doi.org/10.2969/aspm/02310105
- Maria Pia Gualdani and Nestor Guillen, Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential, Anal. PDE 9 (2016), no. 8, 1772–1809. MR 3599518, DOI https://doi.org/10.2140/apde.2016.9.1772
- Hélène Guérin, Solving Landau equation for some soft potentials through a probabilistic approach, Ann. Appl. Probab. 13 (2003), no. 2, 515–539. MR 1970275, DOI https://doi.org/10.1214/aoap/1050689592
- Yan Guo, The Landau equation in a periodic box, Comm. Math. Phys. 231 (2002), no. 3, 391–434. MR 1946444, DOI https://doi.org/10.1007/s00220-002-0729-9
- Yan Guo, The Vlasov-Poisson-Landau system in a periodic box, J. Amer. Math. Soc. 25 (2012), no. 3, 759–812. MR 2904573, DOI https://doi.org/10.1090/S0894-0347-2011-00722-4
- Fred L. Hinton, Collisional transport in plasma, Handbook of Plasma Physics, Volume I: Basic Plasma Physics I (M.N. Rosenbluth and R.Z. Sagdeev, eds.), North-Holland Publishing Company, Amsterdam, 1983, pp. 147–197.
- J. Horowitz and R. L. Karandikar, Martingale problems associated with the Boltzmann equation, Seminar on Stochastic Processes, 1989 (San Diego, CA, 1989) Progr. Probab., vol. 18, Birkhäuser Boston, Boston, MA, 1990, pp. 75–122. MR 1042343
- Ling Hsiao and Hongjun Yu, Global classical solutions to the initial value problem for the relativistic Landau equation, J. Differential Equations 228 (2006), no. 2, 641–660. MR 2289548, DOI https://doi.org/10.1016/j.jde.2005.10.022
- M. Lemou, Linearized quantum and relativistic Fokker-Planck-Landau equations, Math. Methods Appl. Sci. 23 (2000), no. 12, 1093–1119. MR 1773932, DOI https://doi.org/10.1002/1099-1476%28200008%2923%3A12%3C1093%3A%3AAID-MMA153%3E3.0.CO%3B2-8
- E. M. Lifshitz and L. P. Pitaevskiĭ, Course of theoretical physics [”Landau-Lifshits“]. Vol. 10, Pergamon International Library of Science, Technology, Engineering and Social Studies, Pergamon Press, Oxford-Elmsford, N.Y., 1981. Translated from the Russian by J. B. Sykes and R. N. Franklin. MR 684990
- Luis Silvestre, Upper bounds for parabolic equations and the Landau equation, J. Differential Equations 262 (2017), no. 3, 3034–3055. MR 3582250, DOI https://doi.org/10.1016/j.jde.2016.11.010
- Robert M. Strain and Yan Guo, Stability of the relativistic Maxwellian in a collisional plasma, Comm. Math. Phys. 251 (2004), no. 2, 263–320. MR 2100057, DOI https://doi.org/10.1007/s00220-004-1151-2
- Robert M. Strain and Yan Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations 31 (2006), no. 1-3, 417–429. MR 2209761, DOI https://doi.org/10.1080/03605300500361545
- Robert M. Strain and Yan Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal. 187 (2008), no. 2, 287–339. MR 2366140, DOI https://doi.org/10.1007/s00205-007-0067-3
- Robert M. Strain and Maja Tasković, Entropy dissipation estimates for the relativistic Landau equation, and applications, J. Funct. Anal. 277 (2019), no. 4, 1139–1201. MR 3959729, DOI https://doi.org/10.1016/j.jfa.2019.04.007
- Robert M. Strain and Keya Zhu, The Vlasov-Poisson-Landau system in $\Bbb {R}^3_x$, Arch. Ration. Mech. Anal. 210 (2013), no. 2, 615–671. MR 3101794, DOI https://doi.org/10.1007/s00205-013-0658-0
- Walter A. Strauss, The relativistic Boltzmann equation, Quantization, nonlinear partial differential equations, and operator algebra (Cambridge, MA, 1994) Proc. Sympos. Pure Math., vol. 59, Amer. Math. Soc., Providence, RI, 1996, pp. 203–209. MR 1392991, DOI https://doi.org/10.1090/pspum/059/1392991
- Hiroshi Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. Verw. Gebiete 46 (1978/79), no. 1, 67–105. MR 512334, DOI https://doi.org/10.1007/BF00535689
- Cédric Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal. 143 (1998), no. 3, 273–307. MR 1650006, DOI https://doi.org/10.1007/s002050050106
- Cédric Villani, Optimal transport, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. Old and new. MR 2459454
- John B. Walsh, An introduction to stochastic partial differential equations, École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 265–439. MR 876085, DOI https://doi.org/10.1007/BFb0074920
- Kung-Chien Wu, Global in time estimates for the spatially homogeneous Landau equation with soft potentials, J. Funct. Anal. 266 (2014), no. 5, 3134–3155. MR 3158719, DOI https://doi.org/10.1016/j.jfa.2013.11.005
- Tong Yang and Hongjun Yu, Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space, J. Differential Equations 248 (2010), no. 6, 1518–1560. MR 2593052, DOI https://doi.org/10.1016/j.jde.2009.11.027
- Tong Yang and Hongjun Yu, Global solutions to the relativistic Landau-Maxwell system in the whole space, J. Math. Pures Appl. (9) 97 (2012), no. 6, 602–634 (English, with English and French summaries). MR 2921603, DOI https://doi.org/10.1016/j.matpur.2011.09.006
- Hongjun Yu, Smoothing effects for classical solutions of the relativistic Landau-Maxwell system, J. Differential Equations 246 (2009), no. 10, 3776–3817. MR 2514726, DOI https://doi.org/10.1016/j.jde.2009.02.021
References
- Radjesvarane Alexandre, Jie Liao, and Chunjin Lin, Some a priori estimates for the homogeneous Landau equation with soft potentials, Kinet. Relat. Models 8 (2015), no. 4, 617–650. MR 3375485, DOI https://doi.org/10.3934/krm.2015.8.617
- A. A. Arsen′ev and N. V. Peskov, The existence of a generalized solution of Landau’s equation, Ž. Vyčisl. Mat. i Mat. Fiz. 17 (1977), no. 4, 1063–1068, 1096 (Russian). MR 0470442
- S. T. Belyaev and G. I. Budker, Relativistic kinetic equation, Dokl. Akad. Nauk SSSR (N.S.) 107 (1956), 807–810 (Russian). MR 0083886
- S. T. Belyaev and G. I. Budker, Boltzmann’s equation for an electron gas in which collisions are infrequent, Plasma Physics and the problem of controlled thermonuclear reactions (M.A. Leontovich, ed.), Pergamon Press, New York, 1961, p. 431.
- Abhay G. Bhatt and Rajeeva L. Karandikar, Invariant measures and evolution equations for Markov processes characterized via martingale problems, Ann. Probab. 21 (1993), no. 4, 2246–2268. MR 1245309
- Alexander Bobylev, Irene M. Gamba, and Chenglong Zhang, On the rate of relaxation for the Landau kinetic equation and related models, J. Stat. Phys. 168 (2017), no. 3, 535–548. MR 3670754, DOI https://doi.org/10.1007/s10955-017-1814-y
- K. Carrapatoso and S. Mischler, Landau equation for very soft and Coulomb potentials near Maxwellians, Ann. PDE 3 (2017), no. 1, Art. 1, 65. MR 3625186, DOI https://doi.org/10.1007/s40818-017-0021-0
- Kleber Carrapatoso, Laurent Desvillettes, and Lingbing He, Estimates for the large time behavior of the Landau equation in the Coulomb case, Arch. Ration. Mech. Anal. 224 (2017), no. 2, 381–420. MR 3614751, DOI https://doi.org/10.1007/s00205-017-1078-3
- L. Desvillettes, Entropy dissipation estimates for the Landau equation in the Coulomb case and applications, J. Funct. Anal. 269 (2015), no. 5, 1359–1403. MR 3369941, DOI https://doi.org/10.1016/j.jfa.2015.05.009
- Laurent Desvillettes, Entropy dissipation estimates for the Landau equation: general cross sections, From particle systems to partial differential equations. III, Springer Proc. Math. Stat., vol. 162, Springer, [Cham], 2016, pp. 121–143. MR 3557719, DOI https://doi.org/10.1007/978-3-319-32144-8_6
- Laurent Desvillettes and Cédric Villani, On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness, Comm. Partial Differential Equations 25 (2000), no. 1-2, 179–259. MR 1737547, DOI https://doi.org/10.1080/03605300008821512
- Laurent Desvillettes and Cédric Villani, On the spatially homogeneous Landau equation for hard potentials. II. $H$-theorem and applications, Comm. Partial Differential Equations 25 (2000), no. 1-2, 261–298. MR 1737548, DOI https://doi.org/10.1080/03605300008821513
- François Golse, Cyril Imbert, Clément Mouhot, and Alexis F. Vasseur, Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19 (2019), no. 1, 253–295. MR 3923847
- Nicolas Fournier, Uniqueness of bounded solutions for the homogeneous Landau equation with a Coulomb potential, Comm. Math. Phys. 299 (2010), no. 3, 765–782. MR 2718931, DOI https://doi.org/10.1007/s00220-010-1113-9
- Nicolas Fournier and Hélène Guérin, On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity, J. Stat. Phys. 131 (2008), no. 4, 749–781. MR 2398952, DOI https://doi.org/10.1007/s10955-008-9511-5
- Nicolas Fournier and Hélène Guérin, Well-posedness of the spatially homogeneous Landau equation for soft potentials, J. Funct. Anal. 256 (2009), no. 8, 2542–2560. MR 2502525, DOI https://doi.org/10.1016/j.jfa.2008.11.008
- Tadahisa Funaki, The diffusion approximation of the spatially homogeneous Boltzmann equation, Duke Math. J. 52 (1985), no. 1, 1–23. MR 791288, DOI https://doi.org/10.1215/S0012-7094-85-05201-9
- R. T. Glassey and W. A. Strauss, Asymptotic stability of the relativistic Maxwellian via fourteen moments, Transport Theory Statist. Phys. 24 (1995), no. 4-5, 657–678. MR 1321370, DOI https://doi.org/10.1080/00411459508206020
- Robert T. Glassey and Walter A. Strauss, On the derivatives of the collision map of relativistic particles, Transport Theory Statist. Phys. 20 (1991), no. 1, 55–68. MR 1105532, DOI https://doi.org/10.1080/00411459108204708
- Robert T. Glassey and Walter A. Strauss, Asymptotic stability of the relativistic Maxwellian, Publ. Res. Inst. Math. Sci. 29 (1993), no. 2, 301–347. MR 1211782, DOI https://doi.org/10.2977/prims/1195167275
- Robert T. Glassey and Walter A. Strauss, The relativistic Boltzmann equation near equilibrium, Spectral and scattering theory and applications, Adv. Stud. Pure Math., vol. 23, Math. Soc. Japan, Tokyo, 1994, pp. 105–111. MR 1275397, DOI https://doi.org/10.2969/aspm/02310105
- Maria Pia Gualdani and Nestor Guillen, Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential, Anal. PDE 9 (2016), no. 8, 1772–1809. MR 3599518, DOI https://doi.org/10.2140/apde.2016.9.1772
- Hélène Guérin, Solving Landau equation for some soft potentials through a probabilistic approach, Ann. Appl. Probab. 13 (2003), no. 2, 515–539. MR 1970275, DOI https://doi.org/10.1214/aoap/1050689592
- Yan Guo, The Landau equation in a periodic box, Comm. Math. Phys. 231 (2002), no. 3, 391–434. MR 1946444, DOI https://doi.org/10.1007/s00220-002-0729-9
- Yan Guo, The Vlasov-Poisson-Landau system in a periodic box, J. Amer. Math. Soc. 25 (2012), no. 3, 759–812. MR 2904573, DOI https://doi.org/10.1090/S0894-0347-2011-00722-4
- Fred L. Hinton, Collisional transport in plasma, Handbook of Plasma Physics, Volume I: Basic Plasma Physics I (M.N. Rosenbluth and R.Z. Sagdeev, eds.), North-Holland Publishing Company, Amsterdam, 1983, pp. 147–197.
- J. Horowitz and R. L. Karandikar, Martingale problems associated with the Boltzmann equation, Seminar on Stochastic Processes, 1989 (San Diego, CA, 1989) Progr. Probab., vol. 18, Birkhäuser Boston, Boston, MA, 1990, pp. 75–122. MR 1042343
- Ling Hsiao and Hongjun Yu, Global classical solutions to the initial value problem for the relativistic Landau equation, J. Differential Equations 228 (2006), no. 2, 641–660. MR 2289548, DOI https://doi.org/10.1016/j.jde.2005.10.022
- M. Lemou, Linearized quantum and relativistic Fokker-Planck-Landau equations, Math. Methods Appl. Sci. 23 (2000), no. 12, 1093–1119. MR 1773932, DOI https://doi.org/10.1002/1099-1476%28200008%2923%3A12%24%5Clangle%241093%3A%3AAID-MMA153%24%5Crangle%243.0.CO%3B2-8
- E. M. Lifshitz and L. P. Pitaevskiĭ, Course of theoretical physics [“Landau-Lifshits”]. Vol. 10, Pergamon International Library of Science, Technology, Engineering and Social Studies, Pergamon Press, Oxford-Elmsford, N.Y., 1981. Translated from the Russian by J. B. Sykes and R. N. Franklin. MR 684990
- Luis Silvestre, Upper bounds for parabolic equations and the Landau equation, J. Differential Equations 262 (2017), no. 3, 3034–3055. MR 3582250, DOI https://doi.org/10.1016/j.jde.2016.11.010
- Robert M. Strain and Yan Guo, Stability of the relativistic Maxwellian in a collisional plasma, Comm. Math. Phys. 251 (2004), no. 2, 263–320. MR 2100057, DOI https://doi.org/10.1007/s00220-004-1151-2
- Robert M. Strain and Yan Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations 31 (2006), no. 1-3, 417–429. MR 2209761, DOI https://doi.org/10.1080/03605300500361545
- Robert M. Strain and Yan Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal. 187 (2008), no. 2, 287–339. MR 2366140, DOI https://doi.org/10.1007/s00205-007-0067-3
- Robert M. Strain and Maja Tasković, Entropy dissipation estimates for the relativistic Landau equation, and applications, J. Funct. Anal. 277 (2019), no. 4, 1139–1201. MR 3959729, DOI https://doi.org/10.1016/j.jfa.2019.04.007
- Robert M. Strain and Keya Zhu, The Vlasov-Poisson-Landau system in $\mathbb {R}^3_x$, Arch. Ration. Mech. Anal. 210 (2013), no. 2, 615–671. MR 3101794, DOI https://doi.org/10.1007/s00205-013-0658-0
- Walter A. Strauss, The relativistic Boltzmann equation, Quantization, nonlinear partial differential equations, and operator algebra (Cambridge, MA, 1994) Proc. Sympos. Pure Math., vol. 59, Amer. Math. Soc., Providence, RI, 1996, pp. 203–209. MR 1392991, DOI https://doi.org/10.1090/pspum/059/1392991
- Hiroshi Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Z. Wahrsch. Verw. Gebiete 46 (1978/79), no. 1, 67–105. MR 512334, DOI https://doi.org/10.1007/BF00535689
- Cédric Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal. 143 (1998), no. 3, 273–307. MR 1650006, DOI https://doi.org/10.1007/s002050050106
- Cédric Villani, Optimal transport, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. Old and new. MR 2459454
- John B. Walsh, An introduction to stochastic partial differential equations, École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 265–439. MR 876085, DOI https://doi.org/10.1007/BFb0074920
- Kung-Chien Wu, Global in time estimates for the spatially homogeneous Landau equation with soft potentials, J. Funct. Anal. 266 (2014), no. 5, 3134–3155. MR 3158719, DOI https://doi.org/10.1016/j.jfa.2013.11.005
- Tong Yang and Hongjun Yu, Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space, J. Differential Equations 248 (2010), no. 6, 1518–1560. MR 2593052, DOI https://doi.org/10.1016/j.jde.2009.11.027
- Tong Yang and Hongjun Yu, Global solutions to the relativistic Landau-Maxwell system in the whole space, J. Math. Pures Appl. (9) 97 (2012), no. 6, 602–634 (English, with English and French summaries). MR 2921603, DOI https://doi.org/10.1016/j.matpur.2011.09.006
- Hongjun Yu, Smoothing effects for classical solutions of the relativistic Landau-Maxwell system, J. Differential Equations 246 (2009), no. 10, 3776–3817. MR 2514726, DOI https://doi.org/10.1016/j.jde.2009.02.021
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Additional Information
Robert M. Strain
Affiliation:
Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
MR Author ID:
746810
ORCID:
0000-0002-1107-8570
Email:
strain@math.upenn.edu
Zhenfu Wang
Affiliation:
Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
MR Author ID:
1183905
Email:
zwang423@math.upenn.edu
Keywords:
Relativistic Landau equation,
weak solutions,
stochastic representation,
uniqueness,
Wasserstein distance.
Received by editor(s):
March 12, 2019
Received by editor(s) in revised form:
May 27, 2019
Published electronically:
July 8, 2019
Additional Notes:
The first author was partially supported by the NSF grants DMS-1500916 and DMS-1764177.
Dedicated:
Dedicated to Professor Walter Strauss on the occasion of his eightieth birthday
Article copyright:
© Copyright 2019
Brown University