Compactness properties and local existence of weak solutions to the Landau equation

Hwang, Hyung Ju; Jang, Jin Woo

doi:10.1090/proc/15173

Compactness properties and local existence of weak solutions to the Landau equation
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by Hyung Ju Hwang and Jin Woo Jang PDF

Proc. Amer. Math. Soc. 148 (2020), 5141-5157 Request permission

Abstract:

We consider the Landau equation nearby the Maxwellian equilibrium. Based on the assumptions on the boundedness of mass, energy, and entropy in the sense of Silvestre [J. Diffential Equations 262 (2017), no. 3, 3034–3055], we enjoy the locally uniform ellipticity of the linearized Landau operator to derive local-in-time $L^\infty _{x,v}$ uniform bounds. Then we establish a compactness theorem for the sequence of solutions using the $L^\infty _{x,v}$ bounds and the standard velocity averaging lemma. Finally, we pass to the limit and prove the local existence of a weak solution to the Cauchy problem. The highlight of this work is in the low-regularity setting where we only assume that the initial condition $f_0$ is bounded in $L^\infty _{x,v}$, whose size determines the maximal time-interval of the existence of the weak solution.

References

V. I. Agoshkov, Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation, Dokl. Akad. Nauk SSSR 276 (1984), no. 6, 1289–1293 (Russian). MR 753365
A. A. Arsen′ev, On a connection between the Boltzmann equation and the Landau-Fokker-Planck equations, Dokl. Akad. Nauk SSSR 305 (1989), no. 2, 322–324 (Russian); English transl., Soviet Phys. Dokl. 34 (1989), no. 3, 212–213. MR 996783
A. V. Bobylëv, The expansion of the Boltzmann collision integral in a Landau series, Dokl. Akad. Nauk SSSR 225 (1975), no. 3, 535–538 (Russian). MR 0434265
Kleber Carrapatoso, On the rate of convergence to equilibrium for the homogeneous Landau equation with soft potentials, J. Math. Pures Appl. (9) 104 (2015), no. 2, 276–310. MR 3365830, DOI 10.1016/j.matpur.2015.02.008
Kleber Carrapatoso, Isabelle Tristani, and Kung-Chien Wu, Cauchy problem and exponential stability for the inhomogeneous Landau equation, Arch. Ration. Mech. Anal. 221 (2016), no. 1, 363–418. MR 3483898, DOI 10.1007/s00205-015-0963-x
Sydney Chapman and T. G. Cowling, The mathematical theory of nonuniform gases, 3rd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases; In co-operation with D. Burnett; With a foreword by Carlo Cercignani. MR 1148892
Yemin Chen, Laurent Desvillettes, and Lingbing He, Smoothing effects for classical solutions of the full Landau equation, Arch. Ration. Mech. Anal. 193 (2009), no. 1, 21–55. MR 2506070, DOI 10.1007/s00205-009-0223-z
Ennio De Giorgi, Sull’analiticità delle estremali degli integrali multipli, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 20 (1956), 438–441 (Italian). MR 82045
Ennio De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43 (Italian). MR 0093649
P. Degond and B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case, Math. Models Methods Appl. Sci. 2 (1992), no. 2, 167–182. MR 1167768, DOI 10.1142/S0218202592000119
L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing, Transport Theory Statist. Phys. 21 (1992), no. 3, 259–276. MR 1165528, DOI 10.1080/00411459208203923
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 10.1016/j.jfa.2015.05.009
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 10.1080/03605300008821512
Yunmei Chen, Dirichlet boundary value problems of Landau-Lifshitz equation, Comm. Partial Differential Equations 25 (2000), no. 1-2, 101–124. MR 1737544, DOI 10.1080/03605300008821509
R. J. DiPerna and P.-L. Lions, On the Fokker-Planck-Boltzmann equation, Comm. Math. Phys. 120 (1988), no. 1, 1–23. MR 972541
R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math. 42 (1989), no. 6, 729–757. MR 1003433, DOI 10.1002/cpa.3160420603
R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math. (2) 130 (1989), no. 2, 321–366. MR 1014927, DOI 10.2307/1971423
R. J. DiPerna, P.-L. Lions, and Y. Meyer, $L^p$ regularity of velocity averages, Ann. Inst. H. Poincaré C Anal. Non Linéaire 8 (1991), no. 3-4, 271–287 (English, with French summary). MR 1127927, DOI 10.1016/S0294-1449(16)30264-5
Renjun Duan, Shuangqian Liu, Shota Sakamoto, and Robert M. Strain, Global mild solutions of the landau and non-cutoff boltzmann equations, arXiv:1904.12086.
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 10.1007/s00220-010-1113-9
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 10.1016/j.jfa.2008.11.008
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
François Golse, Pierre-Louis Lions, Benoît Perthame, and Rémi Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal. 76 (1988), no. 1, 110–125. MR 923047, DOI 10.1016/0022-1236(88)90051-1
Maria Gualdani and Nestor Guillen, On $A_p$ weights and the Landau equation, Calc. Var. Partial Differential Equations 58 (2019), no. 1, Paper No. 17, 55. MR 3884792, DOI 10.1007/s00526-018-1451-6
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 10.2140/apde.2016.9.1772
Yan Guo, The Landau equation in a periodic box, Comm. Math. Phys. 231 (2002), no. 3, 391–434. MR 1946444, DOI 10.1007/s00220-002-0729-9
Yan Guo, Hyung Ju Hwang, Jin Woo Jang, and Zhimeng Ouyang, The Landau equation with the specular reflection boundary condition, Arch. Ration. Mech. Anal. 236 (2020), no. 3, 1389–1454. MR 4076068, DOI 10.1007/s00205-020-01496-5
F. Hérau and K. Pravda-Starov, Anisotropic hypoelliptic estimates for Landau-type operators, J. Math. Pures Appl. (9) 95 (2011), no. 5, 513–552 (English, with English and French summaries). MR 2786222, DOI 10.1016/j.matpur.2010.11.003
Jinoh Kim, Yan Guo, and Hyung Ju Hwang, An $l^2$ to $l^\infty$ framework for the landau equation, Peking Mathematical Journal (2020).
E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Politec. Torino 52 (1994), no. 1, 29–63. Partial differential equations, II (Turin, 1993). MR 1289901
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
P.-L. Lions, On Boltzmann and Landau equations, Philos. Trans. Roy. Soc. London Ser. A 346 (1994), no. 1679, 191–204. MR 1278244, DOI 10.1098/rsta.1994.0018
Jürgen Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457–468. MR 170091, DOI 10.1002/cpa.3160130308
Jürgen Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134. MR 159139, DOI 10.1002/cpa.3160170106
Clément Mouhot, De Giorgi–Nash–Moser and Hörmander theories: new interplays, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. III. Invited lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 2467–2493. MR 3966858
J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954. MR 100158, DOI 10.2307/2372841
Sergio Polidoro, On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type, Matematiche (Catania) 49 (1994), no. 1, 53–105 (1995). MR 1386366
Luis Silvestre, Upper bounds for parabolic equations and the Landau equation, J. Differential Equations 262 (2017), no. 3, 3034–3055. MR 3582250, DOI 10.1016/j.jde.2016.11.010
Barbara Stachurska, On a nonlinear integral inequality, Zeszyty Nauk. Uniw. Jagielloń. Prace Mat. 15 (1971), 151–157. MR 328510
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 10.1080/03605300500361545
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 10.1007/s00205-013-0658-0
Cédric Villani, On the Cauchy problem for Landau equation: sequential stability, global existence, Adv. Differential Equations 1 (1996), no. 5, 793–816. MR 1392006
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 10.1007/s002050050106
WenDong Wang and LiQun Zhang, The $C^\alpha$ regularity of a class of non-homogeneous ultraparabolic equations, Sci. China Ser. A 52 (2009), no. 8, 1589–1606. MR 2530175, DOI 10.1007/s11425-009-0158-8
Wendong Wang and Liqun Zhang, The $C^\alpha$ regularity of weak solutions of ultraparabolic equations, Discrete Contin. Dyn. Syst. 29 (2011), no. 3, 1261–1275. MR 2773175, DOI 10.3934/dcds.2011.29.1261
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 10.1016/j.jfa.2013.11.005