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

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- by Hyung Ju Hwang and Jin Woo Jang
- Proc. Amer. Math. Soc.
**148**(2020), 5141-5157 - DOI: https://doi.org/10.1090/proc/15173
- Published electronically: September 24, 2020
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## 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

## Bibliographic Information

**Hyung Ju Hwang**- Affiliation: Department of Mathematics, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea
- MR Author ID: 672369
- Email: hjhwang@postech.ac.kr
**Jin Woo Jang**- Affiliation: Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang 37673, Republic of Korea
- Address at time of publication: Institute for Applied Mathematics, University of Bonn, 53115 Bonn, Germany
- MR Author ID: 1297316
- ORCID: 0000-0002-3846-1983
- Email: jangjinw@iam.uni-bonn.de
- Received by editor(s): February 19, 2019
- Received by editor(s) in revised form: January 27, 2020
- Published electronically: September 24, 2020
- Additional Notes: The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea NRF- 2017R1E1A1A03070105 and NRF-2019R1A5A1028324.

The second author was supported by the Korean IBS project IBS-R003-D1.

The second author is the corresponding author. - Communicated by: Ryan Hynd
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 5141-5157 - MSC (2010): Primary 35Q84, 35Q20, 82C40, 35B45, 34C29, 35B65
- DOI: https://doi.org/10.1090/proc/15173
- MathSciNet review: 4163828