Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials

Cameron, Stephen; Silvestre, Luis; Snelson, Stanley

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 3, pp. 625-642.

Nous établissons des estimations a priori pour les solutions de l'équation de Landau non homogène en espace, dans le cas de potentiels faiblement mous, pour toute donnée initiale, sous l'hypothèse que la masse, l'énergie et la densité d'entropie restent contrôlées. Nos estimations ponctuelles ont une décroissance polynomiale par rapport à la variable de vitesse. Nous démontrons également que si la donnée initiale est bornée par une gaussienne, alors cette borne est propagée pour tous les temps positifs.

We establish a priori upper bounds for solutions to the spatially inhomogeneous Landau equation in the case of moderately soft potentials, with arbitrary initial data, under the assumption that mass, energy and entropy densities stay under control. Our pointwise estimates decay polynomially in the velocity variable. We also show that if the initial data satisfies a Gaussian upper bound, this bound is propagated for all positive times.

Mots clés : Landau equation, A priori estimates

[1] Alexandre, R.; Liao, J.; Lin, C. Some a priori estimates for the homogeneous Landau equation with soft potentials, Kinet. Relat. Models, Volume 8 (2015) no. 4, pp. 617–650 | MR | Zbl

[2] Alexandre, R.; Villani, C. On the Landau approximation in plasma physics, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 21 (2004) no. 1, pp. 61–95 | DOI | Numdam | MR | Zbl

[3] Bardos, C.; Golse, F.; Levermore, D. Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Stat. Phys., Volume 63 (1991) no. 1–2, pp. 323–344 | MR

[4] Chapman, S.; Cowling, T.G. The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press, 1970 | MR

[5] Desvillettes, L. Entropy dissipation estimates for the Landau equation in the Coulomb case and applications, J. Funct. Anal., Volume 269 (2015) no. 5, pp. 1359–1403 | DOI | MR | Zbl

[6] Desvillettes, L.; Villani, C. On the spatially homogeneous Landau equation for hard potentials part I: existence, uniqueness and smoothness, Commun. Partial Differ. Equ., Volume 25 (2000) no. 1–2, pp. 179–259 | MR | Zbl

[7] Gamba, I.M.; Panferov, V.; Villani, C. Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal., Volume 194 (2009) no. 1, pp. 253–282 | DOI | MR | Zbl

[8] Golse, F.; Imbert, C.; Mouhot, C.; Vasseur, A. Harnack inequality for kinetic Fokker–Planck equations with rough coefficients and application to the Landau equation, Ann. Sc. Norm. Super. Pisa (2017) (in press) | MR | Zbl

[9] Gressman, P.T.; Strain, R.M. Global classical solutions of the Boltzmann equation without angular cut-off, J. Am. Math. Soc., Volume 24 (2011) no. 3, pp. 771–847 | DOI | MR | Zbl

[10] Guo, Y. The Landau equation in a periodic box, Commun. Math. Phys., Volume 231 (2002) no. 3, pp. 391–434 | MR | Zbl

[11] Han, Q.; Lin, F.-H. Elliptic Partial Differential Equations, Courant Lecture Notes, Courant Institute of Mathematical Sciences, New York University, 2011 | MR

[12] Imbert, C.; Silvestre, L. Weak Harnack inequality for the Boltzmann equation without cut-off, J. Eur. Math. Soc. (2017) (in press) | MR

[13] Lifshitz, E.M.; Pitaevskii, L.P. Physical Kinetics, Course of Theoretical Physics, vol. 10, Butterworth-Heinemann, 1981

[14] Liu, S.; Ma, X. Regularizing effects for the classical solutions to the Landau equation in the whole space, J. Math. Anal. Appl., Volume 417 (2014) no. 1, pp. 123–143 | MR | Zbl

[15] Mouhot, C.; Neumann, L. Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, Volume 19 (2006) no. 4, pp. 969 | DOI | MR | Zbl

[16] Pascucci, A.; Polidoro, S. The Moser's iterative method for a class of ultraparabolic equations, Commun. Contemp. Math., Volume 06 (2004) no. 03, pp. 395–417 | DOI | MR | Zbl

[17] Silvestre, L. A new regularization mechanism for the Boltzmann equation without cut-off, Commun. Math. Phys., Volume 348 (2016) no. 1, pp. 69–100 | DOI | MR | Zbl

[18] Silvestre, L. Upper bounds for parabolic equations and the Landau equation, J. Differ. Equ., Volume 262 (2017) no. 3, pp. 3034–3055 | DOI | MR | Zbl

[19] Villani, C. On the Cauchy problem for Landau equation: sequential stability, global existence, Adv. Differ. Equ., Volume 1 (1996) no. 5, pp. 793–816 | MR | Zbl

[20] Villani, C. On the spatially homogeneous Landau equation for Maxwellian molecules, Math. Models Methods Appl. Sci., Volume 08 (1998) no. 06, pp. 957–983 | DOI | MR | Zbl

[21] Wang, W.; Zhang, L. The Cα regularity of a class of non-homogeneous ultraparabolic equations, Sci. China Ser. A, Volume 52 (2009) no. 8, pp. 1589–1606 | DOI | MR | Zbl

[22] Wang, W.; Zhang, L. The Cα regularity of weak solutions of ultraparabolic equations, Discrete Contin. Dyn. Syst., Volume 29 (2011) no. 3, pp. 1261–1275 | DOI | MR | Zbl

[23] Wu, K.-C. Global in time estimates for the spatially homogeneous Landau equation with soft potentials, J. Funct. Anal., Volume 266 (2014) no. 5, pp. 3134–3155 | MR | Zbl