Abstract
We consider the spatially inhomogeneous Landau equation with initial data that is bounded by a Gaussian in the velocity variable. In the case of moderately soft potentials, we show that weak solutions immediately become smooth, and remain smooth as long as the mass, energy, and entropy densities remain under control. For very soft potentials, we obtain the same conclusion with the additional assumption that a sufficiently high moment of the solution in the velocity variable remains bounded. Our proof relies on the iteration of local Schauder-type estimates.
Similar content being viewed by others
Notes
Technically, Theorem 2.12 does not apply to f since it is not sufficiently regular; however, a standard mollification argument allows us to sidestep this potential issue. We omit the details.
References
Alexandre, R., Villani, C.: On the Landau approximation in plasma physics. Annales de l’Institut Henri Poincare (C) Non Linear Anal. 21(1), 61–95, 2004
Bramanti , M., Brandolini , L.: Schauder estimates for parabolic nondivergence operators of Hörmander type. J. Differ. Equ. 234(1), 177–245, 2007
Cameron , S., Silvestre , L., Snelson , S.: Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials. Annales de l’Institut Henri Poincaré (C) Analyse Non Linéare35(3), 625–642, 2018
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, 3rd edn. Cambridge University Press, Cambridge 1970
Chen , Y., Desvillettes , L., He , L.: Smoothing effects for classical solutions of the full Landau equation. Arch. Ration. Mech. Anal. 193(1), 21–55, 2009
Desvillettes , L., Villani , C.: On the spatially homogeneous Landau equation for hard potentials part I: existence, uniqueness and smoothness. Commun. Partial Differ. Equ. 25(1–2), 179–259, 2000
Di Francesco , M., Polidoro , S.: Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form. Adv. Differ. Equ. 11(11), 1261–1320, 2006
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin 2001
Golse , F., Imbert , C., Mouhot , C., Vasseur , A.: Harnack inequality for kinetic Fokker–Planck equations with rough coefficients and application to the Landau equation. Annali della Scuola Normale Superiore di PisaXIX(1), 253–295, 2019
Guo , Y.: The Landau equation in a periodic box. Commun. Math. Phys. 231(3), 391–434, 2002
Han , Q., Lin , F.-H.: Elliptic Partial Differential Equations. Courant Lecture Notes, 2nd edn. Courant Institute of Mathematical Sciences, New York University, New York 2011
He , L., Yang , X.: Well-posedness and asymptotics of grazing collisions limit of Boltzmann equation with Coulomb interaction. SIAM J. Math. Anal. 46(6), 4104–4165, 2014
Hörmander , L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171, 1967
Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Hölder Spaces. Graduate Studies in Mathematics, vol. 12. American Mathematical Society, Providence 1996
Lanconelli , E., Polidoro , S.: On a class of hypoelliptic evolution operators. Rend. Sem. Mat. Univ. Politec. Torino52(1), 29–63, 1994. (Partial differential equations, II (Turin, 1993))
Lifshitz, E.M., Pitaevskii, L.P.: Course of Theoretical Physics: Physical Kinetics, vol. 10, 1st edn. Butterworth-Heinemann, Oxford 1981
Liu , S., Ma , X.: Regularizing effects for the classical solutions to the Landau equation in the whole space. J. Math. Anal. Appl. 417(1), 123–143, 2014
Manfredini , M.: The Dirichlet problem for a class of ultraparabolic equations. Adv. Differ. Equ. 2(5), 831–866, 1997
Mouhot , C., Neumann , L.: Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Nonlinearity19(4), 969, 2006
Silvestre , L.: Upper bounds for parabolic equations and the Landau equation. J. Differ. Equ. 262(3), 3034–3055, 2017
Villani , C.: On the Cauchy problem for Landau equation: sequential stability, global existence. Adv. Differ. Equ. 1(5), 793–816, 1996
Villani , C.: On the spatially homogeneous Landau equation for Maxwellian molecules. Math. Models Methods Appl. Sci. 08(06), 957–983, 1998
Wang , W., Zhang , L.: The \(C^\alpha \) regularity of weak solutions of ultraparabolic equations. Discrete Contin. Dyn. Syst. 29(3), 1261–1275, 2011
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Funding
Both authors were partially supported by National Science Foundation Grant DMS-1246999. CH was partially supported by NSF grant DMS-1907853. SS was partially supported by a Ralph E. Powe Award from ORAU.
Additional information
Communicated by C. Mouhot
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Both authors were partially supported by National Science Foundation grant DMS-1246999. CH was partially supported by NSF Grant DMS-1907853. SS was partially supported by a Ralph E. Powe Award from ORAU
Appendix A. Bounds on the Coefficients of the Landau Equation
Appendix A. Bounds on the Coefficients of the Landau Equation
In this appendix, we collect the available bounds on the coefficients \({{\overline{a}}}\), \({{\overline{b}}}\), and \({{\overline{c}}}\) in the Landau equation (1.3) with soft potentials (\(\gamma \in [-d,0)\)). The estimates in Propositions A.1 and A.2 were derived in [20] and [3]. Earlier, corresponding bounds in the case \(\gamma \geqq 0\) were shown in [6].
Proposition A.1
Let \(f:[0,T_0]\times {\mathbb {R}}^d\times {\mathbb {R}}^d\rightarrow {\mathbb {R}}_+\) satisfy the bounds (1.8), (1.9), and (1.10), and let \({{\overline{a}}}\) be defined by (1.5). If \(\gamma \in [-d,0)\), then for unit vectors \(e\in {\mathbb {R}}^d\),
If \(\gamma \in [-2,0)\), then \({{\overline{a}}}\) satisfies the upper bound
and if \(\gamma \in [-d,-2)\),
The constants c and C depend on d, \(\gamma \), \(m_0\), \(M_0\), \(E_0\), and \(H_0\).
Proposition A.2
Let f be as in Proposition A.1. The coefficients \({{\overline{b}}}\) and \({{\overline{c}}}\) defined by (1.6) and (1.7) respectively, satisfy the upper bounds
and
where the constants depend on d, \(\gamma \), \(M_0\), and \(E_0\).
Finally, we show that when \(\gamma \in [-d,-2]\), the coefficients \({{\overline{a}}}\) and \({{\overline{c}}}\) still have the appropriate decay to prove Theorem 3.4, if sufficiently many moments of f are finite.
Lemma A.3
Let \(\gamma \in [-d,-2]\), and let \(f:[0,T_0]\times {\mathbb {R}}^d\times {\mathbb {R}}^d\rightarrow {\mathbb {R}}\) be a bounded function satisfying (1.8) and (1.9). Assume in addition that
where p is the smallest integer such that \(p>\dfrac{d|\gamma |}{2+\gamma +d}\). Then the upper bounds (A.2) hold, with constants depending on d, \(\gamma \), \(M_0\), \(E_0\), \(P_0\), and \(\Vert f\Vert _{L^\infty ([0,T_0]\times {\mathbb {R}}^{d}\times {\mathbb {R}}^d)}\).
If, in addition, \(\gamma > -d/2-1\), there is an \(\varepsilon >0\) depending on d and \(\gamma \) such that
with C depending on the same quantities.
Proof
For any \(e\in {\mathbb {S}}^{d-1}\), the formula (1.5) implies
Let \(r := \frac{1}{2} |v|^{(\gamma +2)/(\gamma +2+d)}\), \(R = |v|/2\), and define
We have
Our choice of p implies \(-p< d(\gamma +2)/(d+\gamma +2)\), so that \(I_2 \lesssim |v|^{\gamma +2}\). Finally, for \(|w|\geqq |v|/2\), we have \(|w|^{\gamma +2} \lesssim |v|^{\gamma +2}\), and
If e is parallel to v, then proceeding as in [3, Lemma 2.1], we have
where \(\theta \) is the angle between v and z. We may assume \(|v|>2\). Let \(R = |v|/2\) and \(q = \dfrac{p(p-2)}{p+d}\). By our choice of p, we have \((\gamma +2)p/q > -d\). If \(z\in B_R(v)\), then \(|\sin \theta | \leqq |v-z|/|v|\), \(|z|\lesssim |v|\), and
If \(|v-z| \geqq R=|v|/2\), then \(|v-z|^\gamma \lesssim |v|^\gamma \), and we have
For \({{\overline{c}}}\), our choice of p and the restriction that \(\gamma > -d/2 -1\) implies there is an \(\varepsilon >0\) such that \(-p + \dfrac{\gamma (\gamma +2-\varepsilon )}{d+\gamma } < \gamma + 2\). Define \(r = \frac{1}{2} |v|^{(\gamma +2-\varepsilon )/(d+\gamma )}\), \(R = |v|/2\), and \(I_1\), \(I_2\), \(I_3\) as above. The same method implies that \(I_1 + I_2 + I_3 \lesssim |v|^{\gamma +2-\varepsilon }\). \(\quad \square \)
Rights and permissions
About this article
Cite this article
Henderson, C., Snelson, S. \(C^\infty \) Smoothing for Weak Solutions of the Inhomogeneous Landau Equation. Arch Rational Mech Anal 236, 113–143 (2020). https://doi.org/10.1007/s00205-019-01465-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-019-01465-7