Abstract
We provide a general condition on the kernel of an integro-differential operator so that its associated quadratic form satisfies a coercivity estimate with respect to the \(H^s\)-seminorm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alexandre, R.: Some solutions of the Boltzmann equation without angular cutoff. J. Statist. Phys. 104(1–2), 327–358 (2001)
Alexandre, R.: A review of Boltzmann equation with singular kernels. Kinet. Relat. Models 2(4), 551–646 (2009)
Alexandre, R., Desvillettes, L., Villani, C., Wennberg, B.: Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal. 152(4), 327–355 (2000)
Alexandre, R., El Safadi, M.: Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. I. Non-cutoff case and Maxwellian molecules. Math. Models Methods Appl. Sci. 15(6), 907–920 (2005)
Alexandre, R., Elsafadi, M.: Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. II. Non cutoff case and non Maxwellian molecules. Discrete Contin. Dyn. Syst. 24(1), 1–11 (2009)
Alexandre, R., Morimoto, Y., Ukai, S., Xu, C.-J., Yang, T.: The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential. J. Funct. Anal. 262(3), 915–1010 (2012)
Barlow, M.T., Bass, R.F., Chen, Z.-Q., Kassmann, M.: Non-local Dirichlet forms and symmetric jump processes. Trans. Am. Math. Soc. 361(4), 1963–1999 (2009)
Bass, R.F., Levin, D.A.: Transition probabilities for symmetric jump processes. Trans. Am. Math. Soc. 354(7), 2933–2953 (2002). (electronic)
Bjorland, C., Caffarelli, L., Figalli, A.: Non-local gradient dependent operators. Adv. Math. 230(4–6), 1859–1894 (2012)
Bux, K.-U., Kassmann, M., Schulze, T.: Quadratic forms and Sobolev spaces of fractional order. Proc. Lond. Math. Soc. 119(3), 841–866 (2019)
Caffarelli, L., Chan, C.H., Vasseur, A.: Regularity theory for parabolic nonlinear integral operators. J. Am. Math. Soc. 24(3), 849–869 (2011)
Chen, Y., He, L.: Smoothing estimates for Boltzmann equation with full-range interactions: spatially homogeneous case. Arch. Ration. Mech. Anal. 201(2), 501–548 (2011)
Dyda, B., Kassmann, M.: Regularity estimates for elliptic nonlocal operators. Anal. PDE 13(2), 317–370 (2020)
Felsinger, M., Kassmann, M.: Local regularity for parabolic nonlocal operators. Commun. Partial Differ. Equ. 38(9), 1539–1573 (2013)
Gressman, P.T., Strain, R.M.: Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production. Adv. Math. 227(6), 2349–2384 (2011)
He, L.-B.: Sharp bounds for Boltzmann and Landau collision operators. Ann. Sci. Éc. Norm. Supér. (4) 51(5), 1253–1341 (2018)
Imbert, C., Silvestre, L.: The weak Harnack inequality for the Boltzmann equation without cut-off. J. Eur. Math. Soc. (JEMS) 22(2), 507–592 (2020)
Kassmann, M.: A priori estimates for integro-differential operators with measurable kernels. Calc. Var. Partial. Differ. Equ. 34(1), 1–21 (2009)
Kassmann, M., Rang, M., Schwab, R.W.: Integro-differential equations with nonlinear directional dependence. Indiana Univ. Math. J. 63(5), 1467–1498 (2014)
Kassmann, M., Schwab, R.W.: Regularity results for nonlocal parabolic equations. Riv. Math. Univ. Parma (N.S.) 5(1), 183–212 (2014)
Komatsu, T.: Uniform estimates for fundamental solutions associated with non-local Dirichlet forms. Osaka J. Math. 32(4), 833–860 (1995)
Krylov, N.V., Safonov, M.V.: A certain property of solutions of parabolic equations with measurable coefficients. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 44(1), 161–175 (1980)
Lions, P.-L.: Régularité et compacité pour des noyaux de collision de Boltzmann sans troncature angulaire. C. R. Acad. Sci. Paris Sér. I Math. 326(1), 37–41 (1998)
Mouhot, C.: Explicit coercivity estimates for the linearized Boltzmann and Landau operators. Commun. Partial Differ. Equ. 31(7–9), 1321–1348 (2006)
Ros-Oton, X., Serra, J.: Regularity theory for general stable operators. J. Differ. Equ. 260(12), 8675–8715 (2016)
Schwab, R.W., Silvestre, L.: Regularity for parabolic integro-differential equations with very irregular kernels. Anal. PDE 9(3), 727–772 (2016)
Silvestre, L.: A new regularization mechanism for the Boltzmann equation without cut-off. Commun. Math. Phys. 348(1), 69–100 (2016)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by O.Savin.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Luis Silvestre is supported in part by NSF grant DMS-1764285. Jamil Chaker is supported by DFG Forschungsstipendium through Project 410407063.
