×

Regularity for the Boltzmann equation conditional to macroscopic bounds. (English) Zbl 1459.35307

Summary: The Boltzmann equation is a nonlinear partial differential equation that plays a central role in statistical mechanics. From the mathematical point of view, the existence of global smooth solutions for arbitrary initial data is an outstanding open problem. In the present article, we review a program focused on the type of particle interactions known as non-cutoff. It is dedicated to the derivation of a priori estimates in \(C^\infty\), depending only on physically meaningful conditions. We prove that the solution will stay uniformly smooth provided that its mass, energy and entropy densities remain bounded, and away from vacuum.

MSC:

35Q20 Boltzmann equations
35B65 Smoothness and regularity of solutions to PDEs
35B45 A priori estimates in context of PDEs
35H10 Hypoelliptic equations
82C22 Interacting particle systems in time-dependent statistical mechanics
35R09 Integro-partial differential equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] R. Alexandre, Some solutions of the Boltzmann equation without angular cutoff,J. Statist. Phys.,104(2001), no. 1-2, 327-358.Zbl 1034.82043 MR 1851391 · Zbl 1034.82043
[2] R. Alexandre, A review of Boltzmann equation with singular kernels,Kinet. Relat. Models, 2(2009), no. 4, 551-646.Zbl 1193.35121 MR 2556715 · Zbl 1193.35121
[3] R. Alexandre, L. Desvillettes, C. Villani, and B. Wennberg, Entropy dissipation and longrange interactions,Arch. Ration. Mech. Anal.,152(2000), no. 4, 327-355.Zbl 0968.76076 MR 1765272 · Zbl 0968.76076
[4] R. Alexandre and M. El Safadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. I. Non-cutoff case and Maxwellian molecules,Math. Models Methods Appl. Sci.,15(2005), no. 6, 907-920.Zbl 1161.35331 MR 2149928 · Zbl 1161.35331
[5] R. Alexandre and M. Elsafadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. II. Non cutoff case and non Maxwellian molecules,Discrete Contin. Dyn. Syst.,24(2009), no. 1, 1-11.Zbl 1168.35326 MR 2476677 · Zbl 1168.35326
[6] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang, Regularizing effect and local existence for the non-cutoff Boltzmann equation,Arch. Ration. Mech. Anal.,198(2010), no. 1, 39-123.Zbl 1257.76099 MR 2679369 · Zbl 1257.76099
[7] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang, The Boltzmann equation without angular cutoff in the whole space: II. Global existence for hard potential,Anal. Appl. (Singap.), 9(2011), no. 2, 113-134.Zbl 1220.35110 MR 2793203 · Zbl 1220.35110
[8] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang, The Boltzmann equation without angular cutoff in the whole space: qualitative properties of solutions,Arch. Ration. Mech. Anal.,202(2011), no. 2, 599-661.Zbl 1426.76660 MR 2847536 · Zbl 1426.76660
[9] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang, The Boltzmann equation without angular cutoff in the whole space: I. Global existence for soft potential,J. Funct. Anal.,262 (2012), no. 3, 915-1010.Zbl 1232.35110 MR 2863853 · Zbl 1232.35110
[10] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang, Smoothing effect of weak solutions for the spatially homogeneous Boltzmann equation without angular cutoff,Kyoto J. Math.,52(2012), no. 3, 433-463.Zbl 1247.35085 MR 2959943 · Zbl 1247.35085
[11] R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions,Comm. Pure Appl. Math.,55(2002), no. 1, 30-70.Zbl 1029.82036 MR 1857879 · Zbl 1029.82036
[12] D. Arsénio and N. Masmoudi, Regularity of renormalized solutions in the Boltzmann equation with long-range interactions,Comm. Pure Appl. Math.,65(2012), no. 4, 508-548. Zbl 1234.35172 MR 2877343 · Zbl 1234.35172
[13] C. Bardos, F. Golse, and D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations,J. Statist. Phys.,63(1991), no. 1-2, 323-344.MR 1115587 · Zbl 0737.76064
[14] M. T. Barlow, R. F. Bass, Z.-Q. Chen, and M. Kassmann, Non-local Dirichlet forms and symmetric jump processes,Trans. Amer. Math. Soc.,361(2009), no. 4, 1963-1999. Zbl 1166.60045 MR 2465826 · Zbl 1166.60045
[15] R. F. Bass and M. Kassmann, Harnack inequalities for non-local operators of variable order, Trans. Amer. Math. Soc.,357(2005), no. 2, 837-850.Zbl 1052.60060 MR 2095633 · Zbl 1052.60060
[16] R. F. Bass and M. Kassmann, Hölder continuity of harmonic functions with respect to operators of variable order,Comm. Partial Differential Equations,30(2005), no. 7-9, 1249- 1259.Zbl 1087.45004 MR 2180302 · Zbl 1087.45004
[17] R. F. Bass and D. A. Levin, Harnack inequalities for jump processes,Potential Anal.,17 (2002), no. 4, 375-388.Zbl 0997.60089 MR 1918242 · Zbl 0997.60089
[18] R. F. Bass and D. A. Levin, Transition probabilities for symmetric jump processes,Trans. Amer. Math. Soc.,354(2002), no. 7, 2933-2953.Zbl 0993.60070 MR 1895210 · Zbl 0993.60070
[19] C. Bjorland, L. Caffarelli, and A. Figalli, Non-local gradient dependent operators,Adv. Math., 230(2012), no. 4-6, 1859-1894.Zbl 1252.35099 MR 2927356 · Zbl 1252.35099
[20] T. Buckmaster, S. Shkoller, and V. Vicol, Formation of point shocks for 3D compressible Euler, 2019.arXiv:1912.04429 · Zbl 1427.35200
[21] T. Buckmaster, S. Shkoller, and V. Vicol, Formation of shocks for 2D isentropic compressible Euler, 2019.arXiv:1907.03784 · Zbl 1427.35200
[22] K.-u. Bux, M. Kassmann, and T. Schulze, Quadratic forms and Sobolev spaces of fractional order,Proc. Lond. Math. Soc. (3),119(2019), no. 3, 841-866.Zbl 07119250 MR 3960670 · Zbl 07119250
[23] L. Caffarelli, C. H. Chan, and A. Vasseur, Regularity theory for parabolic nonlinear integral operators,J. Amer. Math. Soc.,24(2011), no. 3, 849-869.Zbl 1223.35098 MR 2784330 · Zbl 1223.35098
[24] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations,Comm. Pure Appl. Math.,62(2009), no. 5, 597-638.Zbl 1170.45006 MR 2494809 · Zbl 1170.45006
[25] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasigeostrophic equation,Ann. of Math. (2),171(2010), no. 3, 1903-1930.Zbl 1204.35063 MR 2680400 · Zbl 1204.35063
[26] S. Cameron, L. Silvestre, and S. Snelson, Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials,Ann. Inst. H. Poincaré Anal. Non Linéaire, 35(2018), no. 3, 625-642.Zbl 1407.35036 MR 3778645 · Zbl 1407.35036
[27] S. Cameron and S. Snelson, Velocity decay estimates for Boltzmann equation with hard potentials,Nonlinearity,33(2020), no. 6, 2941-2958.MR 4105382 · Zbl 1483.35265
[28] T. Carleman, Sur la théorie de l’équation intégrodifférentielle de Boltzmann (French),Acta Math.,60(1933), no. 1, 91-146.Zbl 59.0404.02 MR 1555365 · JFM 59.0404.02
[29] T. Carleman,Problèmes mathématiques dans la théorie cinétique des gaz(French), Publ. Sci. Inst. Mittag-Leffler, 2, Almqvist & Wiksells Boktryckeri Ab, Uppsala, 1957.Zbl 0077.23401 MR 98477 · Zbl 0077.23401
[30] J. Chaker and L. Silvestre, Coercivity estimates for integro-differential operators,Calc. Var. Partial Differential Equations,59(2020), no. 4, Paper No. 106, 20pp.Zbl 1444.47058 MR 4111815 · Zbl 1444.47058
[31] H. Chang Lara and G. Dávila, Regularity for solutions of nonlocal, nonsymmetric equations, Ann. Inst. H. Poincaré Anal. Non Linéaire,29(2012), no. 6, 833-859.Zbl 1317.35278 MR 2995098 · Zbl 1317.35278
[32] Y. Chen and L. He, Smoothing estimates for Boltzmann equation with full-range interactions: Spatially homogeneous case,Arch. Ration. Mech. Anal.,201(2011), no. 2, 501-548. Zbl 1318.76018 MR 2820356 · Zbl 1318.76018
[33] Y. Chen and L. He, Smoothing estimates for Boltzmann equation with full-range interactions: Spatially inhomogeneous case,Arch. Ration. Mech. Anal.,203(2012), no. 2, 343-377. Zbl 06101988 MR 2885564 · Zbl 1452.35130
[34] D. Christodoulou,The formation of shocks in 3-dimensional fluids, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2007.Zbl 1117.35001 MR 2284927 · Zbl 1138.35060
[35] P. Constantin and V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications,Geom. Funct. Anal.,22(2012), no. 5, 1289-1321.Zbl 1256.35078 MR 2989434 · Zbl 1256.35078
[36] L. Desvillettes and C. Mouhot, Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions,Arch. Ration. Mech. Anal.,193(2009), no. 2, 227-253.Zbl 1169.76054 MR 2525118 · Zbl 1169.76054
[37] L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation,Invent. Math.,159(2005), no. 2, 245-316. Zbl 1162.82316 MR 2116276 · Zbl 1162.82316
[38] L. Desvillettes and B. Wennberg, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff,Comm. Partial Differential Equations,29(2004), no. 1-2, 133-155.Zbl 1103.82020 MR 2038147 · Zbl 1103.82020
[39] M. Di Francesco and S. Polidoro, Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form,Adv. Differential Equations, 11(2006), no. 11, 1261-1320.Zbl 1153.35312 MR 2277064 · Zbl 1153.35312
[40] H. Dong, T. Jin, and H. Zhang, Dini and Schauder estimates for nonlocal fully nonlinear parabolic equations with drifts,Anal. PDE,11(2018), no. 6, 1487-1534.Zbl 1392.35048 MR 3803717 · Zbl 1392.35048
[41] R. Duan, S. Liu, S. Sakamoto, and R. M. Strain, Global mild solutions of the Landau and non-cutoff Boltzmann equations,Comm. Pure Appl. Math., (2020). DOI:10.1002/cpa.21920 · Zbl 1476.35161
[42] B. Dyda and M. Kassmann, Regularity estimates for elliptic nonlocal operators,Anal. PDE, 13(2020), no. 2, 317-370.Zbl 1437.35175 MR 4078229 · Zbl 1437.35175
[43] S. D. Eidelman, S. D. Ivasyshen, and H. P. Malytska, A modified Levi method: development and application,Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, (1998), no. 5, 14-19.Zbl 0912.35093 MR 1693717 · Zbl 0912.35093
[44] M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators,Comm. Partial Differential Equations,38(2013), no. 9, 1539-1573.Zbl 1277.35090 MR 3169755 · Zbl 1277.35090
[45] F. Golse, M. P. Gualdani, C. Imbert, and A. Vasseur, Partial regularity in time for the space homogeneous Landau equation with Coulomb potential, 2019.arXiv:1906.02841
[46] F. Golse, C. Imbert, C. Mouhot, and A. F. Vasseur, Harnack inequality for kinetic Fokker0Planck equations with rough coefficients and application to the Landau equation,Ann. Sc. Norm. Super. Pisa Cl. Sci. (5),19(2019), no. 1, 253-295.Zbl 1431.35016 MR 3923847 · Zbl 1431.35016
[47] P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off,J. Amer. Math. Soc.,24(2011), no. 3, 771-847.Zbl 1248.35140 MR 2784329 · Zbl 1248.35140
[48] P. T. Gressman and R. M. Strain, Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production,Adv. Math.,227(2011), no. 6, 2349-2384. Zbl 1234.35173 MR 2807092 · Zbl 1234.35173
[49] M. Gualdani and N. Guillen, Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential,Anal. PDE,9(2016), no. 8, 1772-1809.Zbl 1378.35325 MR 3599518 · Zbl 1378.35325
[50] Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains,Arch. Ration. Mech. Anal.,197(2010), no. 3, 713-809.Zbl 1291.76276 MR 2679358 · Zbl 1291.76276
[51] Y. Guo, C. Kim, D. Tonon, and A. Trescases, BV-regularity of the Boltzmann equation in nonconvex domains,Arch. Ration. Mech. Anal.,220(2016), no. 3, 1045-1093.Zbl 1334.35200 MR 3466841 · Zbl 1334.35200
[52] Y. Guo, C. Kim, D. Tonon, and A. Trescases, Regularity of the Boltzmann equation in convex domains,Invent. Math.,207(2017), no. 1, 115-290.Zbl 1368.35199 MR 3592757 · Zbl 1368.35199
[53] L. He, Well-posedness of spatially homogeneous Boltzmann equation with full-range interaction,Comm. Math. Phys.,312(2012), no. 2, 447-476.Zbl 1253.35093 MR 2917172 · Zbl 1253.35093
[54] L.-B. He, Sharp bounds for Boltzmann and Landau collision operators,Ann. Sci. Éc. Norm. Supér. (4),51(2018), no. 5, 1253-1341.Zbl 1428.35266 MR 3942041 · Zbl 1428.35266
[55] C. Henderson, S. Snelson, and A. Tarfulea, Local well-posedness of the Boltzmann equation with polynomially decaying initial data,Kinet. Relat. Models,13(2020), no. 4, 837-867. Zbl 1442.35288 MR 4112183 · Zbl 1442.35288
[56] C. Henderson, S. Snelson, and A. Tarfulea, Self-generating lower bounds and continuation for the Boltzmann equation,Calc. Var. Partial Differential Equations,59(2020), no. 6, Paper No. 191, 13pp.MR 4163318 · Zbl 1475.35225
[57] Z. Huo, Y. Morimoto, S. Ukai, and T. Yang, Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff,Kinet. Relat. Models,1(2008), no. 3, 453-489. Zbl 1158.35332 MR 2425608 · Zbl 1158.35332
[58] C. Imbert, T. Jin, and R. Shvydkoy, Schauder estimates for an integro-differential equation with applications to a nonlocal Burgers equation,Ann. Fac. Sci. Toulouse Math. (6),27 (2018), no. 4, 667-677.Zbl 06984157 MR 3884608 · Zbl 1471.45008
[59] C. Imbert, C. Mouhot, and L. Silvestre, Gaussian lower bounds for the Boltzmann equation without cutoff,SIAM J. Math. Anal.,52(2020), no. 3, 2930-2944.Zbl 1442.35050 MR 4112729 · Zbl 1442.35050
[60] C. Imbert, C. Mouhot, and L. Silvestre, Decay estimates for large velocities in the Boltzmann equation without cutoff,J. Éc. polytech. Math.,7(2020), 143-184.Zbl 1427.35278 MR 4033752 · Zbl 1427.35278
[61] C. Imbert and L. Silvestre, The weak Harnack inequality for the Boltzmann equation without cut-off,J. Eur. Math. Soc. (JEMS),22(2020), no. 2, 507-592.Zbl 07174139 MR 4049224 · Zbl 1473.35077
[62] C. Imbert and L. Silvestre, The Schauder estimate for kinetic integral equations, to appear in Analysis and PDEs.arXiv:1812.11870 · Zbl 1344.35049
[63] C. Imbert and L. Silvestre, The weak Harnack inequality for the Boltzmann equation without cut-off,J. Eur. Math. Soc. (JEMS),22(2020), no. 2, 507-592.Zbl 07174139 MR 4049224 · Zbl 1473.35077
[64] T. Jin and J. Xiong, Schauder estimates for solutions of linear parabolic integro-differential equations,Discrete Contin. Dyn. Syst.,35(2015), no. 12, 5977-5998.Zbl 1334.35370 MR 3393263 · Zbl 1334.35370
[65] M. Kassmann, A priori estimates for integro-differential operators with measurable kernels, Calc. Var. Partial Differential Equations,34(2009), no. 1, 1-21.Zbl 1158.35019 MR 2448308 · Zbl 1158.35019
[66] M. Kassmann and A. Mimica, Intrinsic scaling properties for nonlocal operators,J. Eur. Math. Soc. (JEMS),19(2017), no. 4, 983-1011.Zbl 1371.35316 MR 3626549 · Zbl 1371.35316
[67] M. Kassmann, M. Rang, and R. W. Schwab, Integro-differential equations with nonlinear directional dependence,Indiana Univ. Math. J.,63(2014), no. 5, 1467-1498.Zbl 1311.35047 MR 3283558 · Zbl 1311.35047
[68] M. Kassmann and R. W. Schwab, Regularity results for nonlocal parabolic equations,Riv. Math. Univ. Parma (N.S.),5(2014), no. 1, 183-212.Zbl 1329.35095 MR 3289601 · Zbl 1329.35095
[69] C. Kim, Formation and propagation of discontinuity for Boltzmann equation in non-convex domains,Comm. Math. Phys.,308(2011), no. 3, 641-701.Zbl 1237.35126 MR 2855537 · Zbl 1237.35126
[70] C. Kim and D. Lee, The Boltzmann equation with specular boundary condition in convex domains,Comm. Pure Appl. Math.,71(2018), no. 3, 411-504.Zbl 1384.35053 MR 3762275 · Zbl 1384.35053
[71] A. Kolmogoroff, Zufällige Bewegungen (zur Theorie der Brownschen Bewegung) (German), Ann. of Math. (2),35(1934), no. 1, 116-117.Zbl 0008.39906 MR 1503147 · Zbl 0008.39906
[72] T. Komatsu, Uniform estimates for fundamental solutions associated with non-local Dirichlet forms,Osaka J. Math.,32(1995), no. 4, 833-860.Zbl 0867.35123 MR 1380729 · Zbl 0867.35123
[73] N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients (Russian),Izv. Akad. Nauk SSSR Ser. Mat.,44(1980), no. 1, 161- 175, 239.Zbl 0464.35035 MR 563790
[74] O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva,Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R.I., 1968. Zbl 0164.12302 MR 241822 · Zbl 0174.15403
[75] H. C. Lara and G. Dávila, Regularity for solutions of non local parabolic equations,Calc. Var. Partial Differential Equations,49(2014), no. 1-2, 139-172.Zbl 1292.35068 MR 3148110 · Zbl 1292.35068
[76] J. Luk and J. Speck, Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity,Invent. Math.,214(2018), no. 1, 1-169.Zbl 1409.35142 MR 3858399 · Zbl 1409.35142
[77] A. Lunardi, Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients inRn,Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4),24(1997), no. 1, 133-164.Zbl 0887.35062 MR 1475774 · Zbl 0887.35062
[78] M. Manfredini, The Dirichlet problem for a class of ultraparabolic equations,Adv. Differential Equations,2(1997), no. 5, 831-866.Zbl 1023.35518 MR 1751429 · Zbl 1023.35518
[79] F. Merle, P. Raphael, I. Rodnianski, and J. Szeftel, On smooth self similar solutions to the compressible Euler equations, 2019.arXiv:1912.10998
[80] F. Merle, P. Raphael, I. Rodnianski, and J. Szeftel, On the implosion of a three dimensional compressible fluid, 2019.arXiv:1912.11009
[81] R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the martingale problem,Potential Anal., 40(2014), no. 4, 539-563.Zbl 1296.45009 MR 3201992 · Zbl 1296.45009
[82] Y. Morimoto and S. Ukai, Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff,J. Pseudo-Differ. Oper. Appl.,1(2010), no. 1, 139-159.Zbl 1207.35015 MR 2679746 · Zbl 1207.35015
[83] Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff,Discrete Contin. Dyn. Syst.,24 (2009), no. 1, 187-212.Zbl 1169.35315 MR 2476686 · Zbl 1169.35315
[84] Y. Morimoto and T. Yang, Local existence of polynomial decay solutions to the Boltzmann equation for soft potentials,Anal. Appl. (Singap.),13(2015), no. 6, 663-683.Zbl 1326.35225 MR 3376931 · Zbl 1326.35225
[85] C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations,31(2006), no. 7-9, 1321-1348.Zbl 1101.76053 MR 2254617 · Zbl 1101.76053
[86] A. Pascucci and S. Polidoro, The Moser’s iterative method for a class of ultraparabolic equations,Commun. Contemp. Math.,6(2004), no. 3, 395-417.Zbl 1096.35080 MR 2068847 · Zbl 1096.35080
[87] E. V. Radkevich, Equations with nonnegative characteristic form. II (Russian),Sovrem. Mat. Prilozh., (2008), no. 56, Differentsial’nye Uravneniya s Chastnymi Proizvodnymi, 3-147. Zbl 1200.35157 MR 2675371
[88] J. I. Šatyro, The smoothness of the solutions of certain degenerate second order equations (Russian),Mat. Zametki,10(1971), 101-111.Zbl 0216.12202 MR 288423
[89] R. W. Schwab and L. Silvestre, Regularity for parabolic integro-differential equations with very irregular kernels,Anal. PDE,9(2016), no. 3, 727-772.Zbl 1349.47079 MR 3518535 · Zbl 1349.47079
[90] P. Sergio, Recent results on Kolmogorov equations and applications, inProceedings of the workshop on second order subelliptic equations and applications (Cortona, Italy, June 16- 22, 2003), 129-143, S.I.M. Lecture Notes of Seminario Interdisciplinare di Matematica, 3, Dipartimento di Matematica, Università degli Studi della Basilicata, Potenza, 2004. Zbl 1153.35317 · Zbl 1153.35317
[91] J. Serra, Regularity for fully nonlinear nonlocal parabolic equations with rough kernels,Calc. Var. Partial Differential Equations,54(2015), no. 1, 615-629.Zbl 1327.35170 MR 3385173 · Zbl 1327.35170
[92] T. C. Sideris, Formation of singularities in three-dimensional compressible fluids,Comm. Math. Phys.,101(1985), no. 4, 475-485.Zbl 0606.76088 MR 815196 · Zbl 0606.76088
[93] L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace,Indiana Univ. Math. J.,55(2006), no. 3, 1155-1174.Zbl 1101.45004 MR 2244602 · Zbl 1101.45004
[94] L. Silvestre, On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion,Adv. Math.,226(2011), no. 2, 2020-2039.Zbl 1216.35165 MR 2737806 · Zbl 1216.35165
[95] L. Silvestre, A new regularization mechanism for the Boltzmann equation without cut-off, Comm. Math. Phys.,348(2016), no. 1, 69-100.Zbl 1352.35091 MR 3551261 · Zbl 1352.35091
[96] L. Silvestre, Upper bounds for parabolic equations and the Landau equation,J. Differential Equations,262(2017), no. 3, 3034-3055.Zbl 1357.35066 MR 3582250 · Zbl 1357.35066
[97] R. Song and Z. Vondraček, Harnack inequality for some classes of Markov processes,Math. Z.,246(2004), no. 1-2, 177-202.Zbl 1052.60064 MR 2031452 · Zbl 1052.60064
[98] L. F. Stokols, Hölder continuity for a family of nonlocal hypoelliptic kinetic equations,SIAM J. Math. Anal.,51(2019), no. 6, 4815-4847.Zbl 1430.35065 MR 4039522 · Zbl 1430.35065
[99] C. Villani, Théorème vivant, 2012. · Zbl 1290.01002
[100] W. Wang and L. Zhang, TheC˛regularity of a class of non-homogeneous ultraparabolic equations,Sci. China Ser. A,52(2009), no. 8, 1589-1606.Zbl 1181.35139 MR 2530175 · Zbl 1181.35139
[101] W. Wang and L. Zhang, TheC˛regularity of weak solutions of ultraparabolic equations, Discrete Contin. Dyn. Syst.,29(2011), no. 3, 1261-1275.Zbl 1209.35072 MR 2773175 · Zbl 1209.35072
[102] T.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.