We consider the Cauchy problem for the spatially inhomogeneous non-cutoff Boltzmann equation with polynomially decaying initial data in the velocity variable. We establish short-time existence for any initial data with this decay in a fifth order Sobolev space by working in a mixed $ L^2 $ and $ L^\infty $ space that allows to compensate for potential moment generation and obtaining new estimates on the collision operator that are well-adapted to this space. Our results improve the range of parameters for which the Boltzmann equation is well-posed in this decay regime, as well as relax the restrictions on the initial regularity. As an application, we can combine our existence result with the recent conditional regularity estimates of Imbert-Silvestre (arXiv:1909.12729 [math.AP]) to conclude solutions can be continued for as long as the mass, energy, and entropy densities remain under control. This continuation criterion was previously only available in the restricted range of parameters of previous well-posedness results for polynomially decaying initial data.
Citation: |
[1] | R. Alexandre, Around 3D Boltzmann non linear operator without angular cutoff, a new formulation, M2AN Math. Model. Numer. Anal., 34 (2000), 575-590. doi: 10.1051/m2an:2000157. |
[2] | R. Alexandre, Some solutions of the Boltzmann equation without angular cutoff, J. Statist. Phys., 104 (2001), 327-358. doi: 10.1023/A:1010317913642. |
[3] | R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083. |
[4] | 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), 39-123. doi: 10.1007/s00205-010-0290-1. |
[5] | 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), 113-134. doi: 10.1142/S0219530511001777. |
[6] | 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), 599-661. doi: 10.1007/s00205-011-0432-0. |
[7] | R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Bounded solutions of the Boltzmann equation in the whole space, Kinet. Relat. Models, 41 (2011), 17-40. doi: 10.3934/krm.2011.4.17. |
[8] | R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff, Comm. Math. Phys., 304 (2011), 513-581. doi: 10.1007/s00220-011-1242-9. |
[9] | R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Uniqueness of solutions for the non-cutoff Boltzmann equation with soft potential, Kinet. Relat. Models, 4 (2011), 919-934. doi: 10.3934/krm.2011.4.919. |
[10] | 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), 915-1010. doi: 10.1016/j.jfa.2011.10.007. |
[11] | R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Local existence with mild regularity for the Boltzmann equation, Kinet. Relat. Models, 6 (2013), 1011-1041. doi: 10.3934/krm.2013.6.1011. |
[12] | R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions, Comm. Pure Appl. Math., 55 (2002), 30-70. doi: 10.1002/cpa.10012. |
[13] | R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 21 (2004), 61-95. doi: 10.1016/j.anihpc.2002.12.001. |
[14] | R. Alonso, Y. Morimoto, W. Sun and T. Yang, Non-cutoff Boltzmann equation with polynomial decay perturbation, preprint, 2019, arXiv: 1812.05299. |
[15] | T. Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Acta Math., 60 (1933), 91-146. doi: 10.1007/BF02398270. |
[16] | S. Chaturvedi, Local existence for the Landau equation with hard potentials, preprint, 2019, arXiv: 1910.11866. |
[17] | S. Chaturvedi, Stability of vacuum for the Landau equation with hard potentials, preprint, 2020, arXiv: 2001.07208. |
[18] | Y. Chen and L.-B. He, Smoothing estimates for Boltzmann equation with full-range interactions: Spatially homogeneous case, Arch. Ration. Mech. Anal., 201 (2011), 501-548. doi: 10.1007/s00205-010-0393-8. |
[19] | L. Desvillettes and C. Mouhot, About $L^p$ estimates for the spatially homogeneous Boltzmann equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 22 (2005), 127-142. doi: 10.1016/j.anihpc.2004.03.002. |
[20] | L. Desvillettes and C. Mouhot, Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions, Arch. Ration. Mech. Anal., 193 (2009), 227-253. doi: 10.1007/s00205-009-0233-x. |
[21] | L. Desvillettes and B. Wennberg, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff, Comm. Part. Diff. Eq., 29 (2004), 133-155. doi: 10.1081/PDE-120028847. |
[22] | R. J. DiPerna and P. L. Lions, On the Cauchy Problem for Boltzmann equations: Global existence and weak stability, Ann. Math., 130 (1989), 321-366. doi: 10.2307/1971423. |
[23] | A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. |
[24] | D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-61798-0. |
[25] | T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions, J. Statist. Phys., 89 (1997), 751-776. doi: 10.1007/BF02765543. |
[26] | P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847. doi: 10.1090/S0894-0347-2011-00697-8. |
[27] | L.-B. He and X. Yang, Well-posedness and asymptotics of grazing collisions limit of Boltzmann equation with Coulomb interaction, SIAM J. Math. Anal., 46 (2014), 4104-4165. doi: 10.1137/140965983. |
[28] | L.-B. He and J.-C. Jiang, On the global dynamics of the inhomogeneous Boltzmann equations without angular cutoff: Hard potentials and Maxwellian molecules, preprint, 2017, arXiv: 1710.00315. |
[29] | C. Henderson, S. Snelson and A. Tarfulea, Local existence, lower mass bounds, and a new continuation criterion for the Landau equation, J. Differential Equations, 266 (2019), 1536-1577. doi: 10.1016/j.jde.2018.08.005. |
[30] | C. Henderson, S. Snelson, and A. Tarfulea, Local solutions of the Landau equation with rough, slowly decaying initial data, preprint, 2019, arXiv: 1909.05914. |
[31] | F. Hérau, D. Tonon, and I. Tristani, Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off, arXiv: 1710.01098. |
[32] | C. Imbert, C. Mouhot, and L. Silvestre, Decay estimates for large velocities in the Boltzmann equation without cut-off, preprint, 2019, arXiv: 1804.06135. |
[33] | C. Imbert and L. Silvestre, The Schauder estimate for kinetic integral equations, preprint, 2019, arXiv: 1812.11870. |
[34] | C. Imbert and L. Silvestre, Global regularity estimates for the Boltzmann equation without cut-off, preprint, 2019, arXiv: 1909.12729. |
[35] | C. Imbert and L. Silvestre, Weak Harnack inequality for the Boltzmann equation without cut-off, J. Eur. Math. Soc., 22 (2020), 507-592. doi: 10.4171/JEMS/928. |
[36] | X. Lu and C. Mouhot, On measure solutions of the Boltzmann equation, part I: Moment production and stability estimates, J. Differential Equations, 252 (2012), 3305-3363. doi: 10.1016/j.jde.2011.10.021. |
[37] | J. Luk, Stability of Vacuum for the Landau Equation with Moderately Soft Potentials, Ann. PDE, 5 (2019), 101 pp. doi: 10.1007/s40818-019-0067-2. |
[38] | Y. Morimoto, S. Wang and T. Yang, Measure valued solutions to the spatially homogeneous Boltzmann equation without angular cutoff, J. Stat. Phys., 165 (2016), 866-906. doi: 10.1007/s10955-016-1655-0. |
[39] | Y. Morimoto and T. Yang, Local existence of polynomial decay solutions to the Boltzmann equation for soft potentials, Anal. Appl. (Singap.), 13 (2015), 663-683. doi: 10.1142/S0219530514500079. |
[40] | L. Silvestre, A new regularization mechanism for the Boltzmann equation without cut-off, Comm. Math. Phys., 348 (2016), 69-100. doi: 10.1007/s00220-016-2757-x. |
[41] | C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Ration. Mech. Anal., 143 (1998), 273-307. doi: 10.1007/s002050050106. |
[42] | C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, Vol. I, North-Holland, Amsterdam, (2002), 71–305. doi: 10.1016/S1874-5792(02)80004-0. |