August  2020, 13(4): 837-867. doi: 10.3934/krm.2020029

Local well-posedness of the Boltzmann equation with polynomially decaying initial data

1. 

Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA

2. 

Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA

3. 

Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA

Received  October 2019 Revised  February 2020 Published  May 2020

Fund Project: The first author was partially supported by NSF grant DMS-2003110. The second author was partially supported by a Ralph E. Powe Award from ORAU. The third author was partially supported by NSF grant DMS-2012333

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: Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic & Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029
References:
[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.  Google Scholar

[2]

R. Alexandre, Some solutions of the Boltzmann equation without angular cutoff, J. Statist. Phys., 104 (2001), 327-358.  doi: 10.1023/A:1010317913642.  Google Scholar

[3]

R. AlexandreL. DesvillettesC. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355.  doi: 10.1007/s002050000083.  Google Scholar

[4]

R. AlexandreY. MorimotoS. UkaiC.-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.  Google Scholar

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R. AlexandreY. MorimotoS. UkaiC.-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.  Google Scholar

[6]

R. AlexandreY. MorimotoS. UkaiC.-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.  Google Scholar

[7]

R. AlexandreY. MorimotoS. UkaiC.-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.  Google Scholar

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R. AlexandreY. MorimotoS. UkaiC.-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.  Google Scholar

[9]

R. AlexandreY. MorimotoS. UkaiC.-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.  Google Scholar

[10]

R. AlexandreY. MorimotoS. UkaiC.-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.  Google Scholar

[11]

R. AlexandreY. MorimotoS. UkaiC.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

R. Alonso, Y. Morimoto, W. Sun and T. Yang, Non-cutoff Boltzmann equation with polynomial decay perturbation, preprint, 2019, arXiv: 1812.05299. Google Scholar

[15]

T. Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Acta Math., 60 (1933), 91-146.  doi: 10.1007/BF02398270.  Google Scholar

[16]

S. Chaturvedi, Local existence for the Landau equation with hard potentials, preprint, 2019, arXiv: 1910.11866. Google Scholar

[17]

S. Chaturvedi, Stability of vacuum for the Landau equation with hard potentials, preprint, 2020, arXiv: 2001.07208. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[29]

C. HendersonS. 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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

C. Imbert and L. Silvestre, The Schauder estimate for kinetic integral equations, preprint, 2019, arXiv: 1812.11870. Google Scholar

[34]

C. Imbert and L. Silvestre, Global regularity estimates for the Boltzmann equation without cut-off, preprint, 2019, arXiv: 1909.12729. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

Y. MorimotoS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[2]

R. Alexandre, Some solutions of the Boltzmann equation without angular cutoff, J. Statist. Phys., 104 (2001), 327-358.  doi: 10.1023/A:1010317913642.  Google Scholar

[3]

R. AlexandreL. DesvillettesC. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355.  doi: 10.1007/s002050000083.  Google Scholar

[4]

R. AlexandreY. MorimotoS. UkaiC.-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.  Google Scholar

[5]

R. AlexandreY. MorimotoS. UkaiC.-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.  Google Scholar

[6]

R. AlexandreY. MorimotoS. UkaiC.-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.  Google Scholar

[7]

R. AlexandreY. MorimotoS. UkaiC.-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.  Google Scholar

[8]

R. AlexandreY. MorimotoS. UkaiC.-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.  Google Scholar

[9]

R. AlexandreY. MorimotoS. UkaiC.-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.  Google Scholar

[10]

R. AlexandreY. MorimotoS. UkaiC.-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.  Google Scholar

[11]

R. AlexandreY. MorimotoS. UkaiC.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

R. Alonso, Y. Morimoto, W. Sun and T. Yang, Non-cutoff Boltzmann equation with polynomial decay perturbation, preprint, 2019, arXiv: 1812.05299. Google Scholar

[15]

T. Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Acta Math., 60 (1933), 91-146.  doi: 10.1007/BF02398270.  Google Scholar

[16]

S. Chaturvedi, Local existence for the Landau equation with hard potentials, preprint, 2019, arXiv: 1910.11866. Google Scholar

[17]

S. Chaturvedi, Stability of vacuum for the Landau equation with hard potentials, preprint, 2020, arXiv: 2001.07208. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[29]

C. HendersonS. 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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

C. Imbert and L. Silvestre, The Schauder estimate for kinetic integral equations, preprint, 2019, arXiv: 1812.11870. Google Scholar

[34]

C. Imbert and L. Silvestre, Global regularity estimates for the Boltzmann equation without cut-off, preprint, 2019, arXiv: 1909.12729. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

Y. MorimotoS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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