August  2019, 12(4): 829-884. doi: 10.3934/krm.2019032

Well-posedness of Cauchy problem for Landau equation in critical Besov space

1. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China

2. 

Université de Rouen, CNRS UMR 6085, Laboratoire de Mathématiques, Saint-Etienne du Rouvray, 76801, France

3. 

School of Mathematics and Statistics, South-Central University for Nationalities, 430074, Wuhan, China

* Corresponding author: Hao-Guang Li

Received  July 2018 Published  May 2019

Fund Project: The first author is supported by the China Scholarship Council(CSC). The second author is supported by the Natural Science Foundation of China (11701578). The research of C.-J. Xu is supported by"The Fundamental Research Funds for Central Universities of China". The research of J. Xu is partially supported by the National Natural Science Foundation of China (11871274) and the Fundamental Research Funds for the Central Universities (NE2015005).

We study the Cauchy problem for the inhomogeneous non linear Landau equation with Maxwellian molecules. In perturbation framework, we establish the global existence of solution in spatially critical Besov spaces. Precisely, if the initial datum is a a small perturbation of the equilibrium distribution in the Chemin-Lerner space $\widetilde L_v^2\left( {B_{2,1}^{3/2}} \right)$, then the Cauchy problem of Landau equation admits a global solution belongs to $\widetilde L_t^\infty \widetilde L_v^2\left( {B_{2,1}^{3/2}} \right)$. The spectral property of Landau operator enables us to develop new trilinear estimates, which leads to the global energy estimate.

Citation: Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic and Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032
References:
[1]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, Regularizing effect and local existence for non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal., 198 (2010), 39-123.  doi: 10.1007/s00205-010-0290-1.

[2]

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.

[3]

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., 9 (2011), 113-134.  doi: 10.1142/S0219530511001777.

[4]

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.

[5]

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.

[6]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[7]

C. Baranger and C. Mouhot, Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials, Rev. Mat. Iberoamericana, 21 (2005), 819-841.  doi: 10.4171/RMI/436.

[8]

A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, Soviet Sci. Rev. Sect. C Math. Phys. Rev., 7 (1988), 111-233. 

[9]

K. Carrapatoso, Exponential convergence to equalilibrium for the homogeneous Landau equation with hard potentials, Bull. Sci. Math., 139 (2015), 777-805.  doi: 10.1016/j.bulsci.2014.12.002.

[10]

F. Charve and R. Danchin, A global existence result for the compressible Navier-Stokes equations in the critical Lp framework, Arch. Ration. Mech. Anal., 198 (2010), 233-271.  doi: 10.1007/s00205-010-0306-x.

[11]

J.-Y. Chemin, Théorèmes d'unicité pour le système de Navier-Stokes tridimensionnel, J. Anal. Math., 77 (1999), 27-50.  doi: 10.1007/BF02791256.

[12]

J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314-328.  doi: 10.1006/jdeq.1995.1131.

[13]

Q.-L. ChenC.-X. Miao and Z.-F. Zhang, Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), 1173-1224.  doi: 10.1002/cpa.20325.

[14]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.  doi: 10.1007/s002220000078.

[15]

R. Danchin and J. Xu, Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical Lp framework, Arch. Ration. Mech. Anal., 224 (2017), 53-90.  doi: 10.1007/s00205-016-1067-y.

[16]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials I. Existence, uniqueness and smoothness, Comm. Partial Differential Equations, 25 (2000), 179-259.  doi: 10.1080/03605300008821512.

[17]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials . H-theorem theorem and applications, Comm. Partial Differential Equations, 25 (2000), 261-298.  doi: 10.1080/03605300008821513.

[18]

R.-J. Duan, On the Cauchy problem for the Boltzmann equation in the whole space: Global existence and uniform stability in $L^2_{\xi}(H^N_x)$, J. Differential Equations, 244 (2008), 3204-3234.  doi: 10.1016/j.jde.2007.11.006.

[19]

R.-J. DuanS.-Q. Liu and J. Xu, Global well-posedness in spatially critical Besov space for the Boltzmann Equation, Arch. Ration. Mech. Anal., 220 (2016), 711-745.  doi: 10.1007/s00205-015-0940-4.

[20]

M.-P. Gualdani, S. Mischler and C. Mouhot, Factorization of non-symmetric operators and exponential $H$-theorem, M$\acute{e}$m. Soc. Math. Fr., 153 (2017), 137pp.

[21]

Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434.  doi: 10.1007/s00220-002-0729-9.

[22]

L. Hsiao and H.-J. Yu, On the Cauchy problem of the Boltzmann and Landau equations with soft potentials, Quart. Appl. Math., 65 (2007), 281-315.  doi: 10.1090/S0033-569X-07-01053-8.

[23]

N. LernerY. MorimotoK. Pravda-Starov and C.-J. Xu, Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators, Kinet. Relat. Models, 6 (2013), 625-648.  doi: 10.3934/krm.2013.6.625.

[24]

N. LernerY. MorimotoK. Pravda-Starov and C.-J. Xu, Gelfand-Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff, J. Differential Equations, 256 (2014), 797-831.  doi: 10.1016/j.jde.2013.10.001.

[25]

H.-G. Li and C.-J. Xu, The Cauchy problem for the radially symmetric homogeneous Boltzmann equation with Shubin class initial datum and Gelfand-Shilov smoothing effect, J. Differential Equations, 263 (2017), 5120-5150.  doi: 10.1016/j.jde.2017.06.010.

[26]

H.-G. Li and C.-J. Xu, Cauchy problem for the spatially homogeneous Landau equation with Shubin class initial datum and Gelfand-Shilov smoothing effect, Siam J. Math. Anal., 51 (2019), 532-564.  doi: 10.1137/17M115116X.

[27]

Y. MorimotoK. Pravda-Starov and C.-J. Xu, A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equation, Kinet. Relat. Models, 6 (2013), 715-727.  doi: 10.3934/krm.2013.6.715.

[28]

Y. Morimoto and C.-J. Xu, Ultra-analytic effect of Cauchy problem for a class of kinetic equations, J. Differential Equations, 247 (2009), 596-617.  doi: 10.1016/j.jde.2009.01.028.

[29]

Y. Morimoto and S. Sakamoto, Global solutions in the critical Besov space for the non-cutoff Boltzmann equation, J. Differential Equations, 261 (2016), 4073-4134.  doi: 10.1016/j.jde.2016.06.017.

[30]

C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations, 31 (2006), 1321-1348.  doi: 10.1080/03605300600635004.

[31]

C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules, Math. Models Methods Appl. Sci., 8 (1998), 957-983.  doi: 10.1142/S0218202598000433.

[32]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307.  doi: 10.1007/s002050050106.

[33]

C. Villani, On the Cauchy problem for Landau equation: Sequential stability, global existence, Adv. Differential Equations, 1 (1996), 793-816. 

[34]

J. Xu and S. Kawashima, Global classical solutions for partially dissipative hyperbolic system of balance laws, Arch. Ration. Mech. Anal., 211 (2014), 513-553.  doi: 10.1007/s00205-013-0679-8.

show all references

References:
[1]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, Regularizing effect and local existence for non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal., 198 (2010), 39-123.  doi: 10.1007/s00205-010-0290-1.

[2]

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.

[3]

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., 9 (2011), 113-134.  doi: 10.1142/S0219530511001777.

[4]

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.

[5]

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.

[6]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[7]

C. Baranger and C. Mouhot, Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials, Rev. Mat. Iberoamericana, 21 (2005), 819-841.  doi: 10.4171/RMI/436.

[8]

A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, Soviet Sci. Rev. Sect. C Math. Phys. Rev., 7 (1988), 111-233. 

[9]

K. Carrapatoso, Exponential convergence to equalilibrium for the homogeneous Landau equation with hard potentials, Bull. Sci. Math., 139 (2015), 777-805.  doi: 10.1016/j.bulsci.2014.12.002.

[10]

F. Charve and R. Danchin, A global existence result for the compressible Navier-Stokes equations in the critical Lp framework, Arch. Ration. Mech. Anal., 198 (2010), 233-271.  doi: 10.1007/s00205-010-0306-x.

[11]

J.-Y. Chemin, Théorèmes d'unicité pour le système de Navier-Stokes tridimensionnel, J. Anal. Math., 77 (1999), 27-50.  doi: 10.1007/BF02791256.

[12]

J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314-328.  doi: 10.1006/jdeq.1995.1131.

[13]

Q.-L. ChenC.-X. Miao and Z.-F. Zhang, Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), 1173-1224.  doi: 10.1002/cpa.20325.

[14]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.  doi: 10.1007/s002220000078.

[15]

R. Danchin and J. Xu, Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical Lp framework, Arch. Ration. Mech. Anal., 224 (2017), 53-90.  doi: 10.1007/s00205-016-1067-y.

[16]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials I. Existence, uniqueness and smoothness, Comm. Partial Differential Equations, 25 (2000), 179-259.  doi: 10.1080/03605300008821512.

[17]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials . H-theorem theorem and applications, Comm. Partial Differential Equations, 25 (2000), 261-298.  doi: 10.1080/03605300008821513.

[18]

R.-J. Duan, On the Cauchy problem for the Boltzmann equation in the whole space: Global existence and uniform stability in $L^2_{\xi}(H^N_x)$, J. Differential Equations, 244 (2008), 3204-3234.  doi: 10.1016/j.jde.2007.11.006.

[19]

R.-J. DuanS.-Q. Liu and J. Xu, Global well-posedness in spatially critical Besov space for the Boltzmann Equation, Arch. Ration. Mech. Anal., 220 (2016), 711-745.  doi: 10.1007/s00205-015-0940-4.

[20]

M.-P. Gualdani, S. Mischler and C. Mouhot, Factorization of non-symmetric operators and exponential $H$-theorem, M$\acute{e}$m. Soc. Math. Fr., 153 (2017), 137pp.

[21]

Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434.  doi: 10.1007/s00220-002-0729-9.

[22]

L. Hsiao and H.-J. Yu, On the Cauchy problem of the Boltzmann and Landau equations with soft potentials, Quart. Appl. Math., 65 (2007), 281-315.  doi: 10.1090/S0033-569X-07-01053-8.

[23]

N. LernerY. MorimotoK. Pravda-Starov and C.-J. Xu, Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators, Kinet. Relat. Models, 6 (2013), 625-648.  doi: 10.3934/krm.2013.6.625.

[24]

N. LernerY. MorimotoK. Pravda-Starov and C.-J. Xu, Gelfand-Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff, J. Differential Equations, 256 (2014), 797-831.  doi: 10.1016/j.jde.2013.10.001.

[25]

H.-G. Li and C.-J. Xu, The Cauchy problem for the radially symmetric homogeneous Boltzmann equation with Shubin class initial datum and Gelfand-Shilov smoothing effect, J. Differential Equations, 263 (2017), 5120-5150.  doi: 10.1016/j.jde.2017.06.010.

[26]

H.-G. Li and C.-J. Xu, Cauchy problem for the spatially homogeneous Landau equation with Shubin class initial datum and Gelfand-Shilov smoothing effect, Siam J. Math. Anal., 51 (2019), 532-564.  doi: 10.1137/17M115116X.

[27]

Y. MorimotoK. Pravda-Starov and C.-J. Xu, A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equation, Kinet. Relat. Models, 6 (2013), 715-727.  doi: 10.3934/krm.2013.6.715.

[28]

Y. Morimoto and C.-J. Xu, Ultra-analytic effect of Cauchy problem for a class of kinetic equations, J. Differential Equations, 247 (2009), 596-617.  doi: 10.1016/j.jde.2009.01.028.

[29]

Y. Morimoto and S. Sakamoto, Global solutions in the critical Besov space for the non-cutoff Boltzmann equation, J. Differential Equations, 261 (2016), 4073-4134.  doi: 10.1016/j.jde.2016.06.017.

[30]

C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations, 31 (2006), 1321-1348.  doi: 10.1080/03605300600635004.

[31]

C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules, Math. Models Methods Appl. Sci., 8 (1998), 957-983.  doi: 10.1142/S0218202598000433.

[32]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Rational Mech. Anal., 143 (1998), 273-307.  doi: 10.1007/s002050050106.

[33]

C. Villani, On the Cauchy problem for Landau equation: Sequential stability, global existence, Adv. Differential Equations, 1 (1996), 793-816. 

[34]

J. Xu and S. Kawashima, Global classical solutions for partially dissipative hyperbolic system of balance laws, Arch. Ration. Mech. Anal., 211 (2014), 513-553.  doi: 10.1007/s00205-013-0679-8.

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