April  2016, 36(4): 2229-2256. doi: 10.3934/dcds.2016.36.2229

On the Cauchy problem for the Boltzmann equation in Chemin-Lerner type spaces

1. 

Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China

2. 

Department of Mathematics, South China University of Technology, Guangzhou 510640

Received  March 2015 Revised  May 2015 Published  September 2015

In this paper, motivated by [13], we use the Littlewood-Paley theory to investigate the Cauchy problem of the Boltzmann equation. When the initial data is a small perturbation of an equilibrium state, under the Grad's angular cutoff assumption, we obtain the unique global strong solution to the Boltzmann equation for the hard potential case in the Chemin-Lerner type spaces $C([0,\infty);\widetilde{L}^{2}_{\xi}(B_{2,r}^{s}))$ with $1\leq r\leq2$ and $s>3/2$ or $s=3/2$ and $r=1$. Besides, we also prove the Lipschitz continuity of the solution map. Our results extend some previous works on the Boltzmann equation in Sobolev spaces.
Citation: Hao Tang, Zhengrong Liu. On the Cauchy problem for the Boltzmann equation in Chemin-Lerner type spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2229-2256. doi: 10.3934/dcds.2016.36.2229
References:
[1]

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

[2]

R. Alexandre R. 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. Rational Mech. Anal., 202 (2011), 599.  doi: 10.1007/s00205-011-0432-0.  Google Scholar

[3]

R. Alexandre R. 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.  doi: 10.1016/j.jfa.2011.10.007.  Google Scholar

[4]

R. Alexandre R. 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.  doi: 10.3934/krm.2013.6.1011.  Google Scholar

[5]

D. Arsénio and N. Masmoudi, A new approach to velocity averaging lemmas in Besov spaces,, J. Math.Pures Appl., 101 (2014), 495.  doi: 10.1016/j.matpur.2013.06.012.  Google Scholar

[6]

H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Springer-Verlag, (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar

[7]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases,, Applied Mathematical Sciences, (1994).  doi: 10.1007/978-1-4419-8524-8.  Google Scholar

[8]

J. Chemin, Perfect Incompressible Fluids,, Clarendon Press, (1998).   Google Scholar

[9]

J. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes,, J. Diff. Equs., 121 (1995), 314.  doi: 10.1006/jdeq.1995.1131.  Google Scholar

[10]

R. Duan, The Boltzmann equation near equilibrium states in $R^n$,, Methods Appl. Anal., 14 (2007), 227.  doi: 10.4310/MAA.2007.v14.n3.a2.  Google Scholar

[11]

R. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations,, Nonlinearity, 24 (2011), 2165.  doi: 10.1088/0951-7715/24/8/003.  Google Scholar

[12]

R. Duan, On the Cauchy problem for the Boltzmann equation in the whole space: global existence and uniform stability in $L^2(H_x^N)$,, J. Diff. Equs., 244 (2008), 3204.  doi: 10.1016/j.jde.2007.11.006.  Google Scholar

[13]

R. Duan, S. Liu and J. Xu, Global well-posedness in spatially critical Besov space for the Boltzmann equation,, , ().   Google Scholar

[14]

R. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbbR^3$,, Arch. Ration. Mech. Anal., 199 (2011), 291.  doi: 10.1007/s00205-010-0318-6.  Google Scholar

[15]

R. Duan and T. Yang, Stability of the one species Vlasov-Poisson-Boltzmann system., SIAM J. Math. Anal., 41 (2010), 2353.  doi: 10.1137/090745775.  Google Scholar

[16]

N. Fournier, Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition,, Ann. Appl. Probab., 25 (2015), 860.  doi: 10.1214/14-AAP1012.  Google Scholar

[17]

R. Glassey, The Cauchy Problem in Kinetic Theory,, Society for Industrial and Applied Mathematics (SIAM), (1996).  doi: 10.1137/1.9781611971477.  Google Scholar

[18]

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.  doi: 10.1090/S0894-0347-2011-00697-8.  Google Scholar

[19]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians,, Comm. Pure Appl. Math., 55 (2002), 1104.  doi: 10.1002/cpa.10040.  Google Scholar

[20]

Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff,, Arch. Ration. Mech. Anal., 169 (2003), 305.  doi: 10.1007/s00205-003-0262-9.  Google Scholar

[21]

Y. Guo, The Boltzmann equation in the whole space,, Indiana Univ. Math. J., 53 (2004), 1081.  doi: 10.1512/iumj.2004.53.2574.  Google Scholar

[22]

Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians,, Invent. Math., 153 (2003), 593.  doi: 10.1007/s00222-003-0301-z.  Google Scholar

[23]

T. Kato, Quasi-linear equations of evolution with applications to partial differential equations,, in: Spectral Theory and Differential Equations, 448 (1975), 25.   Google Scholar

[24]

T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation,, Phys. D., 188 (2004), 178.  doi: 10.1016/j.physd.2003.07.011.  Google Scholar

[25]

T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles,, Comm. Math. Phys., 246 (2004), 133.  doi: 10.1007/s00220-003-1030-2.  Google Scholar

[26]

H. J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces,, Wiley in Chichester, (1987).   Google Scholar

[27]

V. Sohinger and R. M. Strain, The Boltzmann equation, Besov spaces, and optimal time decay rates in $\mathbbR_x^n$,, Adv. in Math., 261 (2014), 274.  doi: 10.1016/j.aim.2014.04.012.  Google Scholar

[28]

R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space,, Comm. Math. Phys., 268 (2006), 543.  doi: 10.1007/s00220-006-0109-y.  Google Scholar

[29]

T. Tao, Global well-posedness of the Benjamin-ono equation in $H^1(\mathbbR)$,, J. Hyperbolic Diff. Equ., 1 (2004), 27.  doi: 10.1142/S0219891604000032.  Google Scholar

[30]

H. Tang and Z. Liu, Continuous properties of the solution map for the Euler equations,, J. Math. Phys., 55 (2014).  doi: 10.1063/1.4867622.  Google Scholar

[31]

H. Tang and Z. Liu, Well-posedness of the modfied Camassa-Holm equation in Besov spaces,, Z. Angew. Math. Phys., 66 (2015), 1559.  doi: 10.1007/s00033-014-0483-9.  Google Scholar

[32]

H. Tang, Y. Zhao and Z. Liu, A note on the solution map for the periodic Camassa-Holm equation,, Appl. Anal., 93 (2014), 1745.  doi: 10.1080/00036811.2013.847923.  Google Scholar

[33]

H. Tang, S. Shi and Z. Liu, The dependences on initial data for the b-family equation in critical Besov space,, Monatsh. Math., 177 (2015), 471.  doi: 10.1007/s00605-014-0673-8.  Google Scholar

[34]

H. Triebel, Theory of Function Spaces,, Birkhäuser, (1983).  doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[35]

S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation,, Proc. Japan Acad., 50 (1974), 179.  doi: 10.3792/pja/1195519027.  Google Scholar

[36]

S. Ukai, Solutions of the Boltzmann equation,, Patterns and waves, 18 (1986), 37.  doi: 10.1016/S0168-2024(08)70128-0.  Google Scholar

[37]

S. Ukai and T. Yang, The Boltzmann equation in the space $L^2\cap L^\infty_\beta$: Global and time-periodic solutions,, Anal. Appl. (Singap.), 4 (2006), 263.  doi: 10.1142/S0219530506000784.  Google Scholar

[38]

S. Ukai, T. Yang and H. Zhao, Global solutions to the Boltzmann equation with external forces,, Anal. Appl., 3 (2005), 157.  doi: 10.1142/S0219530505000522.  Google Scholar

[39]

S. Ukai, T. Yang and H. Zhao, Convergence rate to stationary solutions for Boltzmann equation with external force,, Chinese Ann. Math. Ser. B., 27 (2006), 363.  doi: 10.1007/s11401-005-0199-4.  Google Scholar

[40]

C. Villani, A review of mathematical topics in collisional kinetic theory,, in Handbook of Mathematical Fluid Dynamics, I (2002), 71.  doi: 10.1016/S1874-5792(02)80004-0.  Google Scholar

[41]

L. Xiong, T. Wang and L. Wang, Global existence and decay of solutions to the fokker-planck-boltzmann equation,, Kinet. Relat. Models., 7 (2014), 169.  doi: 10.3934/krm.2014.7.169.  Google Scholar

show all references

References:
[1]

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

[2]

R. Alexandre R. 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. Rational Mech. Anal., 202 (2011), 599.  doi: 10.1007/s00205-011-0432-0.  Google Scholar

[3]

R. Alexandre R. 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.  doi: 10.1016/j.jfa.2011.10.007.  Google Scholar

[4]

R. Alexandre R. 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.  doi: 10.3934/krm.2013.6.1011.  Google Scholar

[5]

D. Arsénio and N. Masmoudi, A new approach to velocity averaging lemmas in Besov spaces,, J. Math.Pures Appl., 101 (2014), 495.  doi: 10.1016/j.matpur.2013.06.012.  Google Scholar

[6]

H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Springer-Verlag, (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar

[7]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases,, Applied Mathematical Sciences, (1994).  doi: 10.1007/978-1-4419-8524-8.  Google Scholar

[8]

J. Chemin, Perfect Incompressible Fluids,, Clarendon Press, (1998).   Google Scholar

[9]

J. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes,, J. Diff. Equs., 121 (1995), 314.  doi: 10.1006/jdeq.1995.1131.  Google Scholar

[10]

R. Duan, The Boltzmann equation near equilibrium states in $R^n$,, Methods Appl. Anal., 14 (2007), 227.  doi: 10.4310/MAA.2007.v14.n3.a2.  Google Scholar

[11]

R. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations,, Nonlinearity, 24 (2011), 2165.  doi: 10.1088/0951-7715/24/8/003.  Google Scholar

[12]

R. Duan, On the Cauchy problem for the Boltzmann equation in the whole space: global existence and uniform stability in $L^2(H_x^N)$,, J. Diff. Equs., 244 (2008), 3204.  doi: 10.1016/j.jde.2007.11.006.  Google Scholar

[13]

R. Duan, S. Liu and J. Xu, Global well-posedness in spatially critical Besov space for the Boltzmann equation,, , ().   Google Scholar

[14]

R. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbbR^3$,, Arch. Ration. Mech. Anal., 199 (2011), 291.  doi: 10.1007/s00205-010-0318-6.  Google Scholar

[15]

R. Duan and T. Yang, Stability of the one species Vlasov-Poisson-Boltzmann system., SIAM J. Math. Anal., 41 (2010), 2353.  doi: 10.1137/090745775.  Google Scholar

[16]

N. Fournier, Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition,, Ann. Appl. Probab., 25 (2015), 860.  doi: 10.1214/14-AAP1012.  Google Scholar

[17]

R. Glassey, The Cauchy Problem in Kinetic Theory,, Society for Industrial and Applied Mathematics (SIAM), (1996).  doi: 10.1137/1.9781611971477.  Google Scholar

[18]

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.  doi: 10.1090/S0894-0347-2011-00697-8.  Google Scholar

[19]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians,, Comm. Pure Appl. Math., 55 (2002), 1104.  doi: 10.1002/cpa.10040.  Google Scholar

[20]

Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff,, Arch. Ration. Mech. Anal., 169 (2003), 305.  doi: 10.1007/s00205-003-0262-9.  Google Scholar

[21]

Y. Guo, The Boltzmann equation in the whole space,, Indiana Univ. Math. J., 53 (2004), 1081.  doi: 10.1512/iumj.2004.53.2574.  Google Scholar

[22]

Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians,, Invent. Math., 153 (2003), 593.  doi: 10.1007/s00222-003-0301-z.  Google Scholar

[23]

T. Kato, Quasi-linear equations of evolution with applications to partial differential equations,, in: Spectral Theory and Differential Equations, 448 (1975), 25.   Google Scholar

[24]

T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation,, Phys. D., 188 (2004), 178.  doi: 10.1016/j.physd.2003.07.011.  Google Scholar

[25]

T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles,, Comm. Math. Phys., 246 (2004), 133.  doi: 10.1007/s00220-003-1030-2.  Google Scholar

[26]

H. J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces,, Wiley in Chichester, (1987).   Google Scholar

[27]

V. Sohinger and R. M. Strain, The Boltzmann equation, Besov spaces, and optimal time decay rates in $\mathbbR_x^n$,, Adv. in Math., 261 (2014), 274.  doi: 10.1016/j.aim.2014.04.012.  Google Scholar

[28]

R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space,, Comm. Math. Phys., 268 (2006), 543.  doi: 10.1007/s00220-006-0109-y.  Google Scholar

[29]

T. Tao, Global well-posedness of the Benjamin-ono equation in $H^1(\mathbbR)$,, J. Hyperbolic Diff. Equ., 1 (2004), 27.  doi: 10.1142/S0219891604000032.  Google Scholar

[30]

H. Tang and Z. Liu, Continuous properties of the solution map for the Euler equations,, J. Math. Phys., 55 (2014).  doi: 10.1063/1.4867622.  Google Scholar

[31]

H. Tang and Z. Liu, Well-posedness of the modfied Camassa-Holm equation in Besov spaces,, Z. Angew. Math. Phys., 66 (2015), 1559.  doi: 10.1007/s00033-014-0483-9.  Google Scholar

[32]

H. Tang, Y. Zhao and Z. Liu, A note on the solution map for the periodic Camassa-Holm equation,, Appl. Anal., 93 (2014), 1745.  doi: 10.1080/00036811.2013.847923.  Google Scholar

[33]

H. Tang, S. Shi and Z. Liu, The dependences on initial data for the b-family equation in critical Besov space,, Monatsh. Math., 177 (2015), 471.  doi: 10.1007/s00605-014-0673-8.  Google Scholar

[34]

H. Triebel, Theory of Function Spaces,, Birkhäuser, (1983).  doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[35]

S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation,, Proc. Japan Acad., 50 (1974), 179.  doi: 10.3792/pja/1195519027.  Google Scholar

[36]

S. Ukai, Solutions of the Boltzmann equation,, Patterns and waves, 18 (1986), 37.  doi: 10.1016/S0168-2024(08)70128-0.  Google Scholar

[37]

S. Ukai and T. Yang, The Boltzmann equation in the space $L^2\cap L^\infty_\beta$: Global and time-periodic solutions,, Anal. Appl. (Singap.), 4 (2006), 263.  doi: 10.1142/S0219530506000784.  Google Scholar

[38]

S. Ukai, T. Yang and H. Zhao, Global solutions to the Boltzmann equation with external forces,, Anal. Appl., 3 (2005), 157.  doi: 10.1142/S0219530505000522.  Google Scholar

[39]

S. Ukai, T. Yang and H. Zhao, Convergence rate to stationary solutions for Boltzmann equation with external force,, Chinese Ann. Math. Ser. B., 27 (2006), 363.  doi: 10.1007/s11401-005-0199-4.  Google Scholar

[40]

C. Villani, A review of mathematical topics in collisional kinetic theory,, in Handbook of Mathematical Fluid Dynamics, I (2002), 71.  doi: 10.1016/S1874-5792(02)80004-0.  Google Scholar

[41]

L. Xiong, T. Wang and L. Wang, Global existence and decay of solutions to the fokker-planck-boltzmann equation,, Kinet. Relat. Models., 7 (2014), 169.  doi: 10.3934/krm.2014.7.169.  Google Scholar

[1]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[2]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052

[3]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[4]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[5]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[6]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

[7]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[8]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051

[9]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[10]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[11]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

[12]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

[13]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075

[14]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[15]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[16]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[17]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[18]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351

[19]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[20]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]