July  2019, 18(4): 1769-1782. doi: 10.3934/cpaa.2019083

Global existence for the Boltzmann equation in $ L^r_v L^\infty_t L^\infty_x $ spaces

Independent scholar

Received  June 2018 Revised  June 2018 Published  January 2019

We study the Boltzmann equation near a global Maxwellian. We prove the global existence of a unique mild solution with initial data which belong to the $ L^r_v L^\infty_x $ spaces where $ r \in (1,\infty] $ by using the excess conservation laws and entropy inequality introduced in [5].

Citation: Koya Nishimura. Global existence for the Boltzmann equation in $ L^r_v L^\infty_t L^\infty_x $ spaces. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1769-1782. doi: 10.3934/cpaa.2019083
References:
[1]

R. DuanF. HuangY. Wang and T. Yang, Global well-posedness of the Boltzmann equation with large Amplitude initial data, Arch. Rational Mech. Anal., 225 (2017), 375-424.  doi: 10.1007/s00205-017-1107-2.  Google Scholar

[2]

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

[3]

Yan Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., Vol. LV (2002), 1104-1135.  doi: 10.1002/cpa.10040.  Google Scholar

[4]

Yan Guo, Decay and continuity of Boltzmann equation in bounded domains, Arch. Rational Mech. Anal., 197 (2010), 713-809.  doi: 10.1007/s00205-009-0285-y.  Google Scholar

[5]

Yan Guo, Bounded solutions for the Boltzmann equation, Quart. Appl. Math., LXVIII (2010), 143-148.  doi: 10.1090/S0033-569X-09-01180-4.  Google Scholar

[6]

M. Strain Robert and Keya Zhu, Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}^3_x$, Kinetic and Related Models, 5 (2012), 383-415.  doi: 10.3934/krm.2012.5.383.  Google Scholar

[7]

S. Ukai and T. Yang, The Boltzmann equation in the space $L^2 \cap L^\infty_\beta$: Global and time-periodic solutions, Analysis and Applications, 4 (2006), 263-310.  doi: 10.1142/S0219530506000784.  Google Scholar

show all references

References:
[1]

R. DuanF. HuangY. Wang and T. Yang, Global well-posedness of the Boltzmann equation with large Amplitude initial data, Arch. Rational Mech. Anal., 225 (2017), 375-424.  doi: 10.1007/s00205-017-1107-2.  Google Scholar

[2]

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

[3]

Yan Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., Vol. LV (2002), 1104-1135.  doi: 10.1002/cpa.10040.  Google Scholar

[4]

Yan Guo, Decay and continuity of Boltzmann equation in bounded domains, Arch. Rational Mech. Anal., 197 (2010), 713-809.  doi: 10.1007/s00205-009-0285-y.  Google Scholar

[5]

Yan Guo, Bounded solutions for the Boltzmann equation, Quart. Appl. Math., LXVIII (2010), 143-148.  doi: 10.1090/S0033-569X-09-01180-4.  Google Scholar

[6]

M. Strain Robert and Keya Zhu, Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}^3_x$, Kinetic and Related Models, 5 (2012), 383-415.  doi: 10.3934/krm.2012.5.383.  Google Scholar

[7]

S. Ukai and T. Yang, The Boltzmann equation in the space $L^2 \cap L^\infty_\beta$: Global and time-periodic solutions, Analysis and Applications, 4 (2006), 263-310.  doi: 10.1142/S0219530506000784.  Google Scholar

[1]

Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310

[2]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, 2021, 14 (1) : 115-148. doi: 10.3934/krm.2020051

[3]

Tong Yang, Seiji Ukai, Huijiang Zhao. Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 495-520. doi: 10.3934/dcds.2009.23.495

[4]

Sabine Hittmeir, Laura Kanzler, Angelika Manhart, Christian Schmeiser. Kinetic modelling of colonies of myxobacteria. Kinetic & Related Models, 2021, 14 (1) : 1-24. doi: 10.3934/krm.2020046

[5]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[6]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[7]

Wenbin Lv, Qingyuan Wang. Global existence for a class of Keller-Segel models with signal-dependent motility and general logistic term. Evolution Equations & Control Theory, 2021, 10 (1) : 25-36. doi: 10.3934/eect.2020040

[8]

Zheng Han, Daoyuan Fang. Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020287

[9]

Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021002

[10]

Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021001

[11]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163

[12]

François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221

[13]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003

[14]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020298

[15]

Franck Davhys Reval Langa, Morgan Pierre. A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 653-676. doi: 10.3934/dcdss.2020353

[16]

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

[17]

Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054

[18]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[19]

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

[20]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (94)
  • HTML views (175)
  • Cited by (0)

Other articles
by authors

[Back to Top]