doi: 10.3934/dcdsb.2021147

Global well-posedness and exponential stability for the fermion equation in weighted Sobolev spaces

1. 

School of Mathematics and Information Sciences, Yantai University, Yantai 264005, China

2. 

Department of Mathematics, National Cheng Kung University, Tainan 70101, Taiwan

3. 

National Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan

* Corresponding author: Baoyan Sun

Received  May 2019 Revised  March 2021 Published  May 2021

Fund Project: The first author is supported by the Scientific Research Foundation of Yantai University grant 2219008. The second author is supported by the Ministry of Science and Technology under the grant 110-2636-M-006-005- and National Center for Theoretical Sciences

This work deals with the Cauchy problem and the asymptotic behavior of the solution of the fermion equation in the Sobolev spaces with a polynomial weight in the torus. We first investigate the linearized equation and obtain the optimal exponential decay rate for the associated semigroup. Our strategy is taking advantage of quantitative spectral gap estimates in smaller reference Hilbert space, the factorization method and the enlargement of the functional space. We then turn to the nonlinear equation and prove the global existence and uniqueness of solutions in a close-to-equilibrium regime. Moreover, we prove an exponential stability for such a solution with the optimal decay rate given by the semigroup decay of the linearized equation.

Citation: Baoyan Sun, Kung-Chien Wu. Global well-posedness and exponential stability for the fermion equation in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021147
References:
[1]

M. BisiJ. A. Cañizo and B. Lods, Uniqueness in the weakly inelastic regime of the equilibrium state to the Boltzmann equation driven by a particle bath, SIAM J. Math. Anal., 43 (2011), 2640-2674.  doi: 10.1137/110837437.  Google Scholar

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M. Briant, From the Boltzmann equation to the incompressible Navier-Stokes equations on the torus: A quantitative error estimate, J. Differential Equations, 259 (2015), 6072-6141.  doi: 10.1016/j.jde.2015.07.022.  Google Scholar

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M. BriantS. Merino-Aceituno and C. Mouhot, From Boltzmann to incompressible Navier-Stokes in Sobolev spaces with polynomial weight, Anal. Appl. (Singap.), 17 (2019), 85-116.  doi: 10.1142/S021953051850015X.  Google Scholar

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K. CarrapatosoI. Tristani and K. C. Wu, Cauchy problem and exponential stability for the inhomogeneous Landau equation, Arch. Ration. Mech. Anal., 221 (2016), 363-418.  doi: 10.1007/s00205-015-0963-x.  Google Scholar

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L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation, Comm. Pure Appl. Math., 54 (2001), 1-42.  doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q.  Google Scholar

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L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316.  doi: 10.1007/s00222-004-0389-9.  Google Scholar

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J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for kinetic equations with linear relaxation terms, C. R. Math. Acad. Sci. Paris, 347 (2009), 511-516.  doi: 10.1016/j.crma.2009.02.025.  Google Scholar

[10]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.  Google Scholar

[11]

R. J. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations, Nonlinearity, 24 (2011), 2165-2189.  doi: 10.1088/0951-7715/24/8/003.  Google Scholar

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J. Evans, Hypocoercivity in phi-entropy for the linear relaxation Boltzmann equation on the torus, SIAM J. Math. Anal., 53 (2021), 1357-1378.  doi: 10.1137/19M1277631.  Google Scholar

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M. P. Gualdani, S. Mischler and C. Mouhot, Factorization of non-symmetric operators and exponential H-theorem, Mém. Soc. Math. Fr., 153 (2017), 137 pp. doi: 10.24033/msmf.461.  Google Scholar

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F. Hérau, Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46 (2006), 349-359.   Google Scholar

[15]

F. HérauD. Tonon and I. Tristani, Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off, Comm. Math. Phys., 377 (2020), 697-771.  doi: 10.1007/s00220-020-03682-8.  Google Scholar

[16]

F. C. Li and K.-C. Wu, Semigroup decay of the linearized Boltzmann equation in a torus, J. Differential Equations, 260 (2016), 2729-2749.  doi: 10.1016/j.jde.2015.10.012.  Google Scholar

[17]

S. Mischler and C. Mouhot, Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation, Arch. Ration. Mech. Anal., 221 (2016), 677-723.  doi: 10.1007/s00205-016-0972-4.  Google Scholar

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S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 849-898.  doi: 10.1016/j.anihpc.2015.01.007.  Google Scholar

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C. Mouhot, Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials, Comm. Math. Phys., 261 (2006), 629-672.  doi: 10.1007/s00220-005-1455-x.  Google Scholar

[20]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998.  doi: 10.1088/0951-7715/19/4/011.  Google Scholar

[21]

L. Neumann and C. Schmeiser, Convergence to global equilibrium for a kinetic fermion model, SIAM J. Math. Anal., 36 (2005), 1652-1663.  doi: 10.1137/S0036141003436533.  Google Scholar

[22]

I. Tristani, Boltzmann equation for the granular media with thermal force in a weakly inhomogeneous setting, J. Funct. Anal., 270 (2016), 1922-1970.  doi: 10.1016/j.jfa.2015.09.025.  Google Scholar

[23]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), 141 pp. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[24]

K.-C. Wu, Pointwise behavior of the linearized Boltzmann equation on a torus, SIAM J. Math. Anal., 46 (2014), 639-656.  doi: 10.1137/13090482X.  Google Scholar

show all references

References:
[1]

M. BisiJ. A. Cañizo and B. Lods, Uniqueness in the weakly inelastic regime of the equilibrium state to the Boltzmann equation driven by a particle bath, SIAM J. Math. Anal., 43 (2011), 2640-2674.  doi: 10.1137/110837437.  Google Scholar

[2]

E. BouinJ. DolbeaultS. MischlerC. Mouhot and C. Schmeiser, Hypocoercivity without confinement, Pure Appl. Anal., 2 (2020), 203-232.  doi: 10.2140/paa.2020.2.203.  Google Scholar

[3]

M. Briant, From the Boltzmann equation to the incompressible Navier-Stokes equations on the torus: A quantitative error estimate, J. Differential Equations, 259 (2015), 6072-6141.  doi: 10.1016/j.jde.2015.07.022.  Google Scholar

[4]

M. BriantS. Merino-Aceituno and C. Mouhot, From Boltzmann to incompressible Navier-Stokes in Sobolev spaces with polynomial weight, Anal. Appl. (Singap.), 17 (2019), 85-116.  doi: 10.1142/S021953051850015X.  Google Scholar

[5]

K. Carrapatoso and S. Mischler, Landau equation for very soft and Coulomb potentials near Maxwellians, Ann. PDE, 3 (2017), Paper No. 1, 65 pp. doi: 10.1007/s40818-017-0021-0.  Google Scholar

[6]

K. CarrapatosoI. Tristani and K. C. Wu, Cauchy problem and exponential stability for the inhomogeneous Landau equation, Arch. Ration. Mech. Anal., 221 (2016), 363-418.  doi: 10.1007/s00205-015-0963-x.  Google Scholar

[7]

L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation, Comm. Pure Appl. Math., 54 (2001), 1-42.  doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q.  Google Scholar

[8]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316.  doi: 10.1007/s00222-004-0389-9.  Google Scholar

[9]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for kinetic equations with linear relaxation terms, C. R. Math. Acad. Sci. Paris, 347 (2009), 511-516.  doi: 10.1016/j.crma.2009.02.025.  Google Scholar

[10]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.  Google Scholar

[11]

R. J. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations, Nonlinearity, 24 (2011), 2165-2189.  doi: 10.1088/0951-7715/24/8/003.  Google Scholar

[12]

J. Evans, Hypocoercivity in phi-entropy for the linear relaxation Boltzmann equation on the torus, SIAM J. Math. Anal., 53 (2021), 1357-1378.  doi: 10.1137/19M1277631.  Google Scholar

[13]

M. P. Gualdani, S. Mischler and C. Mouhot, Factorization of non-symmetric operators and exponential H-theorem, Mém. Soc. Math. Fr., 153 (2017), 137 pp. doi: 10.24033/msmf.461.  Google Scholar

[14]

F. Hérau, Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46 (2006), 349-359.   Google Scholar

[15]

F. HérauD. Tonon and I. Tristani, Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off, Comm. Math. Phys., 377 (2020), 697-771.  doi: 10.1007/s00220-020-03682-8.  Google Scholar

[16]

F. C. Li and K.-C. Wu, Semigroup decay of the linearized Boltzmann equation in a torus, J. Differential Equations, 260 (2016), 2729-2749.  doi: 10.1016/j.jde.2015.10.012.  Google Scholar

[17]

S. Mischler and C. Mouhot, Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation, Arch. Ration. Mech. Anal., 221 (2016), 677-723.  doi: 10.1007/s00205-016-0972-4.  Google Scholar

[18]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 849-898.  doi: 10.1016/j.anihpc.2015.01.007.  Google Scholar

[19]

C. Mouhot, Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials, Comm. Math. Phys., 261 (2006), 629-672.  doi: 10.1007/s00220-005-1455-x.  Google Scholar

[20]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998.  doi: 10.1088/0951-7715/19/4/011.  Google Scholar

[21]

L. Neumann and C. Schmeiser, Convergence to global equilibrium for a kinetic fermion model, SIAM J. Math. Anal., 36 (2005), 1652-1663.  doi: 10.1137/S0036141003436533.  Google Scholar

[22]

I. Tristani, Boltzmann equation for the granular media with thermal force in a weakly inhomogeneous setting, J. Funct. Anal., 270 (2016), 1922-1970.  doi: 10.1016/j.jfa.2015.09.025.  Google Scholar

[23]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), 141 pp. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[24]

K.-C. Wu, Pointwise behavior of the linearized Boltzmann equation on a torus, SIAM J. Math. Anal., 46 (2014), 639-656.  doi: 10.1137/13090482X.  Google Scholar

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