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Stability of the Boltzmann equation with potential forces on torus. (English) Zbl 1173.35638

Summary: We are concerned with the stability of solutions to the Cauchy problem of the Boltzmann equation with potential forces on torus. It is shown that the natural steady state with the symmetry of origin is asymptotically stable in the Sobolev space with exponential rate in time for any initially smooth, periodic, origin symmetric small perturbation, which preserves the same total mass, momentum and mechanical energy. For the non-symmetric steady state, it is also shown that it is stable in \(L^{1}\)-norm for any initial data with the finite total mass, mechanical energy and entropy.

MSC:

35Q35 PDEs in connection with fluid mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82B40 Kinetic theory of gases in equilibrium statistical mechanics
35B35 Stability in context of PDEs
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[1] Cercignani, C.; Illner, R.; Pulvirenti, M., The mathematical theory of dilute gases, (Applied Mathematical Sciences, vol. 106 (1994), Springer-Verlag: Springer-Verlag New York) · Zbl 0813.76001
[2] Desvillettes, L.; Villani, C., On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159, 2, 245-316 (2005) · Zbl 1162.82316
[3] Desvillettes, L.; Villani, C., On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation, Comm. Pure Appl. Math., 54, 1-42 (2001) · Zbl 1029.82032
[4] Guo, Y., The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153, 593-630 (2003) · Zbl 1029.82034
[5] Guo, Y., The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55, 9, 1104-1135 (2002) · Zbl 1027.82035
[6] R. Esposito, Y. Guo, R. Marra, Stability of the front under a Vlasov-Fokker-Planck dynamics, Arch. Ration. Mech. Anal. (2009) (in press); R. Esposito, Y. Guo, R. Marra, Stability of the front under a Vlasov-Fokker-Planck dynamics, Arch. Ration. Mech. Anal. (2009) (in press) · Zbl 1273.76372
[7] Carrillo, J.; Jüngel, A.; Markowich, P.; Toscani, G.; Unterreiter, A., Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133, 1-82 (2001) · Zbl 0984.35027
[8] Toscani, G., Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation, Quart. Appl. Math., 57, 521-541 (1999) · Zbl 1034.82041
[9] DiPerna, R. J.; Lions, P.-L., On the Cauchy problem for Boltzmann equation: Global existence and weak stability, Ann. Math., 130, 321-366 (1989) · Zbl 0698.45010
[10] DiPerna, R. J.; Lions, P.-L., Global weak solution of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42, 729-757 (1989) · Zbl 0698.35128
[11] Villani, C., A review of mathematical topics in collisional kinetic theory, (Handbook of Mathematical Fluid Dynamics, Vol. I (2002), North-Holland: North-Holland Amsterdam), 71-305 · Zbl 1170.82369
[12] Guo, Y., Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169, 4, 305-353 (2003) · Zbl 1044.76056
[13] Strain, R. M.; Guo, Y., Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31, 417-429 (2006) · Zbl 1096.82010
[14] Strain, R. M.; Guo, Y., Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187, 2, 287-339 (2008) · Zbl 1130.76069
[15] Duan, R.-J., On the Cauchy problem for the Boltzmann equation in the whole space: Global existence and uniform stability in \(L_\xi^2(H_x^N)\), J. Differential Equations, 244, 3204-3234 (2008) · Zbl 1150.82019
[16] Kawashima, S., The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7, 301-320 (1990) · Zbl 0702.76090
[17] R.-J. Duan, T. Yang, Stability of the one-species Vlasov-Poisson-Boltzmann system (2009) (in press); R.-J. Duan, T. Yang, Stability of the one-species Vlasov-Poisson-Boltzmann system (2009) (in press)
[18] Guo, Y., The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53, 1081-1094 (2004) · Zbl 1065.35090
[19] Liu, T.-P.; Yu, S.-H., Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246, 1, 133-179 (2004) · Zbl 1092.82034
[20] Liu, T.-P.; Yang, T.; Yu, S.-H., Energy method for the Boltzmann equation, Physica D, 188, 3-4, 178-192 (2004) · Zbl 1098.82618
[21] Strain, R. M., The Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Math. Phys., 268, 2, 543-567 (2006) · Zbl 1129.35022
[22] Ukai, S.; Yang, T.; Zhao, H.-J., Global solutions to the Boltzmann equation with external forces, Anal. Appl., 3, 2, 157-193 (2005) · Zbl 1152.76464
[23] Yang, T.; Zhao, H.-J., A new energy method for the Boltzmann equation, J. Math. Phys., 47, 053301 (2006) · Zbl 1111.82048
[24] Ukai, S.; Yang, T.; Zhao, H.-J., Convergence rate to stationary solutions for Boltzmann equation with external force, Chinese Ann. Math. Ser. B, 27, 363-378 (2006) · Zbl 1151.76571
[25] Duan, R.-J.; Ukai, S.; Yang, T.; Zhao, H.-J., Optimal decay estimates on the linearized Boltzmann equation with time-dependent forces and their applications, Comm. Math. Phys., 277, 1, 189-236 (2008) · Zbl 1175.82047
[26] Duan, R.-J., The Boltzmann equation near equilibrium states in \(R^n\), Methods Appl. Anal., 14, 3, 227-250 (2007) · Zbl 1157.76042
[27] Liu, T.-P.; Yu, S.-H., The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math., 57, 1543-1608 (2004) · Zbl 1111.76047
[28] Liu, T.-P.; Yu, S.-H., Diffusion under gravitational and boundary effects, Bull. Inst. Math. Acad. Sin. (N.S.), 3, 167-210 (2008) · Zbl 1388.76337
[29] Liu, T.-P.; Yang, T.; Yu, S.-H.; Zhao, H., Nonlinear stability of rarefaction waves for the Boltzmann equation, Arch. Ration. Mech. Anal., 181, 333-371 (2006) · Zbl 1095.76024
[30] Huang, F. M.; Xin, Z. P.; Yang, T., Contact discontinuity with general perturbations for gas motions, Adv. Math., 219, 4, 1246-1297 (2008) · Zbl 1155.35068
[31] S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Thesis, Kyoto University, 1983; S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Thesis, Kyoto University, 1983
[32] Grad, H., (Laurmann, J. A., Asymptotic Theory of the Boltzmann Equation II. Asymptotic Theory of the Boltzmann Equation II, Rarefied Gas Dynamics, vol. 1 (1963), Academic Press: Academic Press New York), 26-59 · Zbl 0115.45006
[33] Glassey, R.; Strauss, W., Decay of the linearized Boltzmann-Vlasov system, Transport Theory Statist. Phys., 28, 135-156 (1999) · Zbl 0983.82018
[34] Ukai, S., On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50, 179-184 (1974) · Zbl 0312.35061
[35] Nishida, T.; Imai, K., Global solutions to the initial value problem for the nonlinear Boltzmann equation, Publ. Res. Inst. Math. Sci., 12, 229-239 (1976-1977) · Zbl 0344.35003
[36] Glassey, R., The Cauchy Problem in Kinetic Theory (1996), Society for Industrial and Applied Mathematics (SIAM): Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA, pp. xii+241 · Zbl 0858.76001
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