Global Weak Solution to the Boltzmann-Enskog equation

Transcription

1 Global Weak Solution to the Boltzmann-Enskog equation Seung-Yeal Ha 1 and Se Eun Noh 2 1) Department of Mathematical Science, Seoul National University, Seoul , KOREA 2) Department of Mathematical Science, Seoul National University, Seoul , KOREA ABSTRACT We present the regularity theory of renormalized solutions to the Boltzmann-Enskog model with a truncation kernel and L p stability estimate. We use the multi-dimensional Bony type functional to control the time-phase space integral of the collision operator with a truncation. Then following Cercignani s argument [18,19], we show that the renormalized solutions to the Boltzmann-Enskog model are weak solutions in classical distribution sense and generalized Gronwall inequality is used for L p stability. INTRODUCTION The transport phenomena of moderately dense gases with hard sphere molecules can be modeled by the Boltzmann-Enskog equation, which is an evolution equation for a velocity distribution function f taking into account some geometric effects due to the overall dimensions of hard sphere molecules. In the absence of external forces, the distribution function f = f(x, ξ, t) satisfies an integro-differential equation: t f + ξ x f = Q E (f, f), x, ξ R 3, t R +, f(x, ξ, 0) = f 0 (x, ξ). (0.1) Here Q E (f, f) denotes a binary collision operator measuring binary collisions between particles, and whose specific form will be addressed in the next section. We also use auxiliary functions as the functions f and Q E (f, f) evaluated along the particle trajectories: f (x, ξ, t) f(x + tξ, ξ, t) and Q E(f, f)(x, ξ, t) Q E (f, f)(x + tξ, ξ, t). We integrate the equation (0.1) along the particle path (x + sξ, ξ, s) to find a mild form f (x, ξ, t) = f 0 (x, ξ) + t 0 Q E(f, f)(x, ξ, s)ds, t 0. (0.2) We first recall the definitions of mild and classical solutions as follows. Definition A nonnegative function f(x, ξ, t) C([0, T ); L 1 +(R 3 R 3 )) is a mild solution of (0.1) with a nonnegative initial datum f 0 if and only if for all t [0, T ) and a.e (x, ξ) R 3 R 3, f satisfies the integral equation (0.2). 2. A function f = f(x, ξ, t) C(R 3 R 3 [0, T )) is a classical solution of (0.1) with a nonnegative initial datum f 0 if and only if f is continuously differentiable with respect to (x, t) 145

2 and f satisfies the equation (0.1) pointwise. 3. A nonnegative function f(x, ξ, t) L 1 loc((0, ) R 3 R 3 ) is a renormalized solution if and only if f satisfies (i) (1 + δf) 1 Q(f, f) L 1 loc((0, ) R 3 R 3 ), and f solves (ii) ( t + ξ x ) (δ 1 log(1 + δf) ) = (1 + δf) 1 Q(f, f), δ > 0 in the sense of distribution and f is independent of δ. We next recall the exsitence theory of solutions to (0.1). Local existence to the Enskog equation was first studied in [31], while global existence of classical and mild solutions when initial datum is a small perturbation of a vacuum was proved in [19,39]. In contrast, for large initial data, global mild solutions and normalized solutions were obtained in [5,20,24,34,36] for Enskog type equations. On the other hand, asymptotic equivalence between the Enskog equation and the Boltzmann equation was investigated in [7 9]. For further references, we refer to [10,38]. Recently, the first author has developed a nonlinear functional approach to the study of large-time behavior and the L 1 -stability of (0.1) in [27] This nonlinear functional approach in kinetic theory was first initiated by Bony [15] for one-dimensional discrete velocity Boltzmann models, and was further developed to other one-dimensional continuous kinetic models in [16 18,21]. The main spirit of the nonlinear functional approach is to devise a robust Lyapunov functional whose time-decay rate can control the phase space-time integral of the collision operator. L p stability theory is adressed in [6,22,25,26,33,41,42] only for the spatially homogeneous Boltzmann equation. Recently in [29] the uniform L p stability estimate was studied for the spatially inhomogeneous Boltzmann equation with external forces. We apply their approach to the Boltzmann-Enskog equation for the L p -stability. SUMMARY OF THE MAIN RESULTS In this section, we describe the main framework. Because the collision process conserve mass, momentum and energy, pairs of pre-collisional and post-collisional velocities (ξ, ξ ) and (ξ, ξ ) satisfy a collision transformation: ξ = ξ [(ξ ξ ) ω]ω and ξ = ξ + [(ξ ξ ) ω]ω, ω S 2 +, (0.3) where v w is the standard inner product between v and w in R 3 and S 2 + = {ω S 2 : (ξ ξ ) ω 0}. The Boltzmann-Enskog collision operator Q E (f, f) takes the form of Q E (f, f)(x, ξ, t) a 2 R 3 S 2 + where we adopt the truncated collision kernel q(ξ ξ, ω)(f f ff + )dωdξ, (0.4) (ξ ξ ) ω, if (ξ ξ ) ω > ε, q(x, ξ, t) = 0, if (ξ ξ ) ω ε. Here an a is the diameter of a hard sphere molecule and we have used abbreviated notations: f f(x, ξ, t), f f (x aω, ξ, t), f f(x, ξ, t) and f + f(x + aω, ξ, t). 146

3 The first theorem is the existence of the weak solution in distributional sense. We adopt the nonlinear functional approach to get the bound of time-phase space integral of collision operator and follow Cercignani s argument in [17,?]. Theorem 0.1 [16,?] Let f 0 L 1 (R 3 R 3 ) be such that R 6 f 0 (x, v)(1+ x 2 + v 2 )dvdx <, R 6 f 0 (x, ξ)dξdx < 2 4, and R 6 f 0 ln f 0 dvdx <. Then there is a weak solution to (0.1). Now we discuss the functional setting for the weighted L p stability. We choose a local maxwellian as standard bounding functions e α x 2, e β ξ 2 and define a set of functions S α,β Remark 0.1 The existence of classical solution in S α,β,δ can be found in [35]. With this classical solution, we study the L p stability with the weight ϕ r as follows. Theorem 0.2 Let f and f be classical solutions to (0.1) corresponding the initial data f 0 and f 0 in S α,β,δ respectively. Then for (p, r)3 satisfying 1 < p < 3, r > 3(p 1), p we have sup f(t) f(t) L p r C f 0 f 0 a L p, for some a (0, 1). r 0 t< REFERENCES 1. Hughes, T. J. R., The Finite Element Method, Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Engelwood Cliffs, NJ, Koh, H. M., Lee, H. S. and Haber, R. B., Dynamic crack propagation analysis using Eulerian-Lagrangian kinematic descriptions, Computational Mechanics, Vol. 3, 1988, pp Riedel, H., Nucleation of Creep Cavities/Basic Theories, Chapter 7, Fracture at High Temperatures, Springer-Verlag, Berlin, Lee, H.S. and Koh, H.M., A Moving-Grid Finite Element Method for the Prediction of Dynamic Crack Propagation in Brittle Materials, Proc. of the Second International Conference on Computer Aided Assessment and Control of Localized Damage, Vol. 2, pp , Southampton U.K., July Arkeryd, L. On the Enskog equation with large initial data. SIAM J. Math. Anal. 21, (1990) 6. Arkeryd, L. Stability in L1 for the spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal. 103, (1988) 7. Arkeryd, L. and Cercignani, C. Global existence in L 1 for the Enskog equation and convergence of the solutions to solutions of the Boltzmann equation. J. Stat. Phys. 59, (1990) 8. Arkeryd, L. and Cercignani, C. On the convergence of solutions of the Enskog equation to solutions of the Boltzmann equation. Comm. Partial Differential Equations 14, (1989) 147

4 9. Bellomo, N. and Lachowicz, M. On the asymptotic equivalence between the Enskog and the Boltzmann equaitons. J. Stat. Phys. 51, (1988) 10. Bellomo, N. and Lachowicz, M. Kinetic equation for dense gases. A review of physical and mathematical results. Int. J. Mod. Phys. B 13, (1987) 11. Bellomo, N., Lachowicz, M., Polewczak, J. and Toscani, G. Mathematical Topics in nonlinear kinetic theory II. Series on Advances in Mathematics for Applied Sciences, Vol. 1, World Scientific. Singapore, Bellomo, N., Palczewski, A. and Toscani, G. Mathematical topics in nonlinear kinetic theory. World Scientific Publishing Co., Singapore, Bellomo, N., Polewczak, J. and Preziosi, L. Lyapunov functional and qualitative behavior of the solution to the nonlinear Enskog equation. Developments in partial differential equations and applications to mathematical physics (Ferrara, 1991), 1-13 Plenum, New York Biryuk, A., Craig, W. and Panferov, V. Strong solutions of the Boltzmann equation in one spatial dimension. C. R. Math. Acad. Sci. Paris 342, (2006) 15. Bony, J.M. Solutions globales bornées pour les modèles discrete de l équation de Boltzmann en dimension 1 d espace. Actes Journees E.D.P.St. Jean de Monts, n o XVI, (1987) 16. Cercignani, C. Global weak solutions of the Boltzmann equation. J. Stat. Phys. 118, (2005) 17. Cercignani, C. Weak solutions of the Boltzmann equation and energy conservation. Appl. Math. Lett. 8, (1995). See also: Errata, Appl. Math. Lett. 8, (1995) 18. Cercignani, C. A remarkable estimate for the solutions of the Boltzmann equation. Appl. Math. Lett. 5, (1992) 19. Cercignani, C. Small data existence for the Enskog equation in L 1. J. Stat. Phys. 51, (1988) 20. Cercignani, C. Existence of global solutions for the space inhomogeneous Enskog equation. Transport Theory Stat. Phys. 16, (1987) 21. Cercignani, C. and Illner, R. Global weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions. Arch. Rational Mech. Anal. 134, 1-16 (1996) 22. Desvillettes, L. and Mouhot, C. About Lp estimates for the spatially homogeneous Boltzmann equation. Ann. Inst. H. Poincare Anal. Non Lineaire 22, (2005) 23. Enskog, D. Kinetiske theorie, svenska akad., 63 (1921). English translation in kinetic theory, S. Brush Ed., vol 3, Pergamon Press, New York, (1972). New York Esteban, M.J. and Perthame, B. On the mod ified Enskog equation for elastic and inelastic collisions. Models with spin. Ann. Inst. H. Poincaré Anal. Non Linéaire 8, (1991) 25. Gustafsson, T. Global Lp-properties for the spatially homogeneous Boltzmann equation. Arch. Rational. Mech. Anal. 103, 1-38 (1998) 26. Gustafsson, T. Lp-estimates for the nonlinear spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal. 92, (1997) 27. Ha, S.-Y. Lyapunov functionals for the Boltzmann-Enskog equation. Indiana Univ. Math J. 54, (2005) 148

5 28. Ha, S.-Y. and Noh, S. New a priori estimate for the Boltzmann-Enskog equation. Nonlinearity 19, (2006) 29. Ha, S.-Y., Lee, Ho and Yun, S.B. Uniform L p stability theory for the space inhomogeneous boltzmann equation with the external forces. DCDS A 24, (2009). 30. Hoffman, D.K. and Curtis, C.F. Kinetic theory of dense gases IV; transport viral coefficients. Phys. Fluids 8, (1965) 31. Lachowicz, M On the local existence and uniqueness of solution of initial-value problem for the Enskog equation. Bull. Polish. Acad. Sci. 31, (1983) 32. Lu, X. Spatial decay solutions of the Boltzmann equation: converse properties of long time limiting behavior. SIAM J. Math. Anal. 30, (1999) 33. Mouhot, C. and Villani, C. Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch. Rational Mech. Anal. 173, (2004) 34. Polewczak, J.Global existence in L 1 for the generalized Enskog equation. J. Stat. Phys. 59, (1990) 35. Polewczak, J. Global existence and asymptotic behavior for the nonlinear Enskog equation. SIAM J. Appl. Math. 49, (1989) 36. Polewczak, J. Global existence in L 1 for the modified nonlinear Enskog equation in R 3. J. Stat. Phys. 56, (1989) 37. Résibois, P. H-theorem for the modified nonlinear Enskog equation. J. Stat. Phys. 19, (1978) 38. Résibois, P. and Deleener, M Classical kinetic theory of fluids. John Wiley, NewYork Toscani, G., Bellomo, N. The Boltzmann-Enskog equation in the whole space R 3. Some global existence, uniqueness and stability results. Comput. Math. Appl. 13, (1987) 40. Van Beijeren, H. and Ernst, M.H. The modified Enskog equation. Physica 68, (1973) 41. Wennberg, B. Stability and exponential convergence for the Boltzmann equation. Arch. Rational Mech, Anal. 130, (1995) 42. Wennberg, B. Stability and exponential convergence in Lp for the spatially homogeneous Boltzmann equation. Nonlinear Anal. TMA, 20, (1993) 149