日本数学会・２０１１年度年会（早稲田大学）・企画特別講演
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1 日本数学会 2011 年度年会 ( 早稲田大学 ) 企画特別講演 MSJMEETING , (1) ρ t + (ρw) x = 0, (ρw) t + (ρw 2 + p) x = (µw x ) x, (ρ(e + w2 2 )) t + ((ρ(e + w2 2 ) + p)w) x = (κθ x + µww x ) x., ρ, w, θ, µ κ, p e, p, e (2) p = Rρθ, e = R γ 1 θ R γ (> 1 ). (1) Barotropic (3) Barotropic {, p ρ t + (ρw) x = 0, (ρw) t + (ρw 2 + p(ρ)) x = (µw x ) x. p = p(ρ) = aρ γ (a ) 1
2 , γ = 1 γ = 1 Burgers Burgers (4) u t + ( u2 2 ) x = µu xx. (5) u t + f(u) x = (B(u)u x ) x (x R, t > 0)., u = u(t, x), f = f(u) u : R + R R m, f : R m R m,,., B : R m m R m m. (B = 0)(, f df(u) R m Ω ( ) λ 1 (u) < λ 2 (u) < < λ m (u)., λ i (u) r i (u), i Ω (r i u λ i )(u) 0, (r i u λ i )(u) = 0. i, i. B(u),. (Kawashima[13]) B(u)r i (u) 0 (u Ω, i = 1,,, m). u B(u), Kawashima condition, - 3. (5) u ± Ω. (6) u(0, x) = u 0 (x) (x R), lim x ± u(t, x) = u ± (t 0)., (1)(6), u ±.,,, 2
3 .,,, Itaya [11](1976) ((3) γ = 1),.,,,,,.,., u = u + = ū, u = ū, ū.,, Kawashima condition u = ū Sobolev, ū (Kawashima [13],1987). [13],, m., Burgers Hopf [4](1950), Nishida [27](1986), Kawashima [13], Liu [19](1985), Kato [12](2007) )., u ±,, Riemann (Riemann ) u t + f(u) x { = 0 (x R, t > 0), (7) u, x < 0, u(0, x) = u +, x > 0. Riemann 1860 [29], ( (3) µ = Euler ),. Riemann, Lax [18](1957). Lax [18],, Reimann,,, Riemann, Lax ( ), u + u Riemann m.,,., Riemann Il in-oleinik [10](1960). 3
4 , Riemann,, Riemann, ).,, Riemann,,,.,,, Nishihara-M [24](1985), [25](1986), Goodman [3](1986),,. (1) 4. (1),, Euler (1) Lagrange, v = 1/ρ,, v t w x = 0, (8) w t + p x = µ( w x v ) x, (x R 1, t > 0). (e + w2 ) 2 t + (pw) x = (κ θ x v + µ ww x ) v x,, p e. (8) p = Rθ v, e = R γ 1 θ (9) (v, w, θ)(x, 0) = (v 0, w 0, θ 0 )(x) (x R),, (10) lim x ± (v, w, θ)(t, x) = (v ±, w ±, θ ± ) (t > 0)., v ± (> 0), w ±, θ ± (> 0), lim (v 0, w 0, θ 0 )(x) = (v ±, w ±, θ ± ), x ± inf v 0(x) > 0, x R 1 4 inf θ 0(x) > 0 x R 1
5 ., Riemann v t w x = 0, w t + p x = 0, (11) (e + w2 ) 2 t + (pw) x = 0, (x R, { t > 0), (v, w, θ)(x, 0) = (v0 R, w0 R, θ0 R (v, w, θ ), x < 0, )(x) := (v +, w +, θ + ), x > 0. Riemann (11), v θ, 3 λ 1 = γp/v < 0, λ 2 = 0, λ 3 = λ 1 > 0, 1 3, 2., Riemann , z = (v, w, θ), z ± = (v ±, w ±, θ ± ), z, z + z R 3 ( ). 1) Riemann, i (i = 1, 3) z r i (x/t), Kawashima-Nishihara-M [16] (1986), (8)-(10), z r i (x/t). Riemann z r 1(x/t) z r 3(x/t),,, z r 1(x/t) + z r 3(x/t) z m., z m z ±, z r 1(x/t) z z m, z r 3(x/t) z m z +.,. 2) Riemann, 2, Huang-Shi-M [40](2004), (8) z vc 2 (x/ t), Huang-Xin-M [7](2006), (8)-(10), z vc 2 (x/ t).,,,., Huang-Xin-Yang [8](2008),.,, 1 3. Huang-Li-M [5](2010),,, Riemann. 5
6 3) Riemann, i (i = 1, 3) zi s (x s i t) (s i :, s 1 < 0 < s 3 ), (8) z z + zi vs (x s i t),.,, (8)-(10), α i zi vs (x s i t + α i )., Kawashima-M [14](1985), ( α i = 0 ). 2 2 Barotiropic (3) Nishihara-M [24],,,., α i, Liu [19](1985), i ( Burgers,, α i. Liu, Szepessy-Xin [28](1993), εu xx,., (8), Liu [20],[21], Zumbrun [30],[31] (Evans ), 2 2 Barotiropic (3) Mascia-Zumbrun [22](2004) 3.2. Riemann,, 2 2 Barotiropic (3),., Riemann, z s 1(x s 1 t) z s 3(x s 3 t)., z α1,α 3 (x, t) := z vs 1 (x s 1 t + α 1 ) + z vs 3 (x s 3 t + α 3 ) z m., α 1 α 3. z m z ±, z s 1(x s 1 t) z z m, z s 3(x s 3 t) z m z +. Huang-M [6](2009) z 0,0 (x, 0) (8)-(10) 6
7 , α 1, α 3, z α1,α 3 (x, t)., z m 2 (8), Liu [19] α 1, α 3 Kawashima-M [14]., Riemann Riemann,,. 5. (8) [5] Sobolev L 2 L 2. [5] 2 2 Barotropic (3), Nishihara-M [24](1985), Mascia-Zumbrun [22](2004) Liu-Zeng [21](2009) Zumbrun [24] (γ = 1) Barker-Humpherys-Laffite-Rudd-Zumbrun [1] Humpherys-Laffite-Zumbrun [9] 7
8 Wang-M [26] Chapman-Enskog (Chapman-Cowling [2], Kawashima-Matsumura-Nishida [15] ) Zumbrun Mei-M [23] Hashimoto-M [41] L 2 L 2 L 2 6.,,,,, (, 2004) References [1] B. Barker, J. Humpherys, O. Laffite, K. Rudd, K. Zumbrun : Stability of isentropic Navier-Stokes shocks, Appl. Math. Lett., 21 (2008), [2] S. Chapman, T. Cowling : The mathematical theory of non-uniform gases, 3rd ed. London, Cambridge University Press (1970). [3] J. Goodman : Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rat. Mech. Anal., 95 (1986), [4] E. Hopf : The partial differential equations u t + uu x = µu xx, Commun. Pure Appl. Math., 3 (1950),
9 [5] F. Huang, J. Li, A. Matsumura : Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Rat. Mech. Anal., 197 (2010), [6] F. Huang, A. Matsumura : Stability of a composite wave of two viscous shock waves for the full compressible Navier-Stokes equation, Commun. Math. Phys., 289 (2009), [7] F. Huang, A. Matsumura, Z. Xin : Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations, Arch. Rat. Mech. Anal., 179 (2006) [8] F. Huang, T. Yang, Z. Xin : Contact discontinuity with general perturbations for gas motions, Adv. in Math., 219 (2008), [9] J. Humpherys, O. Laffite, K. Zumbrun : Stability of isentropic viscous shock profiles in the high-mach number limit, Comm. Math. Phys., 293 (2010), [10] A. M. Il in, O. A. Oleinik : Asymptotic behavior of the solutions of Cauchy problem for certain quasilinear equations for large time (Russian), Mat. Sbornik, 51 (1960), [11] N. Itaya : A survey on the gereralized Burgers equation with a pressure model term, J. Math. Kyoto Univ., 16 (1976), [12] M. Kato : Large time behavior of solutions to the generalized Burgers equations, Osaka J. Math., 44 (2007) [13] S. Kawashima : Large-time behavior of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh, Sect.A 106 (1987), [14] S. Kawashima, A. Matsumura : Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985) [15] S. Kawashima, A. Matsumura, T. Nishida : On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation, Comm. Math. Phys., 70 (1979), [16] S. Kawashima, A. Matsumura, K. Nishihara : Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas, Proc. Japan Acad., 62, Ser.A (1986)
10 [17] S. Kawashima, S. Nishibata, M. Nishikawa : L p energy method for multidimensional viscous conservation laws and application to the stability of planar waves, J. Hyperbolic Differ. Eqn., 1 (2004), [18] P. D. Lax : Hyperbolic systems of conservation laws II, Commun. Pure Appl. Math., 10 (1957), [19] T.-P. Liu. : Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc., 56, [20] : Pointwise convergence to shock waves for viscous conservation laws, Commun. Pure Appl. Math., 50 (1997), [21] T.-P. Liu, Y. Zeng : Time-asymptotic behavior of wave propagation around a viscous shock profile, Comm. Math. Phys., 290 (2009), [22] C. Mascia, K. Zumbrun. : Stability of small-amplitude shock profiles of symmetric hyperbolic-parabolic systems, Comm. Pure Appl. Math., 57 (2004) [23] A. Matsumura, M. Mei : Nonlinear stability of viscous shock profile for a non-convex system of viscoelasticity, Osaka J. Math., 34 (1997), [24] A. Matsumura, K. Nishihara : On the stability of traveling wave solutions of a one-dimensional model system for compressible viscous gas, Jpn. J. Appl. Math., 2 (1985), [25] : Asymptotics toward the rarefaction waves of the solutions of a onedimensional model system for compressible viscous gas, Jpn. J. Appl. Math., 3 (1986), [26] A. Matsumura, Y. Wang : Asymptotic stability of viscous shock wave for a one-dimensional isentropic model of viscous gas with density dependent viscosity, Meth. Appl. Anal., to appear. [27] T. Nishida : Equations of motion of compressible viscous fluids, in Pattern and Waves edited by Nishida, T., Mimura, M., Fujii, H., Amsterdam, Tokyo: Kinokuniya/North-Holland, 1986, [28] A. Szepessy, Z. Xin. : Nonlinear stability of viscous shock waves, Arch. Rational Mech. Anal., 122 (1993) [29] B. Riemann : Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite (1860), in Bernhard Riemann Collected Papers, Edited by R. Narasimhan, Springer-Verlag,
11 [30] K. Zumbrun : Multidimensional stability of planar viscous shock waves, in Advances in the theory of shock waves, , Progr. Nonlinear Differential Equations Appl., 47, Birkha user Boston, Boston, MA, [31] : Stability of large-amplitude shock waves of compressible Navier-Stokes equations, in Handbook of Mathematical Fluid Dynamics, III, Edited by S.J. Friedlander, D. Serre, Elsevier, ( [32] T.-P. Liu, K. Nishihara : Asymptotic behavior for scalar viscous conservation laws with boundary effect, J. Differential Equations, 133 (1997), [33] T.-P. Liu, A. Matsumura, K. Nishihara : Behaviors of solutions for the Burgers equation with boundary corresponding to rarefaction waves, SIAM J. Math. Anal., 29 (1998), [34] A. Matsumura, M. Mei : Convergence to traveling fronts of solutions of the p-system with viscosity in the presence of a boundary, Arch. Rational Mech. Anal., 146 (1999) [35] A. Matsumura, K. Nishihara : Global asymptotics toward the rarefaction wave for solutions of viscous p-system with boundary effect, Quart. Appl. Math., 58 (2000) [36] A. Matsumura : Inflow and outflow problems in the half space for a onedimensional isentropic model system of compressible viscous gas, Methods Appl. Anal., 8 (2001), [37] A. Matsumura, K. Nishihara : Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible gas, Comm. Math. Phys., 222 (2001), [38] F. Huang, A. Matsumura and X. Shi : Viscous shock wave and boundary layer solution to an inflow problem for compressible viscous gas, Comm. Math. Phys, 239 (2003), [39] S. Kawashima, S. Nishibata, P. Zhu : Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys., 240 (2003), [40] F.Huang, A.Matsumura, X.Shi : On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary, Osaka J. Math., 41 (2004),
12 [41] I. Hashimoto, A. Matsumura : Large-time behavior of solutions to an initialboundary value problem on the half line for scalar viscous conservation law, Meth. Appl. Anal., 14 (2007), [42] T. Nakamura, S. Nishibata, T. Yuge : Convergence rates toward the stationary solution for the compressible Navier Stokes equation in half space, Journal of Differential Equations, 241 (2007), [43] I. Hashimoto, Y. Ueda, S. Kawashima : Convergence rate to the nonlinear waves for viscous conservation laws on the half line, Methods Appl. Anal., 16 (2009), [44] S. Kawashima, P. Zhu : Asymptotic stability of rarefaction wave for the Navier-Stokes equations for a compressible fluid in the half space. Arch. Ration. Mech. Anal., 194 (2009), [45] X. Qin, Y. Wang : Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations, SIAM J. Math. Anal., 41 (2009), [46] T. Nakamura, S. Nishibata, S. Kawashima, P. Zhu : Stationary waves to viscous heat-conductive gases in half space: existence, stability and convergence rate, Math. Models and Methods Appl. Sci., 20 (2010) [47] T. Nakamura, S. Nishibata : Stationary waves to viscous heat-conductive gas in half space with inflow boundary condition, Journal of Hyperbolic Differential Equations, to appear. 12
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Title 混合体モデルに基づく圧縮性流体と移動する固体の熱連成計算手法 Author(s) 鳥生, 大祐 ; 牛島, 省 Citation 土木学会論文集 A2( 応用力学 ) = Journal of Japan Civil Engineers, Ser. A2 (2017), 73 Issue Date 2017 URL http://hdl.handle.net/2433/229150 Right
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