Asymptotics toward the planar rarefaction wave for viscous conservation law in two space dimensions

Nishikawa, Masataka; Nishihara, Kenji

doi:10.1090/S0002-9947-99-02290-4

Asymptotics toward the planar rarefaction wave for viscous conservation law in two space dimensions
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by Masataka Nishikawa and Kenji Nishihara PDF

Trans. Amer. Math. Soc. 352 (2000), 1203-1215 Request permission

Abstract:

This paper is concerned with the asymptotic behavior of the solution toward the planar rarefaction wave $r(\frac {x}{t})$ connecting $u_{+}$ and $u_{-}$ for the scalar viscous conservation law in two space dimensions. We assume that the initial data $u_{0}(x,y)$ tends to constant states $u_{\pm }$ as $x \rightarrow \pm \infty$, respectively. Then, the convergence rate to $r(\frac {x}{t})$ of the solution $u(t,x,y)$ is investigated without the smallness conditions of $|u_{+}-u_{-}|$ and the initial disturbance. The proof is given by elementary $L^{2}$-energy method.

References

Eduard Harabetian, Rarefactions and large time behavior for parabolic equations and monotone schemes, Comm. Math. Phys. 114 (1988), no. 4, 527–536. MR 929127
Youichi Hattori and Kenji Nishihara, A note on the stability of the rarefaction wave of the Burgers equation, Japan J. Indust. Appl. Math. 8 (1991), no. 1, 85–96. MR 1093830, DOI 10.1007/BF03167186
A. M. Il′in and O. A. Oleĭnik, Asymptotic behavior of solutions of the Cauchy problem for some quasi-linear equations for large values of the time, Mat. Sb. (N.S.) 51 (93) (1960), 191–216 (Russian). MR 0120469
Kazuo Ito, Asymptotic decay toward the planar rarefaction waves of solutions for viscous conservation laws in several space dimensions, Math. Models Methods Appl. Sci. 6 (1996), no. 3, 315–338. MR 1388709, DOI 10.1142/S0218202596000109
Akitaka Matsumura and Kenji Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math. 3 (1986), no. 1, 1–13. MR 899210, DOI 10.1007/BF03167088
Akitaka Matsumura and Kenji Nishihara, Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas, Comm. Math. Phys. 144 (1992), no. 2, 325–335. MR 1152375
Akitaka Matsumura and Kenji Nishihara, Asymptotics toward the rarefaction wave of the solutions of Burgers’ equation with nonlinear degenerate viscosity, Nonlinear Anal. 23 (1994), no. 5, 605–614. MR 1297279, DOI 10.1016/0362-546X(94)90239-9
Anders Szepessy and Kevin Zumbrun, Stability of rarefaction waves in viscous media, Arch. Rational Mech. Anal. 133 (1996), no. 3, 249–298. MR 1387931, DOI 10.1007/BF00380894
Zhou Ping Xin, Asymptotic stability of planar rarefaction waves for viscous conservation laws in several dimensions, Trans. Amer. Math. Soc. 319 (1990), no. 2, 805–820. MR 970270, DOI 10.1090/S0002-9947-1990-0970270-8
Zhou Ping Xin, Asymptotic stability of rarefaction waves for $2\times 2$ viscous hyperbolic conservation laws—the two-modes case, J. Differential Equations 78 (1989), no. 2, 191–219. MR 992146, DOI 10.1016/0022-0396(89)90063-6