In this section, we will prove Theorems 1.3-1.5. This will be carried out by several lemmas.
The notation used in this section can be found in section 1.
Lemma 4.1. Assume that the subcharacteristic condition (1.5) is true. Then there exist connected byR1 and S2 is equivalent to the following condition (cf. [22]),
wr≥wl+g2(vl, zr−zl), zl≥zr, (4.1) where (wl, zl) and (wr, zr) are the Riemann invariants corresponding to (vl, ul) and (vr, ur) respectively,g2 is the function in (1.34).
Let ( ¯wl,z¯l) and ( ¯wr,z¯r) be the Riemann invariants corresponding to (vl, ul+s(S−Cf²(ul)2vl) For the simplicity of the notation, we use ∆Ato denoteAr−Alfor a quantityA, for example,
∆w=wr−wl. By virtue of this setting, (4.2) and (4.3) imply
On the other hand, In view of subcharacteristic condition (1.5), we have
1−
We can choose a positive number such that 1−s
²(p
SCfvl+Cfulvl−O(1)δ)>0 provided 0< s < λ².
By virtue of (4.15) and (4.16), one has
¯
zr−z¯l≤0. (4.16)
The next step is to show
∆ ¯w≥g2(vl,∆¯z). (4.17)
We show this by discussing the two cases.
Case 1. ∆(S−Cfu2v)≥0.
In this case, we have
g2(vl,∆¯z)−g2(vl,∆z)
= ∂g2(vl, X)
∂X |X=θ(∆¯z−∆z)
= ∂g2(vl, X)
∂X |X=θs
²∆(S−Cfu2v), (4.18)
hereθ is between ∆z and ∆¯z. Since 0≤ ∂g2∂X(vl,X)|X=θ <1 (cf. (1.37)), we have g2(vl,∆z)≥g2(vl,∆¯z)−s
²∆(S−Cfu2v). (4.19) By virtue of (4.1) and (4.20), we obtain
∆ ¯w
= ∆w+ s
²∆(S−Cfu2v)
≥g2(vl,∆z) +s
²∆(S−Cfu2v)
≥g2(vl,∆¯z). (4.20)
(4.18) follows then in this case.
Case 2. ∆(S−Cfu2v)<0.
We have, in view of (4.9), that
∆w≥c(−∆z)≥0, (4.21)
for some c >0, if we choose δ small. Using (4.8) again, by virtue of (4.11) and (4.21), we
have, for somec1 >0 andc2 >0,
∆ ¯w
= ∆w+s
²∆(S−Cfu2v)
≥∆w+s
²(c1(−∆z)−c2∆w)
≥∆w+s
²(c1(−∆w)/c−c2∆w)
=
³ 1−s
²((c1/c) +c2)
´
∆w. (4.22)
We can chooseλ >0 such that 1−s²((c1/c) +c2)>0 provided 0< s < λ². Sinceδw≥0 (cf. (4.21)), we have δw¯ ≥ 0. On the other hand, we have form (1.37) and (4.17) that g2(vl,∆¯z)≤0, (4.18) follows then in this case, which completes the proof of this lemma.
With Lemma 4.1, we can prove Theorem 4.1 by the following argument. Suppose |us− f(vs)|(x, t) is sufficiently small for x ∈ R1 and t ≥ 0. Then by Lemma 4.1, the Riemann solutions in each time stepTn={(x, t) : x∈R1, ns < t <(n+ 1)s} (n≥0 are allR1 and S2. Thus, vs(x, t) is nondecreasing in x (cf. [22]). Moreover, since v is a parameter for the wave curves of (1.29), we have
T.V.(us(·, t)≤O(1)(T.V.(u0) +T.V.(v0)). (4.23) The smallness of|us−f(vs)|(x, t) can be verified as follows. Let u(t) = limx→−∞us(x, t) and v(t) = limx→−∞vs(x, t). By (1.41) and our scheme, we have
u(t)−f(v(t)) = 0. (4.24)
The smallness of|us−f(vs)|(x, t) thus follows from ( 4.23) and ( 4.24) if the total variation of initial data is small. Theorem 1.3 is then proved.
Once we have Theorem 1.3, the proof of theorems 1.4 and 1.5 becomes standard (cf. [18]) Acknowledgement
Part of this work was completed when Luo was at the University of Michigan. Luo’s re-search was partially supported by the Rackham Fellowship of University of Michigan. Yang’s research was supported by the Strategic Grant #7000968 of City University of Hong Kong.
References
[1] Amadori, D.; Guerra, G., Global BV solutions and relaxation limit for a system of conservation laws. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 1, 1–26.
[2] Amadori, D., Gosse, L. and Guerra, G., Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws. Arch. Ration. Mech. Anal. 162 (2002), no. 4, 327–366.
[3] Chen, G.-Q., Levermore, C.D., Liu, P.-T., Hyperbolic conservation laws with stiff relax-ation terms and entropy, Comm. Pure Appl. Math., XLVII (1994), 1-45.
[4] R. Courant and K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Pub-lishers Inc., New York, 1948.
[5] C. M. Dafermos, A system of hyperbolic conservation laws with frictional damping.
Theoretical, experimental, and numerical contributions to the mechanics of fluids and solids. Z. Angew. Math. Phys. 46 (1995), Special Issue.
[6] C. Dafermos & L. Hsiao, Hyperbolic systems of balance laws with inhomogeneity and dissipation, Indiana Univ. Math. Journal, Vo1. 31, No. 4, (1982).
[7] H. T. Fan, Self-similar solutions for a modified Broadwell model and its hydrodynamic limits, SIAM J. Math. Anal., 28 (1997) 831-851.
[8] L. Hsiao and T. Luo, Nonlinear diffusive phenomena of entropy weak solutions for a system of quasilinear hyperbolic conservation laws with damping. Quart. Appl. Math.
56 (1998), no. 1, 173–189.
[9] L. Hsiao and S. Tang, Construction and qualitative behavior of the solution of the perturbated Riemann problem for the system of one-dimensional isentropic flow with damping. J. Differential Equations 123 (1995), no. 2, 480–503.
[10] S. Jin & M. Katsoulakis. Hyperbolic systems with supercharacteristic relaxations and roll waves, SIAM J. Appl. Math. 61 (2000), no. 1, 273–292 (electronic).
[11] S. JIn & Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm. Pure Appl. Math. 48 (1995), no. 3, 235–276.
[12] J. Johnson, & J. Smoller, Global solutions of hyperbolic systems of conservation laws in two dependent variables. Bull. Amer. Math. Soc. 74 1968 915–918.
[13] T.T. Li and W.C. Yu, Boundary value problems for quasilinear hyperbolic systems.
Duke University, Durham.
[14] Lien, W.-C., Hyperbolic conservation laws with a moving source. Comm. Pure Appl.
Math. 52 (1999), no. 9, 1075–1098.
[15] Liu, T. P. , Quasilinear hyperbolic systems, Comm. Math. Phys. 68 (1979), 141-172.
[16] Liu, T.-P., Hyperbolic conservation laws with relaxation, Comm. Math. Phys., 108, 153-175(1987).
[17] T. Luo & T. Yang, Interaction of elementary waves for the compressible Euler equations with frictional damping, 161 (2000), no. 1, 42–86.
[18] T. Luo, R. Natalini and T. Yang, Global BV solutions to ap-system with relaxation. J.
Differential Equations 162 (2000), no. 1, 174–198.
[19] R. Natalini, Recent results on hyperbolic relaxation problems. Analysis of systems of conservation laws (Aachen, 1997), 128–198, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., 99, Chapman & Hall/CRC, Boca Raton, FL, 1999.
[20] S. Nishibata and S. H.Yu, The asymptotic behavior of the hyperbolic conservation laws with relaxation on the quarter-plane. SIAM J. Math. Anal. 28 (1997), no. 2, 304–321.
[21] Global solution for an initial boundary value problem of a quasilinear hyperbolic system.
Proc. Japan Acad. 44 1968 642–646.
[22] J. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, New York, 1996.
[23] W.C. Wang and Z. Xin, Asymptotic limit of initial-boundary value problems for con-servation laws with relaxational extensions. Comm. Pure Appl. Math. 51 (1998), no. 5, 505–535.
[24] G. B. Whitham, Linear and nonlinear waves, John Wiley & sons, 1974, New York.
[25] Z. Xin and W. Xu, Stiff well-posedness and asymptotic convergence for a class of linear relaxation systems in a quarter plane. J. Differential Equations 167 (2000), no. 2, 388–
437.
[26] T. Yang & C. Zhu,Existence and nonexistence of smooth solutions to p-system with relaxation, JDEs, 161, 321-336 (2000).
[27] W.A. Yong, Singular perturbations of first-order hyperbolic systems with stiff source terms. J. Differential Equations 155 (1999), no. 1, 89–132.
[28] Y.Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation. Arch. Ration. Mech. Anal. 150 (1999), no. 3, 225–279.
[29] Zhang Tong & Guo Yu-fa, A class of initial value problems for systems of aerodynamic equations. Acta Math. Sinica 15 386–396 (Chinese); translated as Chinese Math.–Acta 7 1965 90–101.