Asymptotic Behavior of Solutions to the Cauchy Problem for the Scalar Viscous Conservation Law with Partially Linearly Degenerate Flux

Decay properties of solutions toward a multiwave pattern to the Cauchy problem for the scalar conservation law with degenerate flux and viscosity

“…Furthermore, Yoshida [31] recently showed their precise decay estimates. In the case where the flux function is given as (1.4) and the far field states as (1.5), Yoshida [30] also showed that the similar asymptotics as the one in [20] which tends toward the multiwave pattern of the combination of the rarefaction waves which connect the far field states u − and u + u ± ∈ (−∞, a ] or u ± ∈ [ b, ∞) , and the viscous contact wave which connects u − and u + (u ± ∈ [ a, b ]). In more detail, the viscous contact wave is said to be "contact wave for p-Laplacian type viscosity" and explicitly given by…”

“…More generally, in the case of the flux functions which are not uniformly genuinely nonlinear, when the Riemann solution consists of a single shock wave satisfying Oleȋnik's shock condition, Matsumura-Nishihara [19] showed the asymptotic stability of the corresponding viscous shock wave. Moreover, Matsumura-Yoshida [20] considered the circumstances where the Riemann solution generically forms a complex pattern of multiple nonlinear waves which consists of rarefaction waves and waves of contact discontinuity (refer to [14]), and investigated that the case where the flux function f is smooth and genuinely nonlinear (that is, f is convex function or concave function) on the whole R except a finite interval I := (a, b) ⊂ R, and linearly degenerate on I, that is,…”

Asymptotic Behavior of Solutions Toward the Viscous Shock Waves to the Cauchy Problem for the Scalar Conservation Law with Nonlinear Flux and Viscosity

“…Under the conditions p = 1, u − < u + , (1.8) and (1.9), it has been proved by Matsumura-Yoshida [37] that the unique global in time solution to the Cauchy problem (1.1) globally tends toward the multiwave pattern of the combination of the rarefaction waves and the viscous contact wave as time goes to infinity, where the viscous contact wave is given by 11) which is constructed by the linear heat kernel (in detail, see Section 10, see also [52]). Also the precise time-decay estimates of above stability was obtained by Yoshida [46].…”

“…Proof of Lemma 2.1. The proofs of (1)-(3) in Lemma 2.1 are wellknown and given in [13], [14], [29], [33], [37], [46], and so on, instead of the decay estimate ∂ t w L r in (2). However, this estimate is immediately given by…”

“…x U r 1 and ∂ 3 x U r 1 are sharper than the estimates of ∂ 2 x U r 2 and ∂ 3 x U r 2 , respectively, although the L r -decay estimate of the difference | U r 2 − u r | is sharper than the estimate of | U r 1 − u r | (we also note that there have not been known the pointwise estimates of | U r − u r | with 1/2 < q ≤ 3/2 so far, see [29], [34], [35], [38], cf. [13], [14], [33], [37], [46], and so on). In fact, by the similar arguments as in Section 8 (in particular, we note the fifth term on the right-hand side of (8.13) which is led from the nonlinear viscous flux term σ ′ ∂ x U r 2 ∂ 2 x U r 2 ), we can obtain the time-decay estimate for ∂ x ψ L 2 correspond to (8.14) in Lemma 8.4 as…”