Decay properties of solutions to the Cauchy problem for the scalar conservation law with nonlinearly degenerate viscosity

Abstract

In this paper, we study the decay rate in time to solutions of the Cauchy problem for the one-dimensional viscous conservation law where the far field states are prescribed. Especially, we deal with the case that the flux function which is convex and also the viscosity is a nonlinearly degenerate one (p-Laplacian type viscosity). As the corresponding Riemann problem admits a Riemann solution as the constant state or the single rarefaction wave, it has already been proved by Matsumura-Nishihara that the solution to the Cauchy problem tends toward the constant state or the single rarefaction wave as the time goes to infinity. We investigate that the decay rate in time of the corresponding solutions. Furthermore, we also investigate that the decay rate in time of the solution for the higher order derivative. These are the first result concerning the asymptotic decay of the solutions to the Cauchy problem of the scalar conservation law with nonlinear viscosity. The proof is given by L 1 , L 2 -energy and time-weighted L q -energy methods.

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

“…31) Substituting (5.29), (5.30) and (5.31) into (5.27), we complete the proof of Proposition 5.3. In particular, we have…”

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

“…Therefore, we finally show Lemma 8.4 by applying the arguments in Yoshida [47], [50]. Taking q = 1 to (8.13), we get…”

“…The timedecay results, Theorems 3.5-3.8 are obtained by the technical time-weighted energy method (cf. [46], [47], [49]) with the help of the uniform boundedness of . Moreover, in order to obtain the time-decay estimates for the derivatives, Theorem 3.8, we apply the bootstrap argument by Yoshida [47], [49] (see the process (8.16)-(8.23)).…”

“…[46], [47], [49]) with the help of the uniform boundedness of . Moreover, in order to obtain the time-decay estimates for the derivatives, Theorem 3.8, we apply the bootstrap argument by Yoshida [47], [49] (see the process (8.16)-(8.23)). Since the proof of Theorem 3.7 is similarly given as the proofs of Theorems 3.5 and 3.6, we only show Theorems 3.5, 3.6 and 3.8 in Sections 6-9.…”

Asymptotic behavior of solutions toward the rarefaction waves to the Cauchy problem for the generalized Benjamin-Bona-Mahony-Burgers equation with dissipative term

“…where p > 1 and the viscosity µ | ∂ x u | p−1 ∂ x u is the so-called Ostwald-de Waeletype viscosity advocated by de Waele [9] and Ostwald [43] (which is a typical example for the non-Newtonian viscosity, see also [6], [7], [8], [19], [25], [30], [31], [48] and so on), Matsumura-Nishihara [33] first investigated and proved the global stability of the rarefaction wave by using the technical energy method. Yoshida [54] further obtained its precise time-decay estimates by using the time-weighted energy method (for the stabilities of the multiwave pattern, see [55], [56], [57]). Furthermore, Matsumura-Yoshida [36] considered the following Cauchy problem of the non-Newtonian viscous conservation law…”