Convergence to Travelling Fronts of Solutions of the p -System with Viscosity in the Presence of a Boundary

Matsumura, Akitaka; Mei, Ming

doi:10.1007/s002050050134

Cited by 130 publications

(93 citation statements)

References 13 publications

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“…Because of the lack of monotonicity, the equation doesn't possess the comparison principle, and we cannot expect global stability. But we may still be able to get local stability, because the weighted energy method doesn't require the monotonicity of the equations and works for any nonmonotone equation if it possesses some viscosity or a damping or relaxation effect [3,14,20,26] …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Exponential Stability of Nonmonotone Traveling Waves for Nicholson's Blowflies Equation

Cheng

Lin

Lin³

et al. 2014

SIAM J. Math. Anal.

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Abstract. This paper is concerned with Nicholson's blowflies equation, a kind of time-delayed reaction-diffusion equation. It is known that when the ratio of birth rate coefficient and death rate coefficient satisfies 1 < p d ≤ e, the equation is monotone and possesses monotone traveling wavefronts, which have been intensively studied in previous research. However, when p d > e, the equation losses its monotonicity, and its traveling waves are oscillatory when the time-delay r or the wave speed c is large, which causes the study of stability of these nonmonotone traveling waves to be challenging. In this paper, we use the technical weighted energy method to prove that when e <

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Exponential Stability of Nonmonotone Traveling Waves for Nicholson's Blowflies Equation

Cheng

Lin

Lin³

et al. 2014

SIAM J. Math. Anal.

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“…For the initial value problems of systems of viscous conservation laws, the stability theory of viscous shock profiles was extensively studied by many authors in the past decade, see e.g., [26][27][28][29]. For the initial boundary problems, see also [30][31][32][33][34][35][36][37] for recent progress. By using the matching method, Tang-Teng [23] proved that, for convex conservation laws whose entropy solution consists of finitely many discontinuities, the L 1 -error between the viscosity solution and the inviscid solution is bounded by O(ε|lnε|), and the error bound is improved to O(ε) if there is neither central rarefaction wave nor spontaneous shock included in the inviscid solution (see also [38]).…”

Section: Introductionmentioning

confidence: 99%

Construction of Solutions and L 1–error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

Liu

Pan

2005

Acta Math Sinica

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This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux-Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type: an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L 1 -error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O(ε 1/2 ) in L 1 -norm; otherwise, as in the initial value problem, the L 1 -error bound is O(ε| ln ε|).

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“…When u+ < u-= 0, see Matsumura and Mei [6]. Here we note that the condition u_ = 0 is not necessarily assumed.…”

Section: Introductionmentioning

confidence: 99%

Global asymptotics toward the rarefaction wave for solutions of viscous 𝑝-system with boundary effect

Matsumura¹,

Nishihara²

2000

Quart. Appl. Math.

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Abstract.The initial-boundary value problem on the half-line R+ = (0, oo) for a system of barotropic viscous flow vt -ux = 0, ut + p(v)x = n{y^)x is investigated, where the pressure p(v) = v~y (7 > 1) for the specific volume v > 0. Note that the boundary value at x = 0 is given only for the velocity u, say u_, and that the initial data (vq,ilq)(x) have the constant states (u+,w+) at x = +00 with vq(x) > 0, v+ > 0. If < u+, then there is a unique i>_ such that (f+,u+) G (the 2-rarefaction curve) and hence there exists the 2-rarefaction wave uf)(x/t) connecting (u_,u_) with (v+,u+). Our assertion is that, if w_ < u+, then there exists a global solution (v,u)(t,x) in C°([0,00); Jcf1(R+)), which tends to the 2-rarefaction wave We consider the initial-boundary value problem on R+ = (0, oo) for a system of the barotropic viscous flow in the Lagrangean coordinate:

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Convergence to Travelling Fronts of Solutions of the p -System with Viscosity in the Presence of a Boundary

Cited by 130 publications

References 13 publications

Exponential Stability of Nonmonotone Traveling Waves for Nicholson's Blowflies Equation

Exponential Stability of Nonmonotone Traveling Waves for Nicholson's Blowflies Equation

Construction of Solutions and L 1–error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

Global asymptotics toward the rarefaction wave for solutions of viscous 𝑝-system with boundary effect

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