Large-time behaviour of the entropy solution of a scalar conservation law with boundary conditions

AbstractWe consider a scalar conservation law with zero-flux boundary conditions imposed on the boundary of a rectangular multidimensional domain. We study monotone schemes applied to this problem. For the Godunov version of the scheme, we simply set the boundary flux equal to zero. For other monotone schemes, we additionally apply a simple modification to the numerical flux. We show that the approximate solutions produced by these schemes converge to the unique entropy solution, in the sense of [7], of the conservation law. Our convergence result relies on a BV bound on the approximate numerical solution. In addition, we show that a certain functional that is closely related to the total variation is nonincreasing from one time level to the next. We extend our scheme to handle degenerate convection-diffusion equations and for the one-dimensional case we prove convergence to the unique entropy solution.

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Regularity and large time behaviour of solutions of a conservation law without convexity

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500014256 ◽

1985 ◽

Vol 99 (3-4) ◽

pp. 201-239 ◽

Cited By ~ 52

Author(s):

C. M. Dafermos

Keyword(s):

Initial Data ◽

Conservation Law ◽

Large Time ◽

Decay Rates ◽

Time Behaviour ◽

Large Time Behaviour ◽

Jump Discontinuities ◽

Contact Discontinuities ◽

Finite Set ◽

Line Characteristic

SynopsisUsing the method of generalized characteristics, we discuss the regularity and large time behaviour of admissible weak solutions of a single conservation law, in one space variable, with one inflection point.We show that when the initial data are C∞ then, generically, the solution is C∞ except: (a) on a finite set of C∞ arcs across which it experiences jump discontinuities (genuine shocks or left contact discontinuities); (b) on a finite set of straight line characteristic segments across which its derivatives of order m, m = 1, 2,…, experience jump discontinuities (weak waves of order m); and (c) on the finite set of points of interaction of shocks and weak waves. Weak waves of order 1 are triggered by rays grazing upon contact discontinuities. Weak waves of order m, m ≥ 2, are generated by the collision of a weak wave of order m − 1 with a left contact discontinuity.We establish sharp decay rates for solutions with initial data of the following types: (a) with bounded primitive; (b) with primitive having sublinear growth; (c) in L1; (d) of compact support; and (e) periodic.

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Large-time behavior of solutions to a scalar conservation law in several space dimensions

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1986-0857450-6 ◽

1986 ◽

Vol 298 (1) ◽

pp. 401-401 ◽

Cited By ~ 6

Author(s):

Patricia Bauman ◽

Daniel Phillips

Keyword(s):

Conservation Law ◽

Large Time ◽

Large Time Behavior ◽

Time Behavior ◽

Scalar Conservation Law ◽

Scalar Conservation

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A NONLOCAL CONSERVATION LAW WITH NONLINEAR "RADIATION" INHOMOGENEITY

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891608001465 ◽

2008 ◽

Vol 05 (01) ◽

pp. 1-23 ◽

Cited By ~ 9

Author(s):

MARCO DI FRANCESCO ◽

KLEMENS FELLNER ◽

HAILIANG LIU

Keyword(s):

Conservation Law ◽

Radiation Effects ◽

Large Time ◽

Heat Radiation ◽

Nonlinear Radiation ◽

Scalar Conservation Law ◽

Threshold Conditions ◽

Global Existence And Uniqueness ◽

Self Similar ◽

Scalar Conservation

A scalar conservation law with a nonlinear dissipative inhomogeneity, which serves as a simplified model for nonlinear heat radiation effects in high-temperature gases is studied. Global existence and uniqueness of weak entropy solutions along with L1 contraction and monotonicity properties of the solution semigroup is established. Explicit threshold conditions ensuring formation of shocks within finite time is derived. The main result proves — under further assumptions on the nonlinearity and on the initial datum — large time convergence in L1 to the self-similar N-waves of the homogeneous conservation law.

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