Asymptotic Time-Behaviour of Solutions to Scalar Conservation Law with a Convex Flux Function

We consider the long-time behaviour of solutions of the Cauchy problem for a quasilinear equation ( ) 0 t x u f u + = with a strictly convex flux function ( ) f u and initial function ( ) 0 u x having the one-sided limiting mean values u that are uniform with respect to translations. The estimates of the rates of convergence to solutions of the Riemann problem depending on the behaviour of the integrals ( ) ( ) 0 a y a u x u dx + ± − ∫ as y →±∞ are established. The similar results are obtained for solutions of the mixed problem in the domain 0, 0 x t > > with a constant boundary data u and initial data having limiting mean value u .


Introduction
In the half-plane   1 , 0 x R t ∈ > (1) with the initial condition ( ) The articles concerning the asymptotic properties of the generalized entropy solutions of the problem (1), (2) for large values of time are reviewed in [1]. New interesting results are obtained in the last years for the case of non-convex flux function ( ) f u (see [2], [3], [4] for example).
It is assumed throughout this paper that the function ( ) there exists a constant 0 С > such that for a.e. 1 1 A generalized solution of problem (1), (2) in the sense of this definition is unique and exists for any bounded measurable initial function ( ) 0 u x (see [4]).

Cauchy problem
Assume that the initial function ( ) 0 u x has the one-sided limiting mean values that are uniform with respect to translations: The following theorems 1 and 2 are proved in [8]. .
In this paper the rates of convergence of the solution ( ) 0 const 1, for t T > and Remark. P.-D.Lax [6] in the case u u u Dafermos [7] proved, that if The proofs of the theorems 1-3 are based on the explicit representation of the generalized solutions of the problem (1), (2) (see [5], [6]). Assume at first that Here the function This set is compact, because the functional ( )

, , I t x y
is continuous for all its arguments and ( ) Lemma( [5], [6]). If 1 y t x is right semi-continuous in x and booth functions have one and the same no more than countable set of points of discontinuity and coincide everywhere outside this set.
x u x x is the solution of the equation , 0 , 0 as is proved in [4] , and x R ∈ (see [1]).

Mixed problem
The mixed problem is set up in [8] as follows : The condition (28) is the entropy condition.
The existence and uniqueness of the solution ( ) , u t x of the problem (25)-(28) are established in [8].
Assume that (4) holds, S t x are defined in (5).

6.
Let u u − + < . Then  In the case 0 u + = and 1 q = this assertion is stated in [9]. This work was carried out using equipment provided by the Center of Collective Use of MSUT "STANKIN".