Stability and Total Variation Estimates on General Scalar Balance Laws
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Stability and Total Variation Estimates on General Scalar Balance Laws Rinaldo M. Colombo, Magali Mercier?and Massimiliano D. Rosini† Department of Mathematics, Brescia University Via Branze 38, 25133 Brescia Italy October 28, 2008 Abstract Consider the general scalar balance law ∂tu+Divf(t, x, u) = F (t, x, u) in several space dimensions. The aim of this note is to estimate the dependence of its solutions from the flow f and from the source F . To this aim, a bound on the total variation in the space variables of the solution is obtained. This result is then applied to obtain well posedness and stability estimates for a balance law with a non local source. 2000 Mathematics Subject Classification: 35L65. Keywords: Multi-dimensional scalar conservation laws, Kruzˇkov entropy solutions. 1 Introduction The Cauchy problem for a scalar balance law in N space dimensions { ∂tu+Divf(t, x, u) = F (t, x, u) (t, x) ? R+ ? RN u(0, x) = uo(x) x ? RN (1.1) is well known to admit a unique weak entropy solution, as proved in the classical result by Kruzˇkov [12, Theorem 5]. The same paper also provides the basic stability estimate on the dependence of solutions from the initial data, see [12, Theorem 1].
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- provide stability
- consider now
- uo ?
- key intermediate result
- balance law
- multi-dimensional scalar
- conservation law
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Stability and Total Variation Estimates on General Scalar Balance Laws
Rinaldo M. Colombo, Magali Mercier∗and Massimiliano D. Rosini† Department of Mathematics, Brescia University Via Branze 38, 25133 Brescia Italy
October 28, 2008
Abstract Consider the general scalar balance law∂tu+ Divf(t x u) =F(t x u) in several space dimensions. The aim of this note is to estimate the dependence of its solutions from the flowfand from the sourceF this aim, a bound . Toon the total variation in the space variables of the solution is obtained. This result is then applied to obtain well posedness and stability estimates for a balance law with a non local source. 2000 Mathematics Subject Classification:35L65. Keywords:nertkzvorKˇuwa,s.ionsolutopysdii-nsmenaiocalscralesnotavrlnoiuMtl
Introduction
The Cauchy problem for a scalar balance law inNspace dimensions (u∂t(u0+xv=iD)fuo((tx)x u) =F(t x u) (txx)∈∈RRN+×RN(1.1) is well known to admit a unique weak entropy solution, as proved in the classical result byKruˇzkov[12,Theorem5].Thesamepaperalsoprovidesthebasicstabilityestimateon the dependence of solutions from the initial data, see [12, Theorem 1]. In the same setting established in [12], we provide here an estimate on the dependence of the solutions to (1.1) from the flowf, from the sourceFand recover the known estimate on the dependence from the initial datumuo. A key intermediate result is a bound on the total variation of the solution to (1.1), which we provide in Theorem 2.5. In the case of a conservation law, i.e.F= 0, and with a flowfindependent fromt x, the dependence of the solution fromfwas already considered in [3], where also other results were presented. In this case, the TV bound is obvious, since TVu(t)≤TV(uo). The estimate provided by Theorem 2.5 slightly improves the analogous result in [3, Theorem 3.1] (that was already known, see [6, 16]), which reads (for a suitable absolute constantC) u(t)−v(t)L1(RN;R)≤ kuo−vokL1(RN;R)+CTV(uo)Lip(f−g)t ∗n11umevo1erb;81962692deLeoy,nnUvireist´eLyon1;43,bd.drePdatnenamUns:esdrt´sieriv Villeurbanne Cedex †Supported by INdAM.
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R.M. Colombo, M. Mercier & M.D. Rosini
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Our result, given by Theorem 2.6, reduces to this inequality whenfandgare not dependent ont xandF=G= 0, but withC= 1. A flow dependent also onxwas considered in [4, 9], though in the special casef(x u) = l(x)g(u), but with a source term containing a possibly degenerate parabolic operator. There, estimates on theL1distance between solutions in terms of the distance between the flows were obtained, but dependent from ana prioriunknown bound on TVu(t). Here, with no parabolic operators in the source term, we provide fully explicit bounds both on TVu(t)and on the distance between solutions. Indeed, remark that with no specific assumptions on the flow, TVu(t)may well blow up to +∞att= 0+, as in the simple casef(x u) = cosxwith zero initial datum. Both the total variation and the stability estimates proved below turn out to be optimal in some simple cases, in which optimal estimates are known. As an example of a possible application, we consider in Section 3 a toy model for a radiating gas. This system was already considered in [5, 8, 10, 11, 13, 14, 15, 17]. It consists of a balance law of the type (1.1), but with a source that contains also a non local term, due to the convolution of the unknown with a suitable kernel. Thanks to the present results, we prove the well posedness of the model extending [8, Theorem 2.4] to more general flows, sources and convolution kernels. Stability and total variation estimates are also provided. This paper is organized as follows: in Section 2, we introduce the notation, state the main results and compare them with those found in the literature. Section 3 is devoted to an application to a radiating gas model. Finally, in sections 4 and 5 the detailed proofs of theorems 2.5 and 2.6 are provided.
2 Notation and Main Results DenoteR+= [0+∞[ andR+= ]0+∞[. Below,Nis a positive integer, Ω =R+×RN×R, B(x r) denotes the ball inRNwith centerx∈RNand radiusr >0. The volume of the unit ballB(01) isωN notational simplicity, we set. Forω0 following relation The= 1. can be proved using the expression ofωNin terms of the Wallis integralWN: ωωN−1= 2WNwhereWN=Z0π2(cosθ)Ndθ (2.1) N
In the present work,1Ais the characteristic function of the setAandδtis the Dirac measure centered att for a vector valued function. Besides,f=f(x u) withu=u(x), Divf On the other hand, divstands for the total divergence.f, respectively∇f, denotes the partial divergence, respectively gradient, with respect to the space variables. Moreover, ∂uand∂tare the usual partial derivatives. Div Thus,f= divf+∂uf ∇u. Recall the definition of weak entropy solution to (1.1), see [12, Definition 1]. Definition 2.1A functionu∈L∞(R+×RN;R)is a weak entropy solution to (1.1) if: 1. for any constantk∈Rand any test functionϕ∈Cc∞(R+×RN;R+) ZR+ZRNh(u−k)∂tϕ+f(t x u)−f(t x k) ∇ϕ+F(t x u)−divf(t x k)ϕi(2.2) ×sign(u−k) dxdt≥0;
