18 Adimurthi & G. D. Veerappa Gowda & J. Jaffr´e
family includes the Force **and** proposes an alternative **for** the Musta **schemes**. **For** all these **schemes** we proved convergence **of** the approximate solution to the **entropy** solution **of** the continuous problem. We also gave hints on how to extend them to systems **and** high resolution **schemes**. In forthcoming papers we will extend these **schemes** to higher resolution **schemes** **and** to the discontinuous **flux** case. We will also give an example **of** application to a 2 × 2 system **of** **conservation** **laws** representing a problem **of** polymer flooding.

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unknown **for** some triangular systems **of** **conservation** **laws** e.g., a pressure swing adsorption system. We answer this question **for** a class **of** 1-D triangular systems **and** the multi-D Keyfitz-Kranzer system in section 5 **and** 6 respectively.
To provide an answer to the question ( Q ) we recall some **of** the previously constructed examples [ 2 , 3 , 4 ]. In Section 3 , Theorem 3.1 provides the direct answer to ( Q ) **for** a power-law type **flux** function f (u) = (1 + p) −1 |u| 1+p . We have discussed before that convex **flux** function with p- degeneracy (i.e., satisfying ( 5 )) gives a regularizing effect in BV s with s = 1/p [ 13 ]. We construct an **entropy** solution u to ( 1 ) such that T V s+ε (u(·, t)) = ∞ **for** all t > 0 **and** ε > 0 with s = 1/p. Following the constructions in [ 3 , 4 ] we build the **entropy** solution u, consisting infinitely many shock profiles in a compact interval. These shock profiles are named Asymptotically Single Shock Packet (ASSP) in [ 2 ]. Loosely speaking an ASSP is a solution with a special structure between two parallel lines in the half plane R x × R + t such that in large time only one shock curve appears

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y 0 = 0. (14)
Denote by R the standard (i.e., without the constraint ( 1c )) Riemann solver **for** ( 1a )-( 1b )-( 14 ), i.e., the (right continuous) map (t, x) 7→ R(ρ L , ρ R )(x/t) given by
the standard weak **entropy** solution, see **for** instance [ 7 , Chapter 5]. Moreover, assume that ˙ y is constant **and** let ˇ ρ = ˇ ρ( ˙ y) **and** ˆ ρ = ˆ ρ( ˙ y), with ˇ ρ ≤ ˆ ρ, be the intersection points **of** the **flux** function f (ρ) with the line f α (ρ α ) + ˙ y(ρ − ρ α ) (see

Riemann problems associated to this system when gravity is neglected **and** therefore the fractional flow function is an increasing function **of** the unknown. In this case, the eigenvalues **of** the corresponding Jaco- bian matrix are positive **and** hence it is less difficult to construct Godunov type **schemes** which turn out to be upwind **schemes**. When the above model with gravity effects is considered, then the **flux** function is not necessarily monotone **and** hence the eigenvalues can change sign. This makes the construction **of** Godunov type **schemes** more difficult as it involves exact solutions **of** Riemann problems with a nonmonotonous fractional flow function. Therefore in section 3 we solve the Riemann problems in the general case when gravity terms are taken into account so the **flux** function is not anymore monotone. This will allow to compare our method with that using an exact Riemann solver. In section 4 we consider Godunov type finite volume **schemes**. We present the DFLU scheme **for** the system **of** polymer flooding **and** compare it to the Godunov scheme whose **flux** is given by the exact solution **of** the Riemann problem. We also present several other possible **numerical** fluxes, centered like Lax-Friedrichs or Force, or upstream like the upstream mobility **flux** commonly used in reservoir engineering [4, 5, 16]. Finally in section 5 we compare numerically the DFLU method with these fluxes.

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1I−ε≤x≤ε + kR1Iε<x, ε > 0.
The kinetic formulation **of** scalar **conservation** **laws** is well adapted to the analy- sis **of** the (Perthame-Tadmor) BGK approximation **of** scalar **conservation** **laws**. Developed in [PT91], this equation is a continuous version **of** the Transport- Collapse method **of** Brenier [Bre81, Bre83]. BGK models have also been used **for** gas dynamics **and** the construction **of** **numerical** **schemes**. See **for** example the book **of** Perthame [Per02] **for** a survey **of** this field.

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while in [ 55 , 57 ], the uniqueness **of** weak solutions is obtained directly from the fixed point theorem, without prescribing any kind **of** **entropy** condition.
The **numerical** approximation **of** the solutions **of** non-local models is challenging due to the high non-linearity **of** the system **and** the dependence **of** the **flux** function on convolution terms, which highly impacts the computational cost. There are different ways to numerically integrate non-local **conservation** **laws**. In [ 2 , 5 , 10 , 11 , 49 ] a first order Lax-Friedrichs-type **numerical** scheme is used to approximate the problem **and** to prove the existence **of** solutions. Indeed, in this setting, the **numerical** scheme is not just a tool **for** numerically investigating the behaviour **of** solutions, but it is important from the analytical point **of** view, because it allows to construct a sequence **of** approximate solutions in order to apply an adapted version **of** Helly’s theorem, see [ 44 ]. Another first order **numerical** scheme is proposed in [ 46 ], where a Godunov-type **numerical** scheme **for** a specific class **of** non-local **flux** problems is presented. This scheme allows **for** physically reasonable solutions, meaning that negative velocities as well as negative fluxes are avoided. Moreover, in comparison with Lax-Friedrichs scheme, Godunov scheme is less diffusive. Concerning high-order **numerical** **schemes**, it is worth citing the papers [ 19 , 45 ]. In [ 19 ], the authors propose discontinuous Galerkin **and** finite volume WENO **schemes** to obtain high-order approximations **of** non-local scalar **conservation** **laws** in one space dimension, where the velocity function depends on a weighted mean **of** the conserved quantity. The discontinuos Galerkin **schemes** give the best results, but under a very restrictive Courant-Friedrichs-Lewy (CFL) condition. On the contrary, finite volume WENO **schemes** can be implemented on larger time steps. In [ 45 ], central WENO **schemes** are proposed, which, in contrast to the other high-order **schemes** **for** non-local **conservation** **laws**, neither require a restrictive CFL condition nor an additional reconstruction step.

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u t + (f (u)) x = 0 **for** (t, x) ∈ R + × R, (1.3)
where the **flux** function f is regular from Ω to R n . Typically, t is the
time **and** x is the position.
The general problem **of** controllability is the following. Consider the problem posed in the interval [0, 1] rather than in R. In such a case one needs **of** course to prescribe boundary conditions on [0, T ] × {0, 1}: here boundary conditions will be considered as a control, that is, a way to influence the system to make it behave in a prescribed way. Let us call u(t, ·) the state **of** the system at time t. The question is: given two possible states **of** the system, say u 0 **and** u 1 , can we choose the

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Stability estimates **for** non-local scalar **conservation** **laws**
Felisia Angela Chiarello 1 Paola Goatin 1 Elena Rossi 1
Abstract
We prove the stability **of** **entropy** weak solutions **of** a class **of** scalar **conservation** **laws** with non-local **flux** arising in traffic modelling. We obtain an estimate **of** the dependence **of** the solution with respect to the kernel function, the speed **and** the initial datum. Stability is obtained from the **entropy** condition through doubling **of** variable technique. We finally provide some **numerical** simulations illustrating the dependencies above **for** some cost functionals derived from traffic flow applications.

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1. Introduction. The **numerical** approximation **of** the weak solutions **of** hyper- bolic systems **of** **conservation** **laws** was widely studied during the last three decades, with a special attention to the so-called Euler equations. Several strategies, coming from finite volume methods, have been introduced. Our purpose is not to detail these techniques, but let us refer to the most famous **of** them: the Godunov scheme [18, 25], the HLL scheme [21], the HLLC scheme [31], the Roe scheme [27], the Osher scheme [13], the relaxation **schemes** [7, 22, 12, 3, 1], the VFRoe scheme [9, 5, 17, 26], BGK scheme [18, 23, 8]... **Of** course, this list stays exhaustive **and** the reader is referred, **for** instance, to Godlewski-Raviart [18] or Toro [32] or LeVeque [25] **and** references therein. These kind **of** **schemes** have also been applied to general fluid equations, **for** instance to 10-moment equations system [3, 28] or radiative transfer equations [4].

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A weak solution **of** (1.1) is called an **entropy** satisfying solution if **and** only if the **entropy** inequality (1.4) holds **for** any pair **entropy**-**entropy** **flux** (η, G). Integrating in space the **entropy** inequality (1.4) results in a global **entropy** stability inequality,
d dt

1 x ≥ 1,
with zero-**flux** boundary conditions in the interval [0, 1], **and** compute the solution at time T = 1, with parameters α = 1, n = 3 **and** a = 0.025. We set ∆x = 1/400 **and** compute the solutions with different RKDG **and** FV-WENO **schemes**, including the first-order Lax-Friedrichs scheme used in [8]. The results displayed in Fig. 3 are compared to a reference solution computed with FV-WENO5 **and** ∆x = 1/3200. Compared to the reference solution, we observe that RKDG1 is more accurate than FV-WENO3, **and** FV-WENO5 more accurate than RKDG3 **and** RKDG2 (Fig. 3). In particular, we observe that the **numerical** solutions obtained with the high-order **schemes** provide better approximations **of** the oscillatory shape **of** the solution than the first order scheme. These oscillations can possibly be explained as layering phenomenon in sedimentation [29], which denotes a traveling staircases pattern, looking as several distinct

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In the present paper, we characterize **and** approximate **entropy** solutions **of** ( 1 - 3 ) in the L ∞ setting. The core **of** the paper is the convergence analysis **for**
finite volume **schemes** adapted to the constrained problem ( 1 - 3 ). The **schemes** are constructed as follows. We consider a classical monotone three-point finite volume scheme (see [ EGH00 ]) **and** denote by g(u, v) the associated **numerical** **flux**; at the interface **of** the mesh which corresponds to the obstacle position {x = 0}, the **numerical** **flux** is replaced by min(g(u, v), F ) in order to comply with the constraint ( 3 ) (see Section 4 **for** more details). Our approach is simpler than the wave-front tracking algorithm devised in [ CG07 ], because we do not need to define explicitly the Riemann solvers at the interface {x = 0} which would fit the constraint at time t. Notice that with our approach, existing finite volume codes **for** the non-constrained **conservation** law ( 1 ) are trivially combined with the constraint ( 3 ).

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All these works use at some point tools from harmonic analysis: Fourier trans- form, Hardy-Littlewood decomposition, Radon transform. Moreover, in all these results, the regularity **of** moments in v **of** the function f depends on γ.
In this section, we give an example **of** velocity averaging result where the reg- ularity **of** the moments in v **of** f is independent **of** γ, at the expense **of** an extra assumption on the v dependence in f . Also, the proof **of** this result is based on the interaction identities (7)-(8) **and** uses only elementary techniques in physical space. Theorem 3.1. Let a ∈ C 1 (R); assume that there exists β ≥ 1 such that, **for** each M > 0, there exists αM > 0 **for** which

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In this paper, the exponential convergence in L 2 -norm is analyzed **for** a class **of** switched linear systems **of** **conservation** **laws**. The boundary conditions are subject to switches. We investigate the problem **of** synthesizing stabilizing switching controllers. By means **of** Lyapunov techniques, three control strategies are developed based on steepest descent selection, possibly combined with a hysteresis **and** a low-pass filter. **For** the first strategy we show the global exponential stabilizability, but no result **for** the existence **and** uniqueness **of** trajectories can be stated. **For** the other ones, the problem is shown to be well posed **and** global exponential convergence can be obtained. Moreover, we consider the robustness issues **for** these switching rules in presence **of** measurement noise. Some **numerical** examples illustrate our approach **and** show the merits **of** the proposed strategies. Particularly, we have developped a model **for** a network **of** open channels, with switching controllers in the gate operations.

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Due to the non-locality **of** the potential, J P (r, t) involves contributions from the two-
body function f 2 taken at points located on each side **of** r, at distances **of** the order
**of** the range **of** W .
The above results **for** the local balance **of** momentum can alternatively be derived through Noether’s approach. The **conservation** **of** the total momentum results from the invariance **of** the Lagrangian under the translation r j 7→ r j + δr j by a small

tinuous with respect to x, several criteria are possible [1]. **For** B(x, u) = k(x)A(u) as above, the choice **of** **entropy** solution is unambiguous [1, remark 4.4] **and** we consider here the selection criterion ﬁrst given in [13]. A kinetic formulation (in the spirit **of** [9]) equivalent to the **entropy** formulation in [13] has been given in [3]. In particular, solutions given by this criterion are limits (almost everywhere (a.e.) **and** in L 1 ) **of** the solutions obtained by monotone regularization **of** the coeﬃcient k in

During the completion **of** this paper, we became aware **of** a study by Y.-J. Kim, who kindly communicated us a preliminary version **of** his work [9]. His approach is based on a detailed study **of** special self-similar solutions. Y.-J. Kim obtains a sharp L 1 -norm
convergence rate **and** we do not. Nevertheless, we believe that our method, based on qualitative results **and** global integral estimates, **and** our results on the convergence in weighted norms are **of** interest.

Chapter 1
Cartan’s structure equations
The goal **of** this chapter is to introduce the fundamental notions **of** differential geometry expressed in moving frames, **and** to establish Cartan’s structure equations. Almost everything revolves around differential forms, **and** thus is expressed in the Cartan formalism. The first section is dedicated to introducing **and** defining the notion **of** a connection **and** its curvature on an arbitrary vector bundle, where the dimension **of** the fiber is not necessarily equal to the dimension **of** the base manifold, **and** the relationship between the connection **and** its curvature is given by Cartan’s second-structure equation. Also shown is an interesting property **of** the connection when the vector bundle is endowed with a metric, **and**, as always in differential geometry, all **of** the transformation rules **for** these objects will be demonstrated. In the second section, a special **and** fundamental class **of** vector bundles is investigated: the tangent bundle **of** a differentiable manifold. The notion **of** torsion **of** a connection appears **and** leads to Cartan’s first-structure equation. An important technical result, the Cartan lemma, is stated because it is useful in several applications. The proofs from the first **and** second sections are given in appendix 1 not only because the calculations are not that difficult **and** almost all **of** the results derive from the definitions, but also to lighten the reading. Finally, since Cartan’s structure equations are ”intensely” used from chapters 3 to 6, the modest purpose **of** section 3 is to pro- vide some useful applications to Cartan’s structure equations in the study **of** surfaces, such as computing Christoffel symbols, computing the Gauss curvature **and** presenting a problem studied by Poincaré pertaining to the conformal metrics **of** constant curvature.

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