0 (T) with small radius whenever this noise is small over sufficiently long time intervals. On the
other hand, the L 1
0 (T)-contraction property ensures that when these two solutions get close to one another
they stay close forever. Hence, each time the noise gets small enough, the two solutions get closer and closer and eventually, they show the same asymptotical behaviour. This idea allows to show that the **law** of two solutions have the same limit as the time goes to infinity. Therefore, starting from two invariant measures leads to the equality of these measures. The same kind of argument was used in [14] for the invariant measure of kinetic solutions of inviscid **scalar** **conservation** laws and in [12] for the **stochastic** Navier-Stokes equations. Let (u(t)) t≥0 and (v(t)) t≥0 be two solutions of (1) driven by the same Q-Wiener process (W Q (t)) t≥0 . For

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i . Two main approaches are actually possible to get this **scalar** conser- vation **law** from ( 1.0.4 ). The initial one in [ 18 ] is based on a use of BV compactness arguments in the multidimensional case and compensated compactness arguments in the one dimensional case, while the second one introduced later in [ 16 ] relies on what is called a kinetic formulation. It consists in obtaining at the macroscopic level a kinetic equation involving a density-like function whose velocity distribution is the equilibrium density.

Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex France Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - [r]

1 Universit´ e Paris-Est, CERMICS, 6 et 8 avenue Blaise Pascal, 77455 Marne-La-Vall´ ee Cedex 2,
France
Abstract
In this paper, we are interested in approximating the solution to **scalar** **conservation** laws using systems of interacting **stochastic** particles. The **scalar** **conservation** **law** may involve a fractional Laplacian term of order α ∈ (0, 2]. When α ≤ 1 as well as in the absence of this term (inviscid case), its solution is characterized by entropic inequalities. The probabilistic interpretation of the **scalar** **conservation** is based on a **stochastic** differential equation driven by an α-stable process and involving a drift nonlinear in the sense of McKean. The particle system is constructed by discretizing this equation in time by the Euler scheme and replacing the nonlinearity by interaction. Each particle carries a signed weight depending on its initial position. At each discretization time we kill the couples of particles with opposite weights and positions closer than a threshold since the contribution of the crossings of such particles has the wrong sign in the derivation of the entropic inequalities. We prove convergence of the particle approximation to the solution of the **conservation** **law** as the number of particles tends to ∞ whereas the discretization step, the killing threshold and, in the inviscid case, the coefficient multiplying the stable increments tend to 0 in some precise asymptotics depending on whether α is larger than the critical level 1.

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the X j ’s being defined by the ODEs (2.1), or the **stochastic** version (2.3).
Strictly speaking, the solution of (2.8) is not of the form (1.3), since it is represented by a convolution with a kernel in time as well as in space. In the nu- merical experiments that follow, we still use the physically relevant form (2.8), even though we restrict the analysis in Section 4 to the case where the signal P has the form (1.3).

Our analysis is strongly connected to the standard theory of hyperbolic PDEs: our FBSDE model appears as a **stochastic** perturbation of a first-order **conservation** **law**. This connection plays a key role in the analysis: the relaxed notion of solvability and the pathological behavior of the solution at the terminal time are consequences of the shock observed in the corresponding **conservation** **law**. In particular, we spend a significant amount of time in determining how the system feels the deterministic shock. Specifically, we compare the intensity of the noise plugged into the system with the typical energy required to escape from the trap resulting from the shock: because of the degeneracy of the forward equation, it turns out that the noise plugged into the system is not strong enough to avoid the collateral effect of the shock. Put differently, the non-standard notion of solution follows from the concomitancy of the degeneracy of the forward equation and of the singularity of the terminal condition.

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with K = ¯ u − 3h 1 ∇ H h 3 ∇ H · ¯ u = ¯ u + 3h 1 ∇ H h 2 Dh Dt . This **conservation** **law** is obtained as an approximation of a general exact balance **law** (22) at the free surface for the full Euler equations in the case of potential motions. Since both (22) and (30) also hold in the the case of rotational flow, a natural question is if the asymptotic equivalence can be extended to this more general case. One possible approach would be to use ideas from [14] where a fully nonlinear long-wave system is derived in the case of non-potential flow. In this approximation, the velocity field below the surface is governed by an additional evolution equation, and this equation could be the basis for finding an approximation to the velocity field in the rotational case. However, this is beyond the scope of the present article.

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from (3.34), since ∆x = ∆t λ and taking λ = 3 L 1 .
Appendix A The local 1D IBVP
We recall below some results concerning the classical (local) one dimensional initial boundary value problem for a **scalar** **conservation** laws. Detailed proofs can be found in [14], which deals with the more general case of a balance **law**.

+ k ∞ k(f 00 ) − k ∞ is replaced
with k(f 00 ) − k ∞ or with k(f 00 ) + k ∞ (or even with any factor of the type O(k(f 00 ) − k ∞ ) or O(k(f 00 ) + k ∞ )).
The cited results in [3], [11] and [13] rely on the theory of generalized characteristics due to Dafermos [6]. Here we rely on a process of selection of binary trees inside front-tracking approximations of the solution of (1)-(2). We recall that the front-tracking method was introduced by Dafermos as well [4]; here we use a version of the algorithm which is an adaptation of a method due to LeFloch and the author [8] (for general hyperbolic systems of **conservation** laws). This method relies on the so-called inner speed variation estimates (also introduced in [8]), which are central in the proof here.

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On peut modéliser cette situation d’un point de vue macroscopique par un système couplant une EDP avec une EDO. L’EDP est une loi de **conservation** scalaire avec une contrainte mobile sur la densité et l’EDO décrive la trajectoire du véhicule plus lent.
Le bus se déplace avec une vitesse qui dépend du trafic routier, c’est-à-dire que le bus voyage à une vitesse constante tant qu’il n’est pas ralenti par les conditions de circulation en aval. Lorsque cela se produit, il adapte sa vitesse à la vitesse moyenne du trafic environnant. A son tour, la circulation est modifiée par la présence du véhicule plus lent. Il y a donc un couplage fort et non trivial.

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[19] C. M. Dafermos, Hyperbolic **conservation** laws in continuum physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2000.
[20] J. I. Diaz and L. V´ eron, Existence theory and qualitative properties of the solutions of some first order quasilinear variational inequalities, Indiana Univ. Math. J., 32 (1983), pp. 319–361. [21] A. F. Filippov, Differential equations with discontinuous righthand sides, vol. 18 of Mathemat- ics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1988. Translated from the Russian.

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(Communicated by the associate editor name)
Abstract. We study well-posedness of **scalar** **conservation** laws with moving flux constraints. In particular, we show the Lipschitz continuous dependence of BV solutions with respect to the initial data and the constraint trajectory. Applications to traffic flow theory are detailed.

[4] Adimurthi, Shyam Sundar Ghoshal, G.D.Veerappa Gowda, Finer analysis of characteristic curves, and its applications to shock profile, exact and optimal controllability of **conservation** **law** with discontinuous fluxes - preprint , 2011.
[5] Adimurthi, G.D. Veerappa Gowda,, **Conservation** **Law** with discontinuous flux, J.Math, Kyoto Univ, 43, no.J , 2003 , 27-70.

[LPT94] P.-L. Lions, B. Perthame, and E. Tadmor, A kinetic formulation of
multidimensional **scalar** **conservation** laws and related equations, J.
Amer. Math. Soc. 7 (1994), no. 1, 169–191.
[Pan08] E. Panov, Generalized solutions of the Cauchy problem for a transport

Remark 1.2. The kinetic formulation of **scalar** **conservation** laws with discontinu- ous spatial dependence of the form ∂ t u + ∂ x (B(x, u)) = 0 (which are more general
than (1.4)) is derived in the last chapter of [2]. We indicate (this would have to be proved rigorously) that, in the case where our approach via the BGK approximation was applied to this problem, the solutions obtained would be the type of entropy solutions considered in [7].

Let u := u ε be the Krushkov solution of (1.1)–(1.2). Consider the geometric optics asymptotic expansion of the solution (1.4). Then the new approach in Section 4 for one-dimensional **conservation** laws requires further refinement for solving the general nonlinear geometric optics for multidimensional **scalar** **conservation** laws. We need a general scaling of variables to recover all the numerous cases. We will perform that with a “quasi” LU factorization depending on the magnitude of all frequencies. We will also use Lemma 3.1 to preserve the L 1

and LIGO-India [78] , will help to break existing degener- acies and allow for increasingly precise polarization measurements.
Long-duration signals offer further opportunities to study gravitational-wave polarizations. Detections of con- tinuous sources like rotating neutron stars [44,45] and the **stochastic** background [42] will offer the ability to directly measure and/or constrain gravitational-wave polarizations, even in the absence of additional detectors. In this Letter, we have conducted a search for a generically polarized **stochastic** background of gravitational waves using data from Advanced LIGO ’s O1 observing run. Although we find no evidence for the presence of a background (of any polarization), we have succeeded in placing the first direct upper limits (listed in Table I ) on the contributions of vector and **scalar** modes to the **stochastic** background.

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ST ´ EPHANE JUNCA
Abstract. The article first studies the propagation of well prepared high frequency waves with small amplitude ε near constant solutions for en- tropy solutions of multidimensional nonlinear **scalar** **conservation** laws. Sec- ond, such oscillating solutions are used to highlight a conjecture of Lions, Perthame, Tadmor, ([ 23 ]), about the maximal regularizing effect for non- linear **conservation** laws. For this purpose, a definition of smooth nonlinear flux is stated and compared to classical definitions. Then it is proved that the uniform smoothness expected by [ 23 ] in Sobolev spaces cannot be ex- ceeded for all smooth nonlinear fluxes.

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maximal smoothing effect for multidimensional **scalar** **conservation** laws , (in preparation).
[5] G.-Q. Chen, S. Junca, M. Rascle, Validity of Nonlinear Geometric Optics for Entropy Solutions of Multidimensional **Scalar** Conser- vation Laws , J. Differential. Equations., 222, (2006), 439–475. [6] K. S. Cheng, The space BV is not enough for hyperbolic conser-

Abstract
This paper deals with a sharp smoothing eect for entropy solutions of one-dimensional **scalar** **conservation** laws with a degenerate convex ux. We briey explain why degenerate uxes are related with the optimal smoothing eect conjectured by Lions, Perthame, Tadmor for entropy solutions of multidimensional **conservation** laws. It turns out that generalized spaces of bounded variation BVΦ are particularly suitable -better than Sobolev spaces- to quantify the regularizing eect and to obtain traces as in BV. The function Φ in question is linked to the degeneracy of the ux. Up to the present, the Lax-Olenik formula has provided optimal results for a uniformly convex ux. This formula is validated in this paper for the more general class of C 1 strictly convex uxes -which contains degenerate convex uxes- and enables the BVΦ smoothing eect in this class. We give a complete proof that for a C 1 strictly convex ux the Lax-Olenik formula provides the unique entropy solution, namely the Kruºkov solution.

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