1. Shocks in finite time, “loss of regularity” [ 44 ]: Even for smooth (compactly sup- ported C ∞ ) initial data, the derivative of the solution may blow-up in finite time, and a discontinuous shock can be created.
After the breaking-down time, classical (strong) **solutions** with classical derivatives can not exist. In order to obtain global (in time) **solutions**, weak **solutions** must be defined, where the derivatives are defined in the sense of distributions. Weak **solutions** do exist globally in time, but no uniqueness is insured, in general. Many “**entropy**” conditions on the solution have been imposed in the literature in order to obtain the uniqueness (for example Lax [ 48 ], Oleinik [ 65 ] and Kruzkov [ 46 ]) for the scalar case. However, for systems, the **entropy** condition is not enough to obtain the uniqueness of the solution [ 25 ]. For some hyperbolic equations, the **entropy** **solutions** enjoy an interesting regularising effect that is directly related to the well-known one-sided Oleinik inequality [ 65 ]

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Abstract
In this article, we investigate the BV stability of 2×2 hyperbolic systems of conserva- tion laws with strictly positive velocities under dissipative boundary conditions. More precisely, we derive sufficient conditions guaranteeing the exponential stability of the system under consideration for **entropy** **solutions** in BV. Our proof is based on a front tracking algorithm used to construct approximate piecewise constants **solutions** whose BV norms are controlled through a Lyapunov functional. This Lyapunov functional is inspired by the one proposed in J. Glimm’s seminal work [16], modified with some suitable weights in the spirit of the previous works [10, 9].

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The advantage of BV s spaces is to recover the fractional Sobolev regularity W σ,p for all σ < s, 1 ≤ p < s −1 and to get the BV like trace properties of **entropy** **solutions** [ 23 , 42 , 43 ]. In one dimension, existence of the **entropy** **solutions** of ( 1 ) in BV s , with BV s data has been done in [ 13 ] and with L ∞ data in the same spaces in [ 13 , 38 , 39 ]. For non-convex fluxes a Lagrangian framework is used [ 8 , 9 ]. For the scalar 1-D case, the BV s smoothing effect corresponds to the optimal smoothing effect conjectured by Lions, Perthame and Tadmor in Sobolev spaces with the same fractional derivative s [ 37 ]. In multi-dimension, for a C 2,γ flux, it has been shown [ 28 ] that **entropy** **solutions** do not need to have fractional derivative s + ε for ε > 0. For multi-dimensional scalar conservation laws, regularizing effect in fractional Sobolev space was first studied in [ 37 ]. We refer [ 44 ] for the best known result in this direction and [ 31 ] for further improvement with some extra assumptions, see also [ 26 ] for such results with a source term. The proof of optimality of the exponent s > 0 is limited to some one-dimensional scalar examples [ 18 , 25 ] and before the nonlinear interaction of waves. It has been extended for the scalar multi-dimensional case in [ 20 , 32 ] but not for all time. Recently, in [ 28 ] it has been shown that in multi-dimension for any C 2 flux f there exists initial data u 0 such that the corresponding **entropy** solution is not in BV for all time t > 0.

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u(x, 0) = u 0 (x), (1.3)
v(x, 0) = v 0 (x). (1.4)
The Pressure Swing Adsorption process (PSA) in chemistry [ 9 ] has such a triangular structure after a change of variables from Euler to Lagrange [ 12 , 42 ]. Such systems are already of mathematical interest due to the coupling of the theory of scalar conservation laws with the theory of transport equations. This system was studied in [ 34 ] in a non hyperbolic setting, f 0 = a, with measure **solutions** for v. That even a strictly hyperbolic setting is not enough to avoid measure **solutions** was shown in [ 24 ]. Here, strengthening the hyperbolicity of the system by a stronger transversality condition, global weak bounded **entropy** **solutions** are provided with an optimal fractional BV regularity for the initial data u 0 .

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in particular **entropy** **solutions**.
To be more precise, in this paper, we will consider **solutions** ` a la Glimm [15], that is, **entropy** **solutions** in the sense above, of small total variation in x for all times. Note that the meaning of the boundary value in this context is intricate, especially when the characteristic speeds are not separated from zero, see in particular the reference of Dubois and LeFloch [13]. Hence the underdetermined version of the problem is particularly well suited here.

s . We extend the smoothing eect proved in [BGJ] for any C
1
strictly convex ux, more general than a ux with a power law behavior.
The rst tool which has provided optimal regularity results is the Lax-Olenik formula for a uni- formly convex ux. In order to get our smoothing eect -in the end part of this article- we rst need to rigorously validate the well-known Lax-Olenik formula for a nonlinear degenerate convex ux. The proof given in this article is self-contained and follows Lax's proof ([La2]). Another possibility was to use the Lax-Hopf formula for the corresponding Hamilton-Jacobi equation ([E]). But to get a ne regularity of the **entropy** **solutions** it is convenient to work on the conservation law instead of the Hamilton-Jacobi equation ([CEL]).

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case requires neither adaptation of the notion of solution (cf. [ 13 ] and [ 40 ] for the pure L 1 theories of, respectively, renormalized and kinetic **solutions**), nor the fine regularization estimates of the super-critical case [ 43 ]. Compared to the subcritical case of [ 44 , 12 ], it requires a careful control of approximations in the existence proof. The precise well-posedness result for the Cauchy problem is given in Theorem 5.1 below. Further, we also investigate the analogous Cauchy-Dirichlet problem in the one-dimensional case, limiting our attention to the Burgers equation. We provide sufficient conditions on boundary data for existence of L 2 **entropy** **solutions**, and give the uniqueness result based upon the automatic boundary regularity of such **solutions**. Concerning the Cauchy-Dirichlet version of our problem, let us point out the related work [ 6 ] of Ancona and Marson where semi- group **solutions** with initial and boundary data of appropriate integrability are considered, though not fully characterized by **entropy** inequalities, and the works [ 41 ] of Porretta and Vovelle, [ 3 ] of Ammar, Carrillo and Wittbold where the problem is treated in the renormalized **solutions** setting, i.e. for a more involved truncated **entropy** formulation.

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imum principle: ess inf u 0 ≤ u ≤ ess sup u 0 .
If α ∈ (1, 2], all **solutions** to (2.7)–(2.8) with bounded initial conditions are smooth and global-in-time (see [8, 16, 17]). On the other hand, the occurrence of discontinuities in finite time of **entropy** **solutions** to (2.7)–(2.8) with α = 1 seems to be unclear. As mentioned in the introduction, regularity results have recently been obtained [16, 7, 18] for a large class of initial conditions which, unfortunately, does not include general L ∞ -initial data. Nevertheless, Theorem 2.2 provides the

a weak sense, the boundary conditions. Furthermore, these **solutions** satisfy the Clausius inequality: the change of the free energy is bounded by the work done by the boundary tension. In this sense they are the correct thermodynamic **solutions**, and we conjecture their uniqueness.
Keywords: hyperbolic conservation laws, quasi-linear wave equation, boundary conditions, weak **solutions**, vanishing viscosity, compensated compactness, **entropy** **solutions**, Clausius inequality. Mathematics Subject Classification numbers: 35L40, 35D40

−c j
| is convergent. §
5. Godunov scheme
There is two diﬀerent approaches to build a Godunov scheme for the system (1.12)-(1.13): the first one uses the “natural” space-time framework (the evolution variable is t), the second one uses the hyperbolicity of the system where the roles of time and space are exchanged, as performed by Rouchon and al. and following the preceding theoretical study (x viewed as the evolution variable). This last option, because of the hyperbolic structure, is the classical one. The corresponding (CFL) condition requires a lower bound for u which is easily obtained thanks to the BV control of ln(u) given in Lemma 4.8. Furthermore, a control of the total variation of u with respect to time is required to show that a subsequence of approximate **solutions** converges toward a weak (**entropy**) solution. This cannot be a priori expected: fix c a constant in [0,1] and u b only in L ∞ , then (c(t,x),u(t,x)) ≡ (c,u b (t)) is always a weak

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Accordingly to Lemma 2 in [23], there exists a family of half-plain supported **entropy**-**entropy** fluxes (η, q) such that η and q are bounded together with their first and second derivatives. These are explicitly given as follows. We define z(r) := R 0 r p τ ′ (ρ)dρ and we define the Riemann coordinates
w 1 = p + z, w 2 = p − z. We also pass from the dependent variables η, q to H,Q as follows:

2 Uniqueness and stability of **entropy** **solutions**
The Lipschitz continuous dependence of **entropy** **solutions** with respect to initial data can be derived using Kruˇ zkov’s doubling of variable technique [13] as in [3, 4, 10].
Theorem 3. Under hypotheses (H), let ρ, σ be two **entropy** **solutions** to (1.1) with initial data ρ 0 , σ 0 respectively. Then, for any T > 0 there holds

1.5. Our second problem: ruling out pathologies in timed automata
An amazing theoretical application of the **entropy** of timed languages is related to a well-known, but not yet sufficiently understood issue of pathological and “normal” behaviors of timed automata. Indeed, timed automata using exact continuous clocks, exact guards and resets are a beautiful mathematical object and a useful model of real-time systems. However, from the very beginning of research on timed automata, it was clear that they are in several aspects too precise, which leads sometimes to strange artifacts, mathematical pathologies or unrealistic models. Several lines of research have partially elucidated these issues.

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of VMs that need to migrate as compared to the amount of available resources only exacerbates these problems. Thus, it is essential that consolidation be as efficient and reactive as possible.
In this paper, we propose a new approach to consolidation in a homogeneous cluster environment that takes into account both the problem of allocating the VMs to the available nodes and the prob- lem of how to migrate the VMs to these nodes. Our consolidation manager, **Entropy**, works in two phases and is based on constraint solving [2, 14]. The first phase, based on constraints describing the set of VMs and their CPU and memory requirements, computes a placement using the minimum number of nodes and a tentative reconfiguration plan to achieve that placement. The second phase, based on a refined set of constraints that take feasible migrations into account, tries to improve the plan, to reduce the number of migrations required. In our experiments, using the NASGrid bench- marks [6] on a cluster of 39 AMD Opteron 2.0GHz CPU uniproces- sors, we find that a solution without consolidation uses 24.31 nodes per hour, consolidation based on the previously-used First Fit De- creasing (FFD) heuristic [3, 17, 18] uses 15.34 nodes per hour, and consolidation based on **Entropy** uses only 11.72 nodes per hour, a savings of more than 50% as compared to the static solution.

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the maximal number of points than can be included in a set ∆ that has **entropy** smaller than B. In other words, it is the maximal number of points in Υ m or Υ m that can be
collected by a path of **entropy** smaller than B. note that we use the different font to be able to differentiate the setting: L, Λ, Υ for the continuous case and L, Λ, Υ for the discrete one. We show the following result—the lower bound is not needed for our applications, but can be found in [4].

tanglement **entropy** is given by S EE = lim n→1 S RE (n) =
− lim n→1 ∂n ∂ tr A (ρ A ) n . From the point of view of holog-
raphy, Ryu and Takayanagi [2] related the entanglement **entropy** with the area of an extremal surface in an anti- de Sitter space, while Maldacena and Susskind [3] pro- posed the ER=EPR conjecture, which claims that a pair of entangled objects are connected by a wormhole. For examples supporting this conjecture, Ref. [4] studied a pair of accelerating quark and anti-quark, and Ref. [5] studied a scattering gluon-gluon pair in the AdS/CFT correspondence. Then one can naturally ask the follow- ing question: How does the entanglement **entropy** of a pair of particles change from an initial state to a final one in an elastic scattering process? Ref. [6] analyzed such change of entanglement **entropy** perturbatively [7] in a field theory with a weak coupling by the use of an S- matrix [8]. In this letter we exploit the S-matrix formal- ism further in order for a non-perturbative understanding of the entanglement **entropy** in a scattering process in a context where inelastic scattering also takes place. This is especially required in the context of strong interaction scattering at high energies where inelastic multi-particle scattering plays an important role which can be analyzed without refereeing explicitly to the underlying quantum field theory [9, 10].

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appearing as weights in the cost functional that can be com- puted explicitly. A renowned example in climate is given by a principle of maximum **entropy** production (MEP) that al- lowed (Paltridge, 1975) to derive the distribution of heat and clouds at the Earth surface with reasonable accuracy, with- out any parameters and with a model of a dozen of degrees of freedom (boxes). Since then, refinements of the Paltridge model have been suggested to increase its generality and range of prediction (Herbert et al., 2011). MEP states that a stationary non-equilibrium system chooses its final state in order to maximize the **entropy** production as is explained in Martyushev and Seleznev (2006). Rigorous justifications of its application have been searched using e.g. information the- ory (Dewar and Maritan, 2014) without convincing success. More recently, we have used the analogy of the climate box model of Paltridge with the asymmetric exclusion Markov process (ASEP) to establish numerically a link between the MEP and the principle of maximum Kolmogorov–Sinai en- tropy (MKS) (Mihelich et al., 2014). The MKS principle is a relatively new concept which extends the classical results of equilibrium physics (Monthus, 2011). This principle ap- plied to Markov chains provides an approximation of the op- timal diffusion coefficient in transport phenomena (Gómez- Gardeñes and Latora, 2008) or simulates random walk on irregular lattices (Burda et al., 2009). It is therefore a good candidate for a physically relevant cost functional in passive scalar modelling.

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in other compact metric spaces that are not manifolds? For instance, do they hold if the space is a Cantor set K?
If the aim were just to construct f ∈ Hom(K) with ergodic mea- sures with infinite metric **entropy**, the answer would be positive. But if the purpose were to prove that such homeomorphisms are generic in Hom(K), the answer would be negative.

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