to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

This monograph explores the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations. It presents typical examples of hyperbolic systems for a wide range of physical engineering applications, allowing readers to understand the concepts in whichever setting is most familiar to them. With these examples, it also illustrates how control boundary conditions may be defined for the most commonly used control devices.
The authors begin with the simple case of systems of two linear conservation laws and then consider the stability of systems under more general boundary conditions that may be differential, nonlinear, or switching. They then extend their discussion to the case of nonlinear conservation laws and demonstrate the use of Lyapunov functions in this type of analysis. Systems of balance laws are considered next, starting with the linear variety before they move on to more general cases of nonlinear ones. They go on to show how the problem of boundary stabilization of systems of two balance laws by both full-state and dynamic output feedback in observer-controller form is solved by using a “backstepping” method, in which the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The final chapter presents a case study on the control of navigable rivers to emphasize the main technological features that may occur in real live applications of boundary feedback control.
Stability and Boundary Stabilization of 1-D Hyperbolic Systems will be of interest to graduate students and researchers in applied mathematics and control engineering. The wide range of applications it discusses will help it to have as broad an appeal within these groups as possible.

Stability and Boundary Stabilization of 1-D Hyperbolic Systems

We propose a general framework for the study of L1 contractive semigroups of solutions to conservation laws with discontinuous flux: 

$$ u_t + \mathfrak{f}(x,u)_x=0, \qquad \mathfrak{f}(x,u)= \left\{\begin{array}{ll} f^l(u),& x 0, \end{array} \right\quad\quad\quad (\rm CL) $$

where the fluxes fl, fr are mainly assumed to be continuous Developing the ideas of a number of preceding works (Baiti and Jenssen in J Differ Equ 140(1):161–185, 1997; Towers in SIAM J Numer Anal 38(2):681–698, 2000; Towers in SIAM J Numer Anal 39(4):1197–1218, 2001; Towers et al in Skr K Nor Vidensk Selsk 3:1–49, 2003; Adimurthi et al in J Math Kyoto University 43(1):27–70, 2003; Adimurthi et al in J Hyperbolic Differ Equ 2(4):783–837, 2005; Audusse and Perthame in Proc Roy Soc Edinburgh A 135(2):253–265, 2005; Garavello et al in Netw Heterog Media 2:159–179, 2007; Burger et al in SIAM J Numer Anal 47:1684–1712, 2009), we claim that the whole admissibility issue is reduced to the selection of a family of “elementary solutions”, which are piecewise constant weak solutions of the form 

$$ c(x)=c^l11_{\left\{{x 0}\right\}} $$

We refer to such a family as a “germ” It is well known that (CL) admits many different L1 contractive semigroups, some of which reflect different physical applications We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions We devote specific attention to the “vanishing viscosity” germ, which is a way of expressing the “Γ-condition” of Diehl (J Hyperbolic Differ Equ 6(1):127–159, 2009) For any given germ, we formulate “germ-based” admissibility conditions in the form of a trace condition on the flux discontinuity line {x = 0} [in the spirit of Vol’pert (Math USSR Sbornik 2(2):225–267, 1967)] and in the form of a family of global entropy inequalities [following Kruzhkov (Math USSR Sbornik 10(2):217–243, 1970) and Carrillo (Arch Ration Mech Anal 147(4):269–361, 1999)] We characterize those germs that lead to the L1-contraction property for the associated admissible solutions Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities “adapted” to the choice of a germ), or for specific germ-adapted finite volume schemes

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A Theory of L 1 -Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux

A nonlinear kinetic equation is constructed and proved to be well-adapted to describe general multidimensional scalar conservation laws. In particular, it is proved to be well-posed uniformly in epsilon - the microscopic scale. It is also shown that the proposed kinetic equation is equipped with a family of kinetic entropy functions - analogous to Boltzmann's microscopic H-function, such that they recover Krushkov-type entropy inequality on the macroscopic scale. Finally, it is proved by both - BV compactness arguments in the one-dimensional case, that the local density of kinetic particles admits a continuum limit, as it converges strongly with epsilon below 0 to the unique entropy solution of the corresponding conservation law.

A kinetic equation with kinetic entropy functions for scalar conservation laws. Final report

When writing can change your life, when writing can enrich you by offering much money, why don't you try it? Are you still very confused of where getting the ideas? Do you still have no idea with what you are going to write? Now, you will need reading. A good writer is a good reader at once. You can define how you write depending on what books to read. This functions of bounded variation and free discontinuity problems can help you to solve the problem. It can be one of the right sources to develop your writing skill.

書評 L.Ambrosio, N.Fusco and D.Pallara: Functions of Bounded Variation and Free Discontinuity Problems

For the scalar conservation laws with discontinuous flux, an infinite family of (A, B)-interface entropies are introduced and each one of them is shown to form an L1-contraction semigroup (see [2]). One of the main unsettled questions concerning conservation law with discontinuous flux is boundedness of total variation of the solution. Away from the interface, boundedness of total variation of the solution has been proved in a recent paper [6]. In this paper, we discuss this particular issue in detail and produce a counterexample to show that the solution, in general, has unbounded total variation near the interface. In fact, this example illustrates that smallness of the BV norm of the initial data is immaterial. We hereby settle the question of determining for which of the aforementioned (A, B) pairs the solution will have bounded total variation in the case of strictly convex fluxes. © 2010 Wiley Periodicals, Inc.

Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux

We consider the scalar conservation law with strict convex flux in one space dimension. In this paper
we study the exact controllability of entropy solution by using initial or boundary data control.
 Some partial results have been obtained in [5],[23]. Here
 we investigate the precise conditions under which, the exact controllability problem admits a solution.
The basic ingredients in the proof of these results are, Lax-Oleinik [15] explicit formula and
 finer properties of the
characteristics curves.

https://www.aimsciences.org/article/exportPdf?id=13c390a1-b231-4d9b-9875-005b39172fcb

Exact controllability of scalar conservation laws with strict convex flux

We consider scalar conservation laws in one space dimension with convex flux and we establish a new structure theorem for entropy solutions by identifying certain shock regions of interest, each of them representing a single shock wave at infinity. Using this theorem, we construct a smooth initial data with compact support for which the solution exhibits infinitely many shock waves asymptotically in time. Our proof relies on Lax–Oleinik explicit formula and the notion of generalized characteristics introduced by Dafermos.

Structure of entropy solutions to scalar conservation laws with strictly convex flux

The dissipative solutions can be seen as a convenient generalization of the concept of weak solution to the isentropic Euler system. They can be seen as expectations of the Young measures a...

On uniqueness of dissipative solutions to the isentropic Euler system

https://hal.archives-ouvertes.fr/hal-00868751/document

Shyam Sundar Ghoshal

Papers

Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux

Exact controllability of scalar conservation laws with strict convex flux

Structure of entropy solutions to scalar conservation laws with strictly convex flux

On uniqueness of dissipative solutions to the isentropic Euler system

Optimal Results On TV Bounds For Scalar Conservation Laws With Discontinuous Flux