G. D. Veerappa Gowda

Bio: G. D. Veerappa Gowda is an academic researcher from Tata Institute of Fundamental Research. The author has contributed to research in topics: Conservation law & Entropy (arrow of time). The author has an hindex of 12, co-authored 38 publications receiving 630 citations. Previous affiliations of G. D. Veerappa Gowda include TIFR Centre for Applicable Mathematics.

Optimal entropy solutions for conservation laws with discontinuous flux-functions

Abstract: We deal with a single conservation law in one space dimension whose flux function is discontinuous in the space variable and we introduce a proper framework of entropy solutions. We consider a large class of fluxes, namely, fluxes of the convex-convex type and of the concave-convex (mixed) type. The alternative entropy framework that is proposed here is based on a two step approach. In the first step, infinitely many classes of entropy solutions are defined, each associated with an interface connection. We show that each of these class of entropy solutions form a contractive semigroup in L1 and is hence unique. Godunov type schemes based on solutions of the Riemann problem are designed and shown to converge to each class of these entropy solutions. The second step is to choose one of these classes of solutions. This choice depends on the Physics of the problem being considered and we concentrate on the model of two-phase flows in a heterogeneous porous medium. We define an optimization problem on the set of admissible interface connections and show the existence of an unique optimal connection and its corresponding optimal entropy solution. The optimal entropy solution is consistent with the expected solutions for two-phase flows in heterogeneous porous media.

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

Abstract: 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.

Exact controllability of scalar conservation laws with strict convex flux

Abstract: 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.

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Operator Theory: Advances and Applications

Abstract: Edited by Joseph A. Ball (Blacksburg, VA, USA) Harry Dym (Rehovot, Israel) Marinus A. Kaashoek (Amsterdam, The Netherlands) Heinz Langer (Vienna, Austria) Christiane Tretter (Bern, Switzerland) Associate Editors: Vadim Adamyan, Wolfgang Arendt, Albrecht Bottcher, B. Malcolm Brown, Raul Curto, Fritz Gesztesy, Pavel Kurasov, Vern Paulsen, Mihai Putinar, Ilya M. Spitkovsky This series is devoted to the publication of current research in operator theory, with particular emphasis on applications to classical analysis and the theory of integral equations, as well as to numerical analysis, mathematical physics and mathematical methods in electrical engineering.

Mathematical Aspects of Subsonic and Transonic Gas Dynamics

Abstract: By Lipman Bers London : Chapman and Hall Ltd. Pp. xv + 164. Price 62s. In the preface to this, the third volume in Surveys in Applied Mathematics, F. J. Weyl the Director of the Mathematical Sciences Division of the Office of Naval Research gives broadly speaking two aims as the basis for the series. These are (i) the need to make readily available up-to-date results in selected areas of research and (ii) the need to relate contributions from both sides of the iron curtain in order to produce a balanced appreciation of the present state of knowledge in any such area.

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

Abstract: 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