327 research outputs found
Existence result for a class of nonconservative and nonstrictly hyperbolic systems
No full textWe consider the class of nonconservative hyperbolic systems partial derivative(t)u+A(u) partial derivative(x) u = 0, partial derivative(t)upsilon + A(u) partial derivative(x) upsilon = 0, where, mu = u(x, t), upsilon = upsilon(x, t) is an element of IRN are the unknowns and A(mu) is a strictly hyperbolic matrix. Relying on the notion of weak solution proposed by Dal Maso, LeFloch, and Murat ("Definition and weak stability of nonconservative products", J. Math. Pures Appl. 74 (1995), 483-548), we establish the existence of weak solutions for the corresponding Cauchy problem, in the class of bounded functions with bounded variation. The main steps in our proof are as follows: (i) We solve the Riemann problem based on a prescribed family of paths. (ii) We derive a uniform bound on the total variation of corresponding wave-front tracking approximations mu(h), upsilon(h). (iii) We justify rigorously the passage. to the limit in the nonconservative product A(mu(4))partial derivative(x)upsilon(h), based on the local uniform convergence properties of mu(h), by extending an argument due to LeFloch and Liu ("Existence theory for nonlinear hyperbolic systems in nonconservative form", Forum Math. 5 (1993), 261-280). Our results provide a generalization to the existence theorem established earlier in the scalar case (N = 1) by the second author ("An existence and uniqueness result for two nonstrictly hyperbolic systems", IMA Volumes in Math. and its Appl. 27,"Nonlinear evolution equations that change type"
Convergence of finite volume methods
No full textCockburn, B.; Coquel, F.; LeFloch, Ph.; Shu, C.W.. (1991). Convergence of finite volume methods. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/1563
BV stability via generalized characteristics for nonclassical solutions of conservation laws
No full text
Nonclassical Shocks and the Cauchy Problem for Nonconvex Conservation Laws
No full textAbstractThe Riemann problem for a conservation law with a nonconvex (cubic) flux can be solved in a class of admissible nonclassical solutions that may violate the Oleinik entropy condition but satisfy a single entropy inequality and a kinetic relation. We use such a nonclassical Riemann solver in a front tracking algorithm, and prove that the approximate solutions remain bounded in the total variation norm. The nonclassical shocks induce an increase of the total variation and, therefore, the classical measure of total variation must be modified accordingly. We prove that the front tracking scheme converges strongly to a weak solution satisfying the entropy inequality
Gas phase formation of the prebiotic molecule formamide: insights from new quantum computations
No full textNew insights into the formation of interstellar formamide, a species of great
relevance in prebiotic chemistry, are provided by electronic structure and
kinetic calculations for the reaction NH2 + H2CO -> NH2CHO + H. Contrarily to
what previously suggested, this reaction is essentially barrierless and can,
therefore, occur under the low temperature conditions of interstellar objects
thus providing a facile formation route of formamide. The rate coefficient
parameters for the reaction channel leading to NH2CHO + H have been calculated
to be A = 2.6x10^-12 cm^3 s^-1, beta = -2.1 and gamma = 26.9 K in the range
of temperatures 10-300 K. Including these new kinetic data in a refined
astrochemical model, we show that the proposed mechanism can well reproduce the
abundances of formamide observed in two very different interstellar objects:
the cold envelope of the Sun-like protostar IRAS16293-2422 and the molecular
shock L1157-B2. Therefore, the major conclusion of this Letter is that there is
no need to invoke grain-surface chemistry to explain the presence of formamide
provided that its precursors, NH2 and H2CO, are available in the gas-phase
Compensated compactness and corrector stress tensor for the Einstein equations in symmetry
No full textWe consider the Einstein equations in symmetry, either for vacuum spacetimes or coupled to the Euler equations for a compressible fluid, and we introduce the notion of areal flows on with finite total energy. By uncovering a hidden structure of the Einstein equations, we establish a compensated compactness framework and solve the global evolution problem for vacuum spacetimes as well as for self-gravitating compressible fluids. We study the stability and instability of such flows and prove that, when the initial data are well-prepared, any family of areal flows is sequentially compact in a natural topology. In order to handle general initial data we propose a relaxed notion of areal flows endowed with a corrector stress tensor (as we call it) which is a bounded measure generated by geometric oscillations and concentrations propagating at the speed of light. This generalizes a result for vacuum spacetimes in: Le Floch B. and LeFloch P.G., Arch. Rational Mech. Anal. 233 (2019), 45-86. In addition, we determine the global geometry of the corresponding future Cauchy developments and we prove that the area of the orbits generically approaches infinity in the future-expanding regime. In the future-contracting regime, the volume of the spacelike slices approaches zero and, for generic initial data, the area of the orbits of symmetry approaches zero in Gowdy symmetric matter spacetimes and in vacuum spacetimes
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