2015
DOI: 10.1007/s00440-015-0648-2
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Scalar conservation laws with monotone pure-jump Markov initial conditions

Abstract: Abstract. In 2010 Menon and Srinivasan published a conjecture for the statistical structure of solutions ρ to scalar conservation laws with certain Markov initial conditions, proposing a kinetic equation that should suffice to describe ρ(x, t) as a stochastic process in x with t fixed. In this article we verify an analogue of the conjecture for initial conditions which are bounded, monotone, and piecewise constant. Our argument uses a particle system representation of ρ(x, t) over 0 ≤ x ≤ L for L > 0, with a s… Show more

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Cited by 13 publications
(14 citation statements)
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“…Although a closure theorem is proved rigorously, many of the calculations in [18] such as the new kinetic equations and derivations of the Lax equation from four different approaches are formal. In [13], (1.15) is rigorously established. Specifically, they consider the Cauchy problem…”
Section: Evolution Of Conservationmentioning
confidence: 99%
“…Although a closure theorem is proved rigorously, many of the calculations in [18] such as the new kinetic equations and derivations of the Lax equation from four different approaches are formal. In [13], (1.15) is rigorously established. Specifically, they consider the Cauchy problem…”
Section: Evolution Of Conservationmentioning
confidence: 99%
“…The conservation law in the form have a wide array of applications among fluid mechanics, shocks, and turbulence. An interesting feature is that even for a smooth flux function H and smooth initial data, discontinuous solutions to the conservation law (1.1) may occur [4,6,10,11,12,14,15,17,18,19,21,22,23,24,25,31,32]. In particular, this is observed in the prototypical case of Burgers' equation, for which H (w) = w 2 /2.…”
Section: Introductionmentioning
confidence: 99%
“…Moreover, there would be no fragmentation because σ ≥ 0. If we also assume that f is independent of x 2 and that K is increasing, then Theorem 1.1 was established in [KR1], confirming affirmatively a conjecture of Menon and Srinivasan [MS]. We may rephrase Theorem 1.1 as follows: If g solves the Hamilton-Jacobi PDE (1.15) in d = 1, and initially g x 1 (x 1 , t 0 ) is a Markov jump process with the rate density f 1 (x 1 , t 0 , ρ − 1 , ρ + 1 ), then for every t > t 0 , the process g x 1 (x 1 , t) is also a Markov jump process with the rate density f 1 (x 1 , t, ρ − 1 , ρ + 1 ).…”
mentioning
confidence: 99%