Novikov S.Y.'s A-Systems, Independent Functions, and Sets Bounded in Spaces PDF

By Novikov S.Y.

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Extra info for A-Systems, Independent Functions, and Sets Bounded in Spaces of Measurable Functions

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We begin with a short discussion of this method. Glimm’s method is a procedure for obtaining solutions of the general initial value problem by constructing a convergent sequence of approximate solutions, the approximation scheme being based on the solution of the Riemann problem. The scheme consists of approximating the solution at a fixed time level by piecewise constant states, so that one can solve the resulting Riemann problems thereby obtaining a sequence of elementary waves at that time level, the goal being to estimate the growth in the amplitude of these elementary waves as the waves interact during the time evolution of the solution.

The solution is unique in the class of rarefaction waves and admissible shock waves. We note that in the case of an isentropic equation of state, (p(ρ) = σ 2 ργ , γ > 1), the sound speed exceeds the speed of light for sufficiently large ρ, except in the limiting case γ = 1. 7 The Initial Value Problem 45 of the isentropic gas model that a big data global existence theorem for the Riemann problem (such as Theorem 3 above) is possible without modifying the equation of state at large densities. 7 The Initial Value Problem In this section we use the Glimm difference scheme to prove Theorem 1.

A calculation using the identities 2 x4 + x2 + 1 = − 1, x2 K and 2 x2 + 1 = , x K leads to {·}∗ = (1 − x2 )(1 − z) 2 23/2 + K K 2z + K(1 − z)2 . 71) we must have s21 − λ21 < 0. 15), the case p = σ 2 ρ. 61). s + r = ln ✻ s= 1 2 ln c+v c−v + 12 K0 ln(ρ) c+v c−v r= ❅ ■ ❅ ✒ ❅ ❅ 1 2 ln c+v c−v − 12 K0 ln(ρ) ❅ ❅ ❅ ❅ ❅ s − r = K0 ln(ρ) ✛ The plane of Riemann invariants. Figure 1 Our main result of this section is that the i-shock curves are independent of the base point (ρL , vL ) in the sense that, when graphed in the rs-plane, all i-shock curves are rigid translations of one another; and moreover, the 1-shock curve based at a given point is the reflection of the 2-shock curve based at the same point about an appropriate axis of rotation.

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A-Systems, Independent Functions, and Sets Bounded in Spaces of Measurable Functions by Novikov S.Y.


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