By Etienne Emmrich, Petra Wittbold

This article incorporates a sequence of self-contained studies at the cutting-edge in numerous components of partial differential equations, provided via French mathematicians. subject matters contain qualitative homes of reaction-diffusion equations, multiscale equipment coupling atomistic and continuum mechanics, adaptive semi-Lagrangian schemes for the Vlasov-Poisson equation, and coupling of scalar conservation legislation.

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**Extra resources for Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series**

**Example text**

Visualization of admissible jumps, II. Let us represent the graph of a flux function f = f (u) (see Fig. 12). 19′ ) means that the chord Ch with the endpoints (u− , f (u−)), (u+, f (u+ )) has a smaller slope (the slope is measured as the inclination of the chord with respect to the positive direction of the u-axis) than the slope of the segment joining the point (u− , f (u− )) with the point (u, f (u)), where u runs over the interval (u− , u+ )). Consequently, the point (u, f (u)) and thus the whole graph of f = f (u) on the interval (u− , u+ ) lies above the chord Ch.

2) whatever be the flux function f = f (u)). Let us check the Rankine–Hugoniot condition on each of the three lines of discontinuity of the first kind (which are x = 0 and x = ±δt): as x = 0, we have u− = −δ , u+ = δ , and 2 dx δ 2 − (−δ ) f (u+ ) − f (u− ) =0= = ; dt δ − (−δ ) u+ − u− 28 Gregory A. Chechkin and Andrey Yu. Goritsky as x = −δt, we have u− = 0, u+ = −δ , and 2 f (u+ ) − f (u− ) dx (−δ ) − 02 = = −δ = ; dt u+ − u− (−δ ) − 0 as x = δt, we have u− = δ , u+ = 0, and 02 − δ 2 f (u+ ) − f (u− ) dx =δ= = .

Also the techniques of existence and uniqueness proofs, techniques deeply rooted in the physical context of the problem, were set up in these papers. 50 Gregory A. Chechkin and Andrey Yu. 2), which is the problem of evolution from a simplest piecewise constant initial datum. 1) where u− and u+ are two arbitrary constant states. The solutions we want to construct will be piecewise smooth in ΠT . 5) and the entropy increase condition on each curve of jump discontinuity. These solutions will converge to the function u0 as t → +0 at all points, except for the point x = 0.

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