# Analytical and Numerical Aspects of Partial Differential by Etienne Emmrich, Petra Wittbold

By Etienne Emmrich, Petra Wittbold

This article encompasses a sequence of self-contained reports at the state-of-the-art in several components of partial differential equations, awarded via French mathematicians. issues 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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40) by ′ E (u). 41) since E ′′ (u) 0 and ε > 0. 11 and integrate it over ΠT . Using the integrationby-parts formula, we transfer all the derivatives to the test function ϕ: u − ΠT f ′ (ξ )E ′ (ξ ) dξ dx dt ϕt E (u) + ϕx k ε ΠT ϕxx E (u) dx dt 49 The Kruzhkov lectures Passing to the limit as ε → +0, we get u ΠT f ′ (ξ )E ′ (ξ ) dξ dx dt ϕt E (u) + ϕx 0. 42) k Let {Em } be a sequence of C 2 -functions approximating the function u → |u − k| uniformly on R. 42) and pass to the limit ′ ′ as m → ∞. We can choose Em in such a way that Em is bounded and Em (ξ ) → sign(ξ − k ) for all ξ ∈ R, ξ = k .

7); this equality of the two areas is a direct consequence of the Rankine–Hugoniot condition. Figure 7. Area-preserving “overturning” of the graph. 7. 2). Denote +∞ S (t) = u(t, x) dx. , S (t) ≡ const. Proof. Indeed, we can write x(t) S (t) = +∞ u(t, x) dx + u(t, x) dx, x(t) −∞ where x = x(t) is the curve of discontinuity of the generalized solution u = u(t, x). As previously, we denote by u± = limx→x(t)±0 u(t, x) the one-sided limits (limits along the x-axis) of the solution u on the discontinuity curve.

Consequently, here the discontinuity is a weak, not a strong one. Now we can solve completely the Riemann problem for the Hopf equation. Here, two substantially different situations should be considered: (i) When u− > u+ , we can construct a shock wave solution, where the two constants u− and u+ are joined across the ray x = u2 +2 u1 t, according to the Rankine– Hugoniot condition (see Fig. 16): u(t, x) = u− for x < u+ for x > u− +u+ t, 2 u− +u+ t. 4) 52 Gregory A. Chechkin and Andrey Yu. Goritsky Figure 16.