By Etienne Emmrich, Petra Wittbold

This article encompasses a sequence of self-contained stories at the state-of-the-art in numerous parts of partial differential equations, awarded by way of French mathematicians. subject matters contain qualitative homes of reaction-diffusion equations, multiscale tools coupling atomistic and continuum mechanics, adaptive semi-Lagrangian schemes for the Vlasov-Poisson equation, and coupling of scalar conservation legislation.

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**Example text**

Kruzhkov, the existence and uniqueness of an admissible generalized solution, in the sense of the new definition, was proved. Let us now give the aforementioned definition. One of the widest spaces of functions in which generalized solutions of our problem can be searched is the space of bounded measurable functions u = u(t, x) defined in the strip ΠT = [0, T ) × Rx . 11. , b ∀ [a, b] ⊂ R, |u(t, x) − u0 (x)| dx = 0. 12. 3). Proof. 1). It follows that for 7 NT — The western literature refers to “Kruzhkov entropy solutions” or merely to “entropy solutions”.

1) irreversible. The readers who sometimes spend vacations at the sea are probably acquainted with this phenomenon. Near the shore, if the sea is calm and the waves are temperate, the sea temperature near the surface is almost the same as the air temperature above. When the wind becomes stronger, waves become foamy, turbulent structures occur; these “broken waves” can be seen as shock waves on the sea surface. In this case, after some time, one can observe that the temperature of the surface layer of the sea has become higher than the air temperature.

39), we obtain the inequality 2 (k − u+ ) cos(ν, t) + (f (k ) − f (u+ )) cos(ν, x) + (u+ − u− ) cos(ν, t) + (f (u+) − f (u− )) cos(ν, x) = 2 (k − u+ ) cos(ν, t) + (f (k ) − f (u+ )) cos(ν, x) 0, which holds for all k ∈ (u+ , u− ). 22). 30) can be derived from the vanishing viscosity approach. 40) and the initial datum u(0, x) = u0 (x). 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.