By Etienne Emmrich, Petra Wittbold

ISBN-10: 3110204479

ISBN-13: 9783110204476

This article incorporates a sequence of self-contained experiences at the state-of-the-art in numerous components of partial differential equations, awarded through 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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**Extra resources for Analytical and numerical aspects of partial differential equations**

**Example text**

These solutions will converge to the function u0 as t → +0 at all points, except for the point x = 0. 1) can be found in [27, Lectures 4–6]; its existence is demonstrated below with an explicit construction. First of all, let us notice that the equation we consider is invariant under the change x → kx, t → kt; moreover, the initial datum also remains unchanged under the action of homotheties x → kx, k > 0. Furthermore, the entropy increase condition is also invariant under the above transformations.

The latter means that the function obtained by the juxtaposition turns out to be continuous on the border ray x = ξt = u3 t, t > 0. 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.

8. 27) such that, in addition, uε , uεx , and uεxx decay to zero as x → ±∞ at a sufficiently high rate, and uniformly in t. Then the full kinetic energy E = E (t) of this solution is a decreasing function of time. Proof. 28) 2 (uεx ) dx 0. 28) only in the case of a function uε that is constant in x. Since we assume that this function decays to zero as x → ∞, we have dE/dt < 0 unless uε ≡ 0. 27); on the latter solutions, the kinetic energy is dissipated. Therefore, it can be expected that also on the limiting solutions u, the kinetic energy does not increase with time.

### Analytical and numerical aspects of partial differential equations by Etienne Emmrich, Petra Wittbold

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