By Etienne Emmrich, Petra Wittbold
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
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