By Etienne Emmrich, Petra Wittbold
This article incorporates a sequence of self-contained stories at the cutting-edge in numerous components of partial differential equations, offered by means of French mathematicians. issues contain qualitative houses 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 info for Analytical and numerical aspects of partial differential equations
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. 9. 1) with one curve of jump discontinuity x = x(t). Then the speed of decrease of the kinetic energy E = E (t) of this solution is equal, at any instant of time t = t0 , to the area S (t0 ) delimited by the graph of the flux function f = f (u) on the segment [u− , u+ ] (or on the segment [u+ , u− ]) and by the chord joining the endpoints (u− , f (u− )) and (u+ , f (u+ )) of this graph (see Fig.
2 6 NT — In the literature on conservation laws, one often speaks of “entropy dissipation conditions”. 42) below. Each of these inequalities states the decrease (the dissipation) and not the increase of another quantity related to various functions called “entropies”. 23) for this “entropy” function S does hold across an admissible shock wave. 13) has the form u− + u+ dx = . 24) 2 dt Since the flux function f (u) = u2 /2 is convex, the jump admissibility condition reduces to the inequality u− − u+ > 0.
Chechkin and Andrey Yu. 20) express analytically the admissibility condition. Now let us interpret this condition geometrically. 19′ ) f (u) − f (u+ ) f (u+ ) − f (u− ) <ω= u − u+ u+ − u− ∀u ∈ (u+ , u− ) if u+ < u− . 20′ ) Figure 12. 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+ )).
Analytical and numerical aspects of partial differential equations by Etienne Emmrich, Petra Wittbold