By Etienne Emmrich, Petra Wittbold
This article features a sequence of self-contained stories at the state-of-the-art in several parts of partial differential equations, awarded via French mathematicians. subject matters comprise 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 laws.
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Additional resources for Analytical and numerical aspects of partial differential equations : notes of a lecture series
5. 15). It is natural to consider as admissible also the pointwise limits of the admissible solutions. Therefore, it is clear that any situation where the graph of f = f (u) touches the chord Ch should also be considered as admissible. 1) may have a jump from u− to u+ (a jump in the direction of increasing x) when the following jump admissibility condition holds: 39 The Kruzhkov lectures • • in the case u− < u+ , the graph of the function f = f (u) on the segment [u− , u+ ] is situated above the chord (in the non-strict sense) with the endpoints (u− , f (u− )) and (u+ , f (u+)); in the case u− > u+ , the graph of the function f = f (u) on the segment [u+ , u− ] is situated below the chord (in the non-strict sense) with the endpoints (u− , f (u− )) and (u+ , f (u+)).
Since 0 the solution also obeys the zero initial datum, the constant C should be the same for the two jumps. Thus both jumps cancel each other, because they occur along one and the same line; thus our piecewise constant solution is in fact equal to zero. 4. 12) with more than three discontinuity lines. 5. Is it possible to construct a solution as in the previous exercise but with an even number of discontinuity lines, each of these lines being a ray originating from the point (0, 0) of the (t, x)-plane?
1 Admissibility condition on discontinuities: the case of a convex flux function Let us make the additional assumption f ′′ 0, f ∈ C 3 ( R) , u 0 ∈ C 2 ( R) . 1. 3, show that in this case, u ∈ C 2 (ΠT ) where [0, T ) is the maximal interval of existence of a classical solution. Now let us exploit the following consideration, which is purely mathematical: we try to reveal such properties of the smooth (for t < T ) solutions that do not weaken (or which are conserved) while time approaches the critical value t = T .
Analytical and numerical aspects of partial differential equations : notes of a lecture series by Etienne Emmrich, Petra Wittbold