By Etienne Emmrich, Petra Wittbold
This article encompasses a sequence of self-contained studies at the cutting-edge in numerous parts of partial differential equations, awarded via French mathematicians. themes 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 legislation.
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Extra info for Analytical and numerical aspects of partial differential equations
8) This solution can be interpreted as follows. The particles with velocities u− and u+ collide when the quicker one (with the velocity u− ) overtakes the slower one (of velocity u+ ); this collision is not elastic, and the two particles agglomerate into one single particle. After the collision, the particles continue to move with the velocity (u+ + u− )/2, creating a shock wave. The velocity of propagation of this wave is calculated with the help of the law of momentum conservation: this velocity is the arithmetic mean of the particles’ velocities before the collision.
14). , 0 < α < β , then the analogous construction yields a non-trivial generalized solution with the initial datum u0 (x) ≡ α. The above construction breaks down in the case where such non-aligned points on the graph of f = f (u) cannot be found. , f (u) = au + b, a, b ∈ R. In the latter case, our quasilinear problem is in fact linear: ut + aux = 0, u|t=0 = u0 (x). 15) In the case where u0 is smooth (this applies, in particular, to u0 ≡ 0), the unique classical solution of this problem is easily constructed by the method of Section 2; the solution takes the form u(t, x) = u0 (x − at).
Lax condition: admissible and non-admissible discontinuity curves. 2. 1) with f (u) = u2 /2. This equation describes the displacement of freely moving particles (see Section 1). , particles with the x-coordinate larger than some sufficiently large value), move with a velocity u+ ; assume that the particles initially located in a neighbourhood of −∞ have a velocity u− ; and let u+ < u− . The latter constraint means that, as time passes, collisions are inevitable, and eventually, a shock wave will form.