By Etienne Emmrich, Petra Wittbold

ISBN-10: 1282296477

ISBN-13: 9781282296473

ISBN-10: 3110204479

ISBN-13: 9783110204476

ISBN-10: 3110212102

ISBN-13: 9783110212105

This article features a sequence of self-contained studies at the cutting-edge in several components of partial differential equations, awarded via French mathematicians. issues comprise qualitative homes of reaction-diffusion equations, multiscale equipment coupling atomistic and continuum mechanics, adaptive semi-Lagrangian schemes for the Vlasov-Poisson equation, and coupling of scalar conservation laws.

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Nonetheless, as we have demonstrated in the previous section, so defined generalized solutions of the Cauchy problem may fail to be unique (even for the case u0 (x) ≡ 0). It is clear that the non-uniqueness stems from the fact that the “wrong” solutions uδ , δ = 0, have discontinuities. One could guess that not all the discontinuities are admissible; but how can we find the appropriate restrictions on the discontinuities? 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) .

11. 22), holds. Proof. Fix a point (t0 , x0 ) ∈ Γ, x0 = x(t0 ), on the discontinuity curve Γ. As usual, denote by u± (t0 , x0 ) the one-sided limits of u(t0 , x) on Γ as x approaches x0 . To be specific, assume that u− (t0 , x0 ) < u+ (t0 , x0 ). 35) u(t, x) > k for (t, x) ∈ O+ ≡ {(t, x) ∈ O | x > x(t)}. 36) This is always possible since we consider a piecewise smooth solution. Moreover, without loss of generality, we can assume that u is smooth in each of the subdomains O+ and O− . 30) it follows that for any test function ϕ ∈ C0∞ (O), ϕ(t, x) holds |u − k|ϕt + sign(u − k ) (f (u) − f (k )) ϕx dx dt 0, there 0.

10) with respect to t, we obtain dx dx = vt (t, x(t)) + vx (t, x(t)) · dt dt Here and in the sequel, ux , vx , ut , vt denote the corresponding limits of the derivatives as the point (t, x) tends to the weak discontinuity curve Γ. 2), we have ut (t, x(t)) + ux (t, x(t)) · dx dx − f ′ (u(t, x(t)))ux = vx (t, x(t)) · − f ′ (v (t, x(t)))vx. 10), we obtain ux (t, x(t)) · ux (t, x(t)) − vx (t, x(t)) dx − f ′ (u(t, x(t)) = 0. 9) follows. 26 Gregory A. Chechkin and Andrey Yu. 1. 2) in the strip ΠT (remind that ΠT = {−∞ < x < +∞, 0 < t < T }), for (i) f (u) = u2 /2, u(t, x) = 0 1 for x < t, for x > t; (ii) f (u) = u2 /2, u(t, x) = 0 2 for x < t, for x > t; (iii) f (u) = u2 /2, u(t, x) = 2 0 for x < t, for x > t; (iv) f (u) = −u2 , u(t, x) = 1 −1 for x < 0, for x > 0; (v) f (u) = −u2 , u(t, x) = −1 1 for x < 0, for x > 0; (vi) f (u) = u3 , u(t, x) = 1 −1 for x < 0, for x > 0; (vii) f (u) = u3 , u(t, x) = −1 1 for x < t, for x > t; (viii) f (u) = u3 , u(t, x) = 1 −1 for x < t, for x > t?

### Analytical and numerical aspects of partial differential equations : notes of a lecture series by Etienne Emmrich, Petra Wittbold

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