By Etienne Emmrich, Petra Wittbold

This article includes a sequence of self-contained reports at the state-of-the-art in numerous parts of partial differential equations, provided through French mathematicians. subject matters 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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7. 1). Proof. Since we have assumed that u(t, ±∞) = 0, we have dE = dt +∞ +∞ uut dx = − u(f (u))x dx −∞ −∞ = −uf (u) x=+∞ x=−∞ +∞ + u(t,+∞) f (u)ux dx = −∞ f (u) du = 0. 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.

14) with f (u) = u3 , then with f (u) = sin u. Is it possible to construct such solutions with more than three discontinuity lines? 4), the only “physically correct” solution of the above problems should be, unquestionably, the solution u(t, x) ≡ 0. Consequently, we should also devise a mathematical condition which would select, among all the generalized solutions, the unique “correct” solution. This condition, called the entropy increase condition, will now be formulated. 2) we encounter the following situation: 1) There exist some bounded smooth (infinitely differentiable) initial data u0 such that the unique classical solution u = u(t, x) remains a smooth function up to some critical instant of time T , but the limit u(T, x) = lim u(t, x) t→T −0 is only a piecewise smooth function with discontinuities of the first kind.

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.