By Etienne Emmrich, Petra Wittbold
This article encompasses a sequence of self-contained experiences at the state-of-the-art in several parts of partial differential equations, provided by means of French mathematicians. themes contain 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 resources for Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series
First of all, let us notice that the equation we consider is invariant under the change x → kx, t → kt; moreover, the initial datum also remains unchanged under the action of homotheties x → kx, k > 0. Furthermore, the entropy increase condition is also invariant under the above transformations. , u(kt, kx) ≡ u(t, x) ∀k > 0. This exactly means that the function u = u(t, x) remains constant on each ray x = ξt, t > 0, issued from the origin (0, 0), so that u(t, x) depends only on the variable ξ = x/t: u(t, x) = u(x/t), t > 0.
Precisely, show that this solution is given by the equality u(t, x) = u0 (x − at). 30 Gregory A. Chechkin and Andrey Yu. Goritsky It can be shown that this solution is unique not only within the class of classical solutions, but also within the class of generalized ones; but this is beyond the scope of these notes. 14) in the case of a linear flux function f = f (u). 7. 14) with f (u) = u3 , then with f (u) = sin u. Is it possible to construct such solutions with more than three discontinuity lines?
Thus, p(t, x(t)) p(0, x(0)) = ux (0, x(0)) sup u′0 (x) =: K0 . x∈R Consequently, at any point (t, x) ∈ ΠT there holds p(t, x) = ux (t, x) K0 . 4) 32 Gregory A. Chechkin and Andrey Yu. 4): u(t, x2 ) − u(t, x1 ) x2 − x1 K0 ∀x1 , x2 . 5) A similar inequality was introduced in the works of O. A. Ole˘ınik (see ); the inequality played the role of the admissibility condition in the theory of generalized solutions. 5) it follows that u(t, x2 ) − u(t, x1 ) K0 (x2 − x1 ) for x1 < x2 ; thus at the limit as x2 → x∗ + 0, x1 → x∗ − 0, where x∗ is a discontinuity point of u(T, x), we have u+ = u(t, x∗ + 0) < u(t, x∗ − 0) = u− .
Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series by Etienne Emmrich, Petra Wittbold