By Etienne Emmrich, Petra Wittbold
This article features a sequence of self-contained reports at the state-of-the-art in numerous components of partial differential equations, offered via French mathematicians. issues 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 resources for Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series (De Gruyter Proceedings in Mathematics)
Let us provide a “physical” explanation of the admissibility condition obtained for the case where the monotonicity of f ′ is strict. At any point of an admissible discontinuity curve x = x(t), consider the slopes f ′ (u+ ) and f ′ (u− ) of the characteristics 33 The Kruzhkov lectures x = f ′ (u± )t + C which impinge at this point from the two sides of the discontinuf (u+ )−f (u− ) ity. Consider also the slope ω = dx of the discontinuity curve (more dt = u+ −u− exactly, the slope of its tangent line); notice that ω is equal to the value f ′ (u˜ ) at some point u˜ which lies strictly between u+ and u− .
In addition to the integrals over the domains O− and O+ , also integrals over their boundaries will arise, that is, we will get integrals over ∂O and over Γ ∩ O. As ϕ is compactly supported in O, the integral over ∂O is zero. Consequently, we obtain ut + (f (u))x ϕ dx dt O− − Γ∩O (u− − k ) cos(ν, t) + (f (u−) − f (k )) cos(ν, x) ϕ dS ut + (f (u))x ϕ dx dt − O+ − Γ∩O (u+ − k ) cos(ν, t) + (f (u+) − f (k )) cos(ν, x) ϕ dS 0. , the outward normal vector to the boundary of O− and, at the same time, the interior normal vector for O+ ).
2), which is the problem of evolution from a simplest piecewise constant initial datum. 1) where u− and u+ are two arbitrary constant states. The solutions we want to construct will be piecewise smooth in ΠT . 5) and the entropy increase condition on each curve of jump discontinuity. These solutions will converge to the function u0 as t → +0 at all points, except for the point x = 0. 1) can be found in [27, Lectures 4–6]; its existence is demonstrated below with an explicit construction. 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.