By Etienne Emmrich, Petra Wittbold

ISBN-10: 3110204479

ISBN-13: 9783110204476

This article incorporates a sequence of self-contained experiences at the state-of-the-art in several components of partial differential equations, provided by way of French mathematicians. subject matters comprise qualitative houses 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 legislation.

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**Extra info for Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series**

**Example text**

DS = 0. 3. 2) has a finite number of discontinuity curves x = xj (t), j = 1, . . , N . 8. 6) is trivially satisfied (indeed, [u] = 0 and, consequently, also [f (u)] = 0). 2) in the whole domain Ω (it is clear that the function u = u(t, x) is not a classical solution in Ω, since it is not differentiable at the points (t, x) ∈ Γ ⊂ Ω). 9. 9) dt on a weak discontinuity curve Γ = {(t, x) | x = x(t)} of u = u(t, x); this means that a weak discontinuity propagates along a characteristic. Let us provide a rigorous justification of this fact.

3) and, in addition, the admissibility condition formulated in the present section. 1. 7. 6 Where does this name come from? The reason is, the equations we study model nonlinear physical phenomena (called “processes” in the sequel) which are time-irreversible, and the function which characterizes this irreversibility is called “entropy”. 1) is, certainly, the simplest model for the displacement of a gas in a tube; in more correct (more precise) models, also the pressure of the gas is present, moreover, the density of the gas enters the equations when the gas is compressible.

14) as ε → +0. (The approach developed below has been suggested by I. M. Gel’fand [18]). 14) under the form uε (t, x) = v (ξ ), ξ= x − ωt . 14), we infer that the function v = v (ξ ) satisfies the equation ′ −ωv ′ + (f (v )) = v ′′ . 12) as ε → +0 if and only if the function v = v (ξ ) satisfies the boundary conditions v (−∞) = u− , v (+∞) = u+ . 4. One cannot hope for uniqueness of such a function v = v (ξ ). 17), then the functions v˜ = v (ξ − ξ0 ) are also solutions of this problem, for all ξ0 ∈ R.

### Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series by Etienne Emmrich, Petra Wittbold

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Categories: Differential Equations