By Etienne Emmrich, Petra Wittbold
This article incorporates a sequence of self-contained stories at the state-of-the-art in several components of partial differential equations, offered by way of French mathematicians. themes 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
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− . These three slopes satisfy the so-called Lax admissibility condition f ′ (u+ ) < ω = f (u+ ) − f (u− ) = f ′ (u˜ ) < f ′ (u− ).
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? 4), the only “physically correct” solution of the above problems should be, unquestionably, the solution u(t, x) ≡ 0.
1) may have a jump from u− to u+ (a jump in the direction of increasing x) when the following jump admissibility condition holds: 39 The Kruzhkov lectures • • in the case u− < u+ , the graph of the function f = f (u) on the segment [u− , u+ ] is situated above the chord (in the non-strict sense) with the endpoints (u− , f (u− )) and (u+ , f (u+)); in the case u− > u+ , the graph of the function f = f (u) on the segment [u+ , u− ] is situated below the chord (in the non-strict sense) with the endpoints (u− , f (u− )) and (u+ , f (u+)).