Abstract
We have so far studied elliptic problems, i.e., stationary problems. We now turn to evolution problems, starting with the archetypal parabolic equation, namely the heat equation. In this chapter, we will present a brief and far from exhaustive theoretical study of the heat equation. We will mostly work in one dimension of space, some of the results having an immediate counterpart in higher dimensions, others not. The study of numerical approximations of the heat equation will be the subject of the next chapter.
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- 1.
The fact that \(\frac{\partial u}{\partial x}(x_0,t_0)=0\) is not useful here.
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Or more accurately a Dirac mass.
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Which is reassuring.
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Beware of the slightly ambiguous notation.
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In the sense of space regularity.
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Observe that the function \(u_k\) is continuous in t.
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Or even more generally, Gårding’s inequality , which reads: for all \(v\in H^1_0(\varOmega )\), \(a(v,v)\ge \alpha |v|^2_{H^1_0(\varOmega )}-\beta \Vert v\Vert ^2_{L^2(\varOmega )}\) with \(\alpha >0\).
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© 2016 Springer International Publishing Switzerland
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Le Dret, H., Lucquin, B. (2016). The Heat Equation. In: Partial Differential Equations: Modeling, Analysis and Numerical Approximation. International Series of Numerical Mathematics, vol 168. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-27067-8_7
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DOI: https://doi.org/10.1007/978-3-319-27067-8_7
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-27065-4
Online ISBN: 978-3-319-27067-8
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