Download Mathematical methods for the Magnetohydrodynamics of Liquid by Jean-Frédéric Gerbeau PDF

By Jean-Frédéric Gerbeau

This finished textual content specializes in mathematical and numerical thoughts for the simulation of magnetohydrodynamic phenomena, with an emphasis laid at the magnetohydrodynamics of liquid metals, and on a prototypical commercial program. aimed toward learn mathematicians, engineers, and physicists, in addition to these operating in undefined, and ranging from a great figuring out of the physics at play, the method is a hugely mathematical one, in response to the rigorous research of the equations to hand, and a high-quality numerical research to discovered the simulations. At each one degree of the exposition, examples of numerical simulations are supplied, first on educational try out situations to demonstrate the procedure, subsequent on benchmarks good documented within the expert literature, and at last, at any time when attainable, on actual commercial circumstances.

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55) for any solution. The particular solution obtained in the previous section by the Galerkin method does satisfy this, by construction. 55), it is sufficient to show that u · ∇u ∈ L2 (0, T ; V ′ ). 9, Ω u · ∇u · v = − Ω u · ∇v · u ≤ u 2 L4 (Ω) ≤C u v L2 (Ω) H1 (Ω) u H1 (Ω) v H1 (Ω) . Since u ∈ L2 (0, T ; V ) ∩ L∞ (0, T ; H), u · ∇u ∈ L2 (0, T ; V ′ ) follows. 13. 12. Let us now denote by w = u1 − u2 the difference of two solutions. 30) that w satisfies ∂w ,v ∂t + V ′ ,V Ω (w · ∇)u1 · v + Ω (u2 · ∇)w · v + η ∇w : ∇v = 0, Ω in the sense of distributions in time, for all v ∈ V , and with zero as initial condition.

Likewise, we may define the weak topology on any Hilbert space with scalar product (·, ·): un weakly converges to u in V if (un , v) n−→+∞ −→ (u, v) ∀v ∈ V. g. that of H 1 (0, 1) : un weakly converges to u in H 1 (0, 1) when 1 0 (u′n v ′ + un v) 1 n−→+∞ −→ 0 (u′ v ′ + uv) ∀v ∈ H 1 (0, 1), or that of H 1 (Ω), for Ω ⊂ IRd , by a straightforward adaptation of the above definition. Another generalization is the notion of weak topology for the Lp spaces, 1 ≤ p ≤ +∞, which are Banach, but (for p = 2) not Hilbert, spaces.

Experts can see the present section as a warm-up for the sequel, or may simply skip it. 1 Basic mathematics tools To start with, we recall the definition of the weak topology of the space L2 (0, 1). 1 A sequence un of functions in L2 (0, 1) is said to weakly converge to the function u ∈ L2 (0, 1) when for any v ∈ L2 (0, 1), 1 1 lim n−→+∞ u v. 3) which weakly converges to the function u ≡ 0 on (0, 1). 2) holds by integration by parts when v is C 1 , and then argue by density of the C 1 functions in L2 .

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