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Asymptotic analysis of some partial differential equations in domains becoming unbounded

Abstract : This thesis has for aim the study of some elliptic problems in some domains becoming unbounded in one or several directions. In the first part of the thesis, we study the problem⌠-div(A ∇uℓ) = f in Ωℓ⌡uℓ = g on ∂Ωℓ.where Ωℓ is the cylinder ω1 x ω2 with ω1and ω2two bounded domains of Rk and Rn-k respectively (with 1 ≤ k ≤ n - 1). We denote by Ω ∞ the infinite cylinder Rk x ω2 and we take f ∈ H-1loc (Ω ∞) and g ∈ H1loc (Ω ∞), so that f ∈ H-1loc (Ω ∞) and g ∈ H1loc (Ω ∞), for any ℓ > 0. This work is based on the methods developed in [23] and [19]. We show that it is possible to indifferently pass to the limit in the sequence of cylinders and then to solve the problem on the infinite cylinder, or to first solve the problem on the cylinder Ωℓ and then to pass to the limit. The limit here is to be understood in the sense of a Saint-Venant type principle, which is to say that the convergence takes place for the restrictions of uℓ to smaller domains Ωℓ (with 0 < ℓ’ < ℓ) contained in Ωℓ. After that, we give some optimality results concerning the domain in which the sequence of solutions uℓ converges to u ∞. In the second chapter, we construct some correctors that enable us to extend the convergence on the whole cylinder. The construction of these correctors is inspired from the ones made in [17] and [18]. In the third chapter of the thesis we prove that, under some decreasing conditions at in_nity for the data f, it is possible to recover the same convergence on the whole cylinder, without the adjunction of correctors. In the last part of the thesis, we study the Stokes problem∫ - µ∆uℓ + ∇pℓ = f dans Ωℓ∫ div u = 0 dans Ωℓ⌡u = 0 sur ∂Ωℓon a domain Ωℓ = Bℓ x ω, where Bℓ ⊂ Rk (1 ≤ k ≤ n-1) is the ball of radius ℓ centered at the origin. Here, an important hypothesis is that f has some radial invariant properties with respect to the first k coordinates. One of the major tools in the proof of the result of this chapter (concerning especially the case k ≥ 2) is a result of the divergence-problem type. More precisely, based on a construction inspired by [9], we prove the following result (here below Dℓ = Ωℓ +1 \ Ωℓ) :If g ∈ W1;p(Dℓ) is a radial function along X1 such that g = 0 on (Bℓ+1 \ Bℓ) x ∂ω and ∫Dℓ gdx = 0. then there exists u ∈ (W1;p 0 (Dℓ))n such that∫div u = g in Dℓ⌡ ||∇u || Lp (Dℓ) ≤ C(|| g || Lp (Dℓ) + ||∇x2g || Lp (Dℓ))the constant C being independent of ℓ for ℓ ≥ 1, (C depends on k, n, p and !). Thanks to this result, used in the case p = 2, we finally prove that in this case, we also have an exponential rate of convergence of uℓ Ωℓ2 to u ∞.
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Submitted on : Friday, September 25, 2020 - 2:13:31 PM
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Adrien Ceccaldi. Asymptotic analysis of some partial differential equations in domains becoming unbounded. Analysis of PDEs [math.AP]. Normandie Université, 2020. English. ⟨NNT : 2020NORMR014⟩. ⟨tel-02949219⟩



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