Regularity of Weakly Well Posed Hyperbolic Mixed Problems With Characteristic Boundary

Morando, Alessandro; Secchi, Paolo

doi:10.1142/s021989161100238x

Cited by 17 publications

(33 citation statements)

References 24 publications

(55 reference statements)

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“…Let us now state our main result: 5 (Ω − ), and ϕ 0 ∈ H m+10 (Γ). Moreover, the initial data satisfy (8), (22), (129) and (130) and are compatible up to order m + 9 in the sense of Definition 20. Then there exists…”

Section: Main Results and Discussionmentioning

confidence: 99%

“…where δ 0 is a fixed constant. Since the basic state in [35] was also assumed to satisfy (21) and the third boundary condition in (18), one can show that the stability condition (22) is equivalently rewritten as…”

Section: 2mentioning

confidence: 99%

“…, for all multi-indices α with |α| ≤ m (see [22,23]). Agreeing with the notations set for the usual Sobolev spaces, for γ ≥ 1, H m tan,γ (Q + T ) will denote the conormal space of order m equipped with the γ−depending norm…”

Section: 2mentioning

confidence: 99%

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Well-posedness of the plasma–vacuum interface problem

2013

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We consider the free-boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the pre-Maxwell dynamics for the magnetic field. At the free interface, driven by the plasma velocity, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasmavacuum system is not isolated from the outside world, because of a given surface current on the fixed boundary that forces oscillations.Under a suitable stability condition satisfied at each point of the initial interface, stating that the magnetic fields on either side of the interface are not collinear, we show the existence and uniqueness of the solution to the nonlinear plasma-vacuum interface problem in suitable anisotropic Sobolev spaces. The proof is based on the results proved in the companion paper (Secchi and Trakhinin 2013 Interfaces Free Boundaries 15 323-57), about the well-posedness of the homogeneous linearized problem and the proof of a basic a priori energy estimate. The proof of the resolution of the nonlinear problem given in the present paper follows from the analysis of the elliptic system for the vacuum magnetic field, a suitable tame estimate in Sobolev spaces for the full linearized equations, and a Nash-Moser iteration.

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Section: Main Results and Discussionmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

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Well-posedness of the plasma–vacuum interface problem

2013

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“…Since U nc n+1 has already been determined, we can apply the well-posedness result of [Cou05], supplemented with the regularity result in [MS11], and construct a solution U n+1 ∈ C ∞ ((−∞, T ]; H +∞ (R d + )) to the above equations (in the interior and on the boundary).…”

Section: Construction Of Correctorsmentioning

confidence: 99%

The Mach stem equation and amplification in strongly nonlinear geometric optics

Coulombel

Williams²

2017

American Journal of Mathematics

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We study highly oscillating solutions to a class of weakly well-posed hyperbolic initial boundary value problems. Weak well-posedness is associated with an amplification phenomenon of oscillating waves on the boundary. In the previous works [CGW14, CW14], we have rigorously justified a weakly nonlinear regime for semilinear problems. In that case, the forcing term on the boundary has amplitude O(ε 2 ) and oscillates at a frequency O(1/ε). The corresponding exact solution, which has been shown to exist on a time interval that is independent of ε ∈ (0, 1], has amplitude O(ε). In this paper, we deal with the exact same scaling, namely O(ε 2 ) forcing term on the boundary and O(ε) solution, for quasilinear problems. In analogy with [CGM03], this corresponds to a strongly nonlinear regime, and our main result proves solvability for the corresponding WKB cascade of equations, which yields existence of approximate solutions on a time interval that is independent of ε ∈ (0, 1]. Existence of exact solutions close to approximate ones is a stability issue which, as shown in [CGM03], highly depends on the hyperbolic system and on the boundary conditions; we do not address that question here.This work encompasses previous formal expansions in the case of weakly stable shock waves [MR83] and two-dimensional compressible vortex sheets [AM87]. In particular, we prove well-posedness for the leading amplitude equation (the "Mach stem equation") of [MR83] and generalize its derivation to a large class of hyperbolic boundary value problems and to periodic forcing terms. The latter case is solved under a crucial nonresonant assumption and a small divisor condition.

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“…The strategy of the proof. We closely follow the techniques developed in [21] (see also [20]). In principle, for given smooth functions u, ψ under the assumptions of Theorem 1, we consider the problem analogous to (1) solved by the functions λ −1,γ (Z)u and λ −1,γ (D )ψ; 5 this problem is obtained by acting on the original BVP solved by (u, ψ) by the operators λ −1,γ (Z), λ −1,γ (D ) and making use of the rules of the symbolic calculus collected in Section 3.1.…”

Section: Proof Of Theoremmentioning

confidence: 99%