Problèmes de Cauchy abstraits et applications à quelques problèmes mixtes

Chazarain, Jacques

doi:10.1016/0022-1236(71)90027-9

Cited by 84 publications

(70 citation statements)

References 10 publications

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“…Proof. By Theorem 3.1, the operator A satisfies the hypothesis of Theorem 1.6 of [3], which explicitly gives this corollary.…”

Section: Stability Near a Convenient Profilementioning

confidence: 67%

“…This is the reason why the notion of distribution semi groups has been introduced by several authors [19], [4] and systematically studied in [3]. It corresponds to a loss of regularity with respect to the initial datum.…”

Section: Stability Near a Convenient Profilementioning

confidence: 99%

“…Proof Reproducing the basic arguments of [3], define g(t, x, v) by the inverse Laplace transform (with λ = γ + iσ, γ = λ > 0):…”

Section: Stability Near a Convenient Profilementioning

confidence: 99%

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A Vlasov equation with Dirac potential used in fusion plasmas

Bardos

Nouri

2012

Journal of Mathematical Physics

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Well-posedness of the Cauchy problem is analyzed for a singular Vlasov equation governing the evolution of the ionic distribution function of a quasineutral fusion plasma. The Penrose criterium is adapted to the linearized problem around a time and space homogeneous distribution function showing (due to the singularity) more drastic differences between stable and unstable situations. This pathology appears on the full nonlinear problem, well-posed locally in time with analytic initial data, but generally ill-posed in the Hadamard sense. Eventually with a very different class of solutions, mono-kinetic, which constrains the structure of the density distribution, the problem becomes locally in time well-posed.

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“…Proof. By Theorem 3.1, the operator A satisfies the hypothesis of Theorem 1.6 of [3], which explicitly gives this corollary.…”

Section: Stability Near a Convenient Profilementioning

confidence: 67%

Section: Stability Near a Convenient Profilementioning

confidence: 99%

See 1 more Smart Citation

A Vlasov equation with Dirac potential used in fusion plasmas

Bardos

Nouri

2012

Journal of Mathematical Physics

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“…Treating Cauchy problems as convolution equations has turned out to be very useful (see [6], [12], [13] and [14]). But in doing so the assumption that D(A) is dense in E is unnecessary in most of the results.…”

Section: Introductionmentioning

confidence: 99%

Distribution semigroups and abstract Cauchy problems

Kunstmann

1999

Trans. Amer. Math. Soc.

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Abstract. We present a new definition of distribution semigroups, covering in particular non-densely defined generators. We show that for a closed operator A in a Banach space E the following assertions are equivalent: (a) A generates a distribution semigroup; (b) the convolution operator δ ⊗ I − δ ⊗ A has a fundamental solution in D (L(E, D)) where D denotes the domain of A supplied with the graph norm and I denotes the inclusion D → E; (c) A generates a local integrated semigroup. We also show that every generator of a distribution semigroup generates a regularized semigroup.

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“…Recently Chazarain [2] has considered certain mixed problems for some fourth order nonstrict equations in the context of distribution semigroups. In this note we deal with mixed problems for a general class of (possibly) nonstrict equations of higher order, including those in [2]. The class of hyperbolic operators does not contain nor is it contained in those of [3], [5].…”

mentioning

confidence: 99%

Mixed boundary value problems for nonstrict hyperbolic equations

Beals¹

1972

Bull. Amer. Math. Soc.

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Let Q c ƒ?", n ^ 2, be a bounded open set with boundary dQ a regularly imbedded C 00 submanifold of R n . Let a(y; £, X) be the polynomial Let a 0 (y;^ X) be the principal part, the terms of total degree p. We assume that, for yeU^eR n 9 the roots of a 0 (y;^, •) are all real and nonzero. This implies p = 2r is even. Let b k (y\ Ç, X) be polynomials of order p k < p, k = 1,2, ...,r, with principal part b k0 (y; 0, or

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Problèmes de Cauchy abstraits et applications à quelques problèmes mixtes

Cited by 84 publications

References 10 publications

A Vlasov equation with Dirac potential used in fusion plasmas

A Vlasov equation with Dirac potential used in fusion plasmas

Distribution semigroups and abstract Cauchy problems

Mixed boundary value problems for nonstrict hyperbolic equations

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