Well-posedness for degenerate Schrödinger equations

Cicognani, Massimo; Reissig, Michael

doi:10.3934/eect.2014.3.15

Cited by 23 publications

(28 citation statements)

References 12 publications

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“…We want to mention that the local well-posedness for degenerate Schrödinger operators has not been intensively studied. We have a significant result due to Cicognani and Reissig in [2] where the local well-posedness of the linear Cauchy problem for degenerate Schrödinger operators with degenerate time-dependent coefficients is considered. Schrödinger operators of the same form considered in [2] have also been analyzed in [9] where it is shown that some weighted smoothing estimates are satisfied by the solutions both of the linear and of the nonlinear problem.…”

Section: Final Remarks and Open Problemsmentioning

confidence: 99%

Local Solvability of Some Partial Differential Operators with Non-smooth Coefficients

Federico

2020

Springer INdAM Series

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In this paper we will analyze the local solvability property of some second order linear degenerate partial differential operators with non-smooth coefficients. We will start by considering some operators with C α,1 coefficients, with α = 0, 1, having a kind of affine structure. Next, we will study operators with a more general structure having C 0,1 or L ∞ coefficients. In both cases the local solvability will be analyzed at multiple characteristic points where the principal symbol may possibly change sign.

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Section: Final Remarks and Open Problemsmentioning

confidence: 99%

Local Solvability of Some Partial Differential Operators with Non-smooth Coefficients

Federico

2020

Springer INdAM Series

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“…where p is a positive integer and a p (t) is a real valued function (cf. [7,8]). This large class of equations includes for instance strictly hyperbolic equations (p = 1) and Schrödinger-type equations (p = 2).…”

Section: Introductionmentioning

confidence: 99%

On the Sharp Gårding Inequality for Operators with Polynomially Bounded and Gevrey Regular Symbols

Junior

Cappiello

2020

Mathematics

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In this paper, we analyze the Friedrichs part of an operator with polynomially bounded symbol. Namely, we derive a precise expression of its asymptotic expansion. In the case of symbols satisfying Gevrey estimates, we also estimate precisely the regularity of the terms in the asymptotic expansion. These results allow new and refined applications of the sharp Gårding inequality in the study of the Cauchy problem for p-evolution equations.

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“…In the more general situation we deal with space-time variable coefficients, at least in the first order part, and we need to replace the standard use of the Fourier transform with the use of pseudo-differential calculus. This will allow us to obtain smoothing estimates for the linear operator (1) and its non homogenous counterpart. Smoothing estimates for an operator such as (1) where α = 0 are by now classical results, see for example [1,2,4,6].…”

Section: Introductionmentioning

confidence: 99%

“…This will allow us to obtain smoothing estimates for the linear operator (1) and its non homogenous counterpart. Smoothing estimates for an operator such as (1) where α = 0 are by now classical results, see for example [1,2,4,6].…”

Section: Introductionmentioning

confidence: 99%

“…In proving smoothing estimates for the operator (1), the main problems are given by the presence of the time degeneracy in the second order term and by the presence of the first order term b(t, x) • ∇ x . In particular, the time degeneracy has to be managed in order to apply a method similar to that of Mizohata [12] and Doi [3] to absorb the spacetime variable coefficients first order term through the application of the Sharp Gårding inequality.…”

Section: Introductionmentioning

confidence: 99%

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Smoothing effect for time-degenerate Schrödinger operators

Federico,

Staffilani

2020

Preprint

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In this work we consider an example of a linear time degenerate Schrödinger operator. We show that with the appropriate assumptions the operator satisfies a Kato smoothing effect. We also show that the solutions to the nonlinear initial value problems involving this operator and polynomial derivative nonlinearities are locally well-posed and their solutions also satisfy the same smoothing estimates as the linear solutions.

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Well-posedness for degenerate Schrödinger equations

Cited by 23 publications

References 12 publications

Local Solvability of Some Partial Differential Operators with Non-smooth Coefficients

Local Solvability of Some Partial Differential Operators with Non-smooth Coefficients

On the Sharp Gårding Inequality for Operators with Polynomially Bounded and Gevrey Regular Symbols

Smoothing effect for time-degenerate Schrödinger operators

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