Wyniki wyszukiwania

1

On blow-up for the Hartree equation

100%

Zheng J.

Colloquium Mathematicum

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2012

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tom 126

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nr 1

111-124

EN

We study the blow-up of solutions to the focusing Hartree equation $iu_{t} + Δu + (|x|^{-γ}*|u|²)u = 0$. We use the strategy derived from the almost finite speed of propagation ideas devised by Bourgain (1999) and virial analysis to deduce that the solution with negative energy (E(u₀) < 0) blows up in either finite or infinite time. We also show a result similar to one of Holmer and Roudenko (2010) for the Schrödinger equations using techniques from scattering theory.

2

The defocusing energy-critical Klein-Gordon-Hartree equation

63%

Miao Q., Zheng J.

Colloquium Mathematicum

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2015

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tom 140

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nr 1

31-58

EN

We study the scattering theory for the defocusing energy-critical Klein-Gordon equation with a cubic convolution $u_{tt} - Δu + u + (|x|^{-4} ∗ |u|²)u = 0$ in spatial dimension d ≥ 5. We utilize the strategy of Ibrahim et al. (2011) derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering can be reduced to disproving the existence of a soliton-like solution. Employing the technique of Pausader (2010), we consider a virial-type identity in the direction orthogonal to the momentum vector to exclude such a solution.

3

An improved maximal inequality for 2D fractional order Schrödinger operators

51%

Miao C., Yang J., Zheng J.

Studia Mathematica

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2015

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tom 230

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nr 2

121-165

EN

The local maximal operator for the Schrödinger operators of order α > 1 is shown to be bounded from $H^{s}(ℝ²)$ to L² for any s > 3/8. This improves the previous result of Sjölin on the regularity of solutions to fractional order Schrödinger equations. Our method is inspired by Bourgain's argument in the case of α = 2. The extension from α = 2 to general α > 1 faces three essential obstacles: the lack of Lee's reduction lemma, the absence of the algebraic structure of the symbol and the inapplicable Galilean transformation in the deduction of the main theorem. We get around these difficulties by establishing a new reduction lemma and analyzing all the possibilities in using the separation of the segments to obtain the analogous bilinear L²-estimates. To compensate for the absence of Galilean invariance, we resort to Taylor's expansion for the phase function. The Bourgain-Guth inequality (2011) is also generalized to dominate the solution of fractional order Schrödinger equations.

Ograniczanie wyników

2 Colloquium Mathematicum

1 Studia Mathematica

3 Zheng J.

1 Miao C.

1 Miao Q.

1 Yang J.

2 2015

1 2012

On blow-up for the Hartree equation

The defocusing energy-critical Klein-Gordon-Hartree equation

An improved maximal inequality for 2D fractional order Schrödinger operators