Global Dispersive Solutions for the Gross–Pitaevskii Equation in Two and Three Dimensions

Gustafson, Stephen J.; Nakanishi, Kenji; Tsai, Tai‐Peng

doi:10.1007/s00023-007-0336-6

Cited by 54 publications

(89 citation statements)

References 27 publications

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“…Jones, S. J. Putterman and P. H. Roberts (see [34])).This would imply that there is no scattering even for small energy initial data when N = 2. Despite these issues, scattering has been obtained in a series of papers by Gustafson, Nakanishi and Tsai in [28,29,30]. For N ≥ 4 they proved scattering for small initial data in H N 2 −1 (R N ), the case N = 3 is much more intricate and requires the data to be small in weighted H 1 (R N ) spaces (to which, nevertheless, traveling waves belong).…”

Section: On the Gross Pitaevskii Equationmentioning

confidence: 99%

“…There exists a constant C such that the following inequality is true: for all s such that −N min( 30) which is called a multilinear Fourier multiplier with symbol B, thus we identify the symbol to the operator.…”

Section: Littlewood-paley Decompositionmentioning

confidence: 99%

“…Indeed due to divisors coming from the phase integrations in the main estimates of theorem 2.2, we shall as in [30] use another bilinear estimates (see [30]). …”

Section: Remarkmentioning

confidence: 99%

“…The main feature is that our solutions have no vacuum for all time. Our construction uses in a crucial way some deep results on the scattering of the Gross-Pitaevskii equation due to Gustafson, Nakanishi and Tsai in [28,29,30]. An important part of the paper is devoted to explain the main technical issues of the scattering in [29] and we give a detailed proof in order to make it more accessible.…”

mentioning

confidence: 99%

See 3 more Smart Citations

From the Gross–Pitaevskii equation to the Euler Korteweg system, existence of global strong solutions with small irrotational initial data

Audiard

Haspot

2017

Annali di Matematica

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In this paper we prove the global well-posedness for small data for the Euler Korteweg system in dimension N ≥ 3, also called compressible Euler system with quantum pressure. It is formally equivalent to the Gross-Pitaevskii equation through the Madelung transform. The main feature is that our solutions have no vacuum for all time. Our construction uses in a crucial way some deep results on the scattering of the Gross-Pitaevskii equation due to Gustafson, Nakanishi and Tsai in [28,29,30]. An important part of the paper is devoted to explain the main technical issues of the scattering in [29] and we give a detailed proof in order to make it more accessible. Bounds for long and short times are treated with special care so that the existence of solutions does not require smallness of the initial data in H s , s > N 2 . The optimality of our assumptions is also discussed.

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Section: On the Gross Pitaevskii Equationmentioning

confidence: 99%

Section: Littlewood-paley Decompositionmentioning

confidence: 99%

“…Indeed due to divisors coming from the phase integrations in the main estimates of theorem 2.2, we shall as in [30] use another bilinear estimates (see [30]). …”

Section: Remarkmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

From the Gross–Pitaevskii equation to the Euler Korteweg system, existence of global strong solutions with small irrotational initial data

Audiard

Haspot

2017

Annali di Matematica

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“…In dimension two, where a scattering theory is excluded due to the existence of small energy traveling-waves, they constructed dispersive solutions with some prescribed data at infinity (see [GNT07]). …”

Section: Using the Madelung Transformation φ(X T) = ρ(X T)ementioning

confidence: 99%

Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity: some results and open problems

Maris¹

2010

Journées Équations Aux Dérivées Partielles

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Long‐term stability of periodic smooth irrotational flows to the electrons Euler–Poisson system in three dimensions

2023

Math Methods in App Sciences

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We are concerned with the one‐fluid Euler–Poisson system for the electrons on the cubes in three dimensions. With the bootstrap estimates, we prove long‐term stability of periodic solutions of this system with small smooth irrotational perturbations of a constant background. Our main result is that the solutions are well‐posed for a time at least ε−12+normalOfalse(1Nfalse)R54−normalOfalse(1Nfalse)$$ {\varepsilon}^{-\frac{1}{2}+\mathrm{O}\left(\frac{1}{N}\right)}{R}^{\frac{5}{4}-\mathrm{O}\left(\frac{1}{N}\right)} $$, where ε$$ \varepsilon $$ is the size of the initial data, the sufficiently large R≫1$$ R\gg 1 $$ is the side length of the cubes, and N$$ N $$ is the index of the classical Sobolev spaces HN$$ {H}^N $$.

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Global Dispersive Solutions for the Gross–Pitaevskii Equation in Two and Three Dimensions

Cited by 54 publications

References 27 publications

From the Gross–Pitaevskii equation to the Euler Korteweg system, existence of global strong solutions with small irrotational initial data

From the Gross–Pitaevskii equation to the Euler Korteweg system, existence of global strong solutions with small irrotational initial data

Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity: some results and open problems

Long‐term stability of periodic smooth irrotational flows to the electrons Euler–Poisson system in three dimensions

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