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Register a new userWellposedness of Cauchy problem for the Fourth Order Nonlinear Schrodinger Equations in Multi-dimensional Spaces
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We study the wellposedness of Cauchy problem for the fourth order nonlinear Schrodinger equations ipartial_t u=-epsDelta u+Delta^2 u+P((partial_x^alpha u)_{abs{alpha}ls 2}, (partial_x^alpha bar{u})_{abs{alpha}ls 2}),quad tin Real, xinReal^n, where $epsin{-1,0,1}$, $ngs 2$ denotes the spatial dimension and $P(cdot)$ is a polynomial excluding constant and linear terms.
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We continue the study on the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schrodinger equation. By considering the renormalized equation, we extend the quasi-invariance results in [30, 27] to Sobolev spaces of negative regularity. Our proof combines the approach introduced by Planchon, Tzvetkov, and Visciglia [35] with the normal form approach in [30, 27].
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