Publications (16)13.38 Total impact
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ABSTRACT: We consider the quadratic derivative nonlinear Schr\"odinger equation (dNLS) on the circle. In particular, we develop an infinite iteration scheme of normal form reductions for dNLS. By combining this normal form procedure with the ColeHopf transformation, we prove unconditional global wellposedness in $L^2(\mathbb{T})$, and more generally in certain FourierLebesgue spaces $\mathcal{F} L^{s, p}(\mathbb{T})$, under the meanzero and smallness assumptions. As a byproduct, we construct an infinite sequence of quantities that are invariant under the dynamics. We also show the necessity of the smallness assumption by explicitly constructing a finite time blowup solution with nonsmall meanzero initial data.  [Show abstract] [Hide abstract]
ABSTRACT: We study the low regularity wellposedness of the 1dimensional cubic nonlinear fractional Schrodinger equations with Levy indices 1 < alpha < 2. We consider both nonperiodic and periodic cases, and prove that the Cauchy problems are locally wellposed in HS for s >= 2alpha/4. This is shown via a trilinear estimate in Bourgain's Xs,Xb space. We also show that nonperiodic equations are illposed in HS for 23 alpha/4(alpha+1) < 2alpha/4 in the sense that the flow map is not locally uniformly continuous. 
Article: The stability of nonlinear Schr\"odinger equations with a potential in high Sobolev norms revisited
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ABSTRACT: We consider the nonlinear Schr\"odinger equations with a potential on $\mathbb T^d$. For almost all potentials, we show the almost global stability in very high Sobolev norms. We apply an iteration of the Birkhoff normal form, as in the formulation introduced by Bourgain \cite{Bo00}. This result reprove a dynamical consequence of the infinite dimensional Birkhoff normal form theorem by Bambusi and Grebert \cite{BG}  [Show abstract] [Hide abstract]
ABSTRACT: We study the fractional Schr\"odinger equations in $\mathbb R^{1+d}, d \geq 3$ of order ${d}/({d1}) < \al < 2$. Under the angular regularity assumption we prove linear and nonlinear profile decompositions which extend the previous results \cite{chkl2} to data without radial assumption. As applications we show blowup phenomena of solutions to masscritical fractional Hartree equations.  [Show abstract] [Hide abstract]
ABSTRACT: We study 1dimensional fractional Schr\"odinger equations with L\'{e}vy index $1 < \alpha < 2$. By using $X^{s,b}$ space, we prove local wellposedness for the equation in $H^s, s \geq \frac {2\alpha}4$. We also show the illposedness when $s < \frac{2\alpha}{4}$.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the fractional Schr\"odinger equations with focusing Hartree type nonlinearity. When the energy is negative, we use a Glassey's virial type argument to show the finite time blowup of solutions. 
Article: Profile decompositions and Blowup phenomena of mass critical fractional Schr\"odinger equations
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ABSTRACT: We study, under the radial symmetry assumption, the solutions to the fractional Schr\"odinger equations of critical nonlinearity in $\mathbb R^{1+d}, d \geq 2$, with L\'{e}vy index ${2d}/({2d1}) < \al < 2$. We firstly prove the linear profile decomposition and then apply it to investigate the properties of the blowup solutions of the nonlinear equations with masscritical Hartree type nonlineartity.  [Show abstract] [Hide abstract]
ABSTRACT: In this short note, we consider the global dynamics of the defocusing generalized KdV equations: u_t + u_{xxx} = (u^{p1}u)_x. We use Tao's theorem that the energy moves faster than mass to prove a moment type dispersion estimate. As an application of the dispersion estimate, we show that there is no solitonlike solutions with decaying assumption.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the Cauchy problem of the fifthorder equation arising from the Kortewegde Vries (KdV) hierarchy u_t + u_{xxxxx} + c_1u_{x} u_{xx} + c_2u u_{x} = 0 x,t \in \R We prove a priori bound of solutions for H^s(\R) with s >= 5/4 and the local wellposedness for s >= 2. The method is a short time X^{s,b} space, which is first developed by IonescuKenigTataru in the context of the KPI equation. In addition, we use a weight on localized X^{s,b} structures to reduce the contribution of highlow frequency interaction where the low frequency has large modulation. As an immediate result from a conservation law, we have the fifthorder equation in the KdV hierarchy is globally wellposed in the energy space H^2. 
Article: PoincaréDulac Normal Form Reduction for Unconditional WellPosedness of the Periodic Cubic NLS
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ABSTRACT: We implement an infinite iteration scheme of PoincaréDulac normal form reductions to establish an energy estimate on the onedimensional cubic nonlinear Schrödinger equation (NLS) in \({C_tL^2(\mathbb{T})}\), without using any auxiliary function space. This allows us to construct weak solutions of NLS in \({C_tL^2(\mathbb{T})}\) with initial data in \({L^2(\mathbb{T})}\) as limits of classical solutions. As a consequence of our construction, we also prove unconditional wellposedness of NLS in \({H^s(\mathbb{T})}\) for \({s \geq \frac{1}{6}}\).  [Show abstract] [Hide abstract]
ABSTRACT: We study growth of higher Sobolev norms of solutions to the onedimensional periodic nonlinear Schrodinger equation (NLS). By a combination of the normal form reduction and the upsidedown Imethod, we establish \u(t)\_{H^s} \lesssim (1+t)^{\alpha (s1)+} with \alpha = 1 for a general power nonlinearity. In the quintic case, we obtain the above estimate with \alpha = 1/2 via the spacetime estimate due to Bourgain [4], [5]. In the cubic case, we concretely compute the terms arising in the first few steps of the normal form reduction and prove the above estimate with \alpha = 4/9. These results improve the previously known results (except for the quintic case.) In Appendix, we also show how Bourgain's idea in [4] on the normal form reduction for the quintic nonlinearity can be applied to other powers.  [Show abstract] [Hide abstract]
ABSTRACT: In this short note, we prove a refinement of bilinear local smoothing estimate to Airy solutions, when the frequency support of two wave are separated. As an application we prove a smoothing property of a bilinear form.  [Show abstract] [Hide abstract]
ABSTRACT: We prove scattering of solutions just below the energy norm of the 3D KleinGordon equation: this result extends those obtained in the energy class and those obtained below the energy norm under the additional assumption of spherical symmetry. It is wellknown that the notion of scattering is closely related to that of decay. The main difficuly is that, unlike the radial case, there is no useful decay estimate available. The idea is to generate some decay estimates, by means of concentration and a lowhigh frequency decomposition. Comment: 41 pages  [Show abstract] [Hide abstract]
ABSTRACT: Bourgain(1993) proved that the periodic modified KdV equation (mKdV) is locally wellposed in Sobolev spave H^s(T), s >= 1/2, by introducing new weighted Sobolev spaces X^s,b, where the uniqueness holds conditionally, namely in the intersection of C([0, T]; H^s) and X^s,b. In this paper, we establish unconditional wellposedness of mKdV in H^s(T), s >= 1/2, i.e. we in addition establish unconditional uniqueness in C([0, T]; H^s), s >= 1/2, of solutions to mKdV. We prove this result via differentiation by parts. For the endpoint case s = 1/2, we perform careful quinti and septilinear estimates after the second differentiation by parts.  [Show abstract] [Hide abstract]
ABSTRACT: We prove an improved version of bilinear local smoothing estimate to Airy solutions. Using this we study a smoothing property of Duhamel part of nonlinear solutions to the masscritical generalized KdV equation.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the masscritical generalized Kortewegde Vries equation $$(\partial_t + \partial_{xxx})u=\pm \partial_x(u^5)$$ for realvalued functions $u(t,x)$. We prove that if the global wellposedness and scattering conjecture for this equation failed, then, conditional on a positive answer to the global wellposedness and scattering conjecture for the masscritical nonlinear Schr\"odinger equation $(i\partial_t + \partial_{xx})u=\pm (u^4u)$, there exists a minimalmass blowup solution to the masscritical generalized KdV equation which is almost periodic modulo the symmetries of the equation. Moreover, we can guarantee that this minimalmass blowup solution is either a selfsimilar solution, a solitonlike solution, or a double hightolow frequency cascade solution. Comment: References added/updated
Publication Stats
65  Citations  
13.38  Total Impact Points  
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Institutions

20092015

Korea Advanced Institute of Science and Technology
 Department of Mathematical Sciences
Sŏul, Seoul, South Korea
