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Soonsik Kwon

Publications and source records attributed to Soonsik Kwon.

36 records · Page 2Linked to original sources
arXiv2016

Normal form approach to global well-posedness of the quadratic derivative nonlinear Schrödinger equation on the circle

We consider the quadratic derivative nonlinear Schrödinger 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 Cole-Hopf transformation, we prove unconditional global well-posedness in $L^2(\mathbb{T})$, and more generally in certain Fourier-Lebesgue spaces $\mathcal{F} L^{s, p}(\mathbb{T})$, under the mean-zero 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 non-small mean-zero initial data.

math.AP↗
arXiv2016

Nonsqueezing property of the coupled KdV type system without Miura transform

We prove the nonsqueezing property of the coupled Korteweg-de Vries (KdV) equation. Relying on Gromov's nonsqueezing theorem for finite dimensional Hamiltonian systems, the argument is to approximate the solutions to the original infinite dimensional Hamiltonian system by a frequency truncated finite dimensional system, and then the nonsqueezing property is transferred to the infinite dimensional system. This is the argument used by Bourgain for the 1D cubic NLS flow, and Colliander et. al. for the KdV flow. One of main ingredients of \cite{Colliander:2005vv} is to use the Miura transform to change the KdV flow to mKdV flow. In this work, we consider the coupled KdV equations for which the Miura transform is not available. Instead of the Miura transform, we use the method of the normal form via the differentiation by parts. Although we present the proof for the coupled KdV equation, the same proof is applicable to the KdV flow, and so provide alternative simplified proof for the KdV flow.

math.AP↗
arXiv2014

The stability of nonlinear Schrödinger equations with a potential in high Sobolev norms revisited

We consider the nonlinear Schrödinger 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}

math.AP↗
arXiv2014

Well-posedness and Ill-posedness for the cubic fractional Schrödinger equations

We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schrödinger equations with Lévy indices $1 < α< 2$. We consider both non-periodic and periodic cases, and prove that the Cauchy problems are locally well-posed in $H^s$ for $s \geq \frac {2-α}4$. This is shown via a trilinear estimate in Bourgain's $X^{s,b}$ space. We also show that non-periodic equations are ill-posed in $H^s$ for $\frac {2 - 3α}{4(α+ 1)} < s < \frac {2-α}4$ in the sense that the flow map is not locally uniformly continuous.

math.AP↗
arXiv2014

Profile decompositions of fractional Schrödinger equations with angularly regular data

We study the fractional Schrödinger equations in $\mathbb R^{1+d}, d \geq 3$ of order ${d}/({d-1}) < \al < 2$. 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 mass-critical fractional Hartree equations.

math.AP↗
arXiv2012

Profile decompositions and Blowup phenomena of mass critical fractional Schrödinger equations

We study, under the radial symmetry assumption, the solutions to the fractional Schrödinger equations of critical nonlinearity in $\mathbb R^{1+d}, d \geq 2$, with Lévy index ${2d}/({2d-1}) < \al < 2$. We firstly prove the linear profile decomposition and then apply it to investigate the properties of the blowup solutions of the nonlinear equations with mass-critical Hartree type nonlineartity.

math.AP↗
arXiv2012

Rough solutions of the fifth-order KdV equations

We consider the Cauchy problem of the fifth-order equation arising from the Korteweg-de 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 well-posedness for s >= 2. The method is a short time X^{s,b} space, which is first developed by Ionescu-Kenig-Tataru in the context of the KP-I equation. In addition, we use a weight on localized X^{s,b} structures to reduce the contribution of high-low frequency interaction where the low frequency has large modulation. As an immediate result from a conservation law, we have the fifth-order equation in the KdV hierarchy is globally well-posed in the energy space H^2.

math.AP↗
arXiv2012

Nonexistence of soliton-like solutions for defocusing generalized KdV equations

In this short note, we consider the global dynamics of the defocusing generalized KdV equations: u_t + u_{xxx} = (|u|^{p-1}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 soliton-like solutions with decaying assumption.

math.AP↗
arXiv2011

A remark on normal forms and the "upside-down" I-method for periodic NLS: growth of higher Sobolev norms

We study growth of higher Sobolev norms of solutions to the one-dimensional periodic nonlinear Schrodinger equation (NLS). By a combination of the normal form reduction and the upside-down I-method, we establish \|u(t)\|_{H^s} \lesssim (1+|t|)^{α(s-1)+} with α= 1 for a general power nonlinearity. In the quintic case, we obtain the above estimate with α= 1/2 via the space-time 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 α= 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.

math.AP↗
arXiv2011

Poincaré-Dulac normal form reduction for unconditional well-posedness of the periodic cubic NLS

We implement an infinite iteration scheme of Poincare-Dulac normal form reductions to establish an energy estimate on the one-dimensional cubic nonlinear Schrodinger equation (NLS) in C_t L^2(T), without using any auxiliary function space. This allows us to construct weak solutions of NLS in C_t L^2(T)$ with initial data in L^2(T) as limits of classical solutions. As a consequence of our construction, we also prove unconditional well-posedness of NLS in H^s(T) for s \geq 1/6.

math.AP↗
arXiv2011

Bilinear local smoothing estimate for Airy equation

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.

math.AP↗
arXiv2011

On unconditional well-posedness of modified KdV

Bourgain(1993) proved that the periodic modified KdV equation (mKdV) is locally well-posed 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 well-posedness 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 septi-linear estimates after the second differentiation by parts.

math.AP↗
arXiv2009

On the mass-critical generalized KdV equation

We consider the mass-critical generalized Korteweg--de Vries equation $$(\partial_t + \partial_{xxx})u=\pm \partial_x(u^5)$$ for real-valued functions $u(t,x)$. We prove that if the global well-posedness and scattering conjecture for this equation failed, then, conditional on a positive answer to the global well-posedness and scattering conjecture for the mass-critical nonlinear Schrödinger equation $(-i\partial_t + \partial_{xx})u=\pm (|u|^4u)$, there exists a minimal-mass blowup solution to the mass-critical generalized KdV equation which is almost periodic modulo the symmetries of the equation. Moreover, we can guarantee that this minimal-mass blowup solution is either a self-similar solution, a soliton-like solution, or a double high-to-low frequency cascade solution.

math.AP↗
arXiv2007

Well-posedness and ill-posedness of the fifth order modifed KdV equation

We consider the initial value problem of the fifth order modified KdV equation on the Sobolev spaces. \partial_t u - \partial_x^5u + c_1\partial_x^3(u^3) + c_2u\partial_x u\partial_x^2 u + c_3uu\partial_x^3 u =0, u(x,0)= u_0(x) where $ u:R\timesR \to R $ and $c_j$'s are real. We show the local well-posedness in H^s(R) for s \geq 3/4 via the contraction principle on $X^{s,b}$ space. Also, we show that the solution map from data to the solutions fails to be uniformly continuous below $H^{3/4}(R)$. The counter example is obtained by approximating the fifth order mKdV equation by the cubic NLS equation.

math.AP↗