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Gyeongha Hwang

Publications and source records attributed to Gyeongha Hwang.

25 records · Page 2Linked to original sources
arXiv2015

Energy concentration of the focusing energy-critical FNLS

We consider the fractional nonlinear Schrödinger equation (FNLS) with general dispersion $|\nabla|^α$ and focusing energy-critical nonlinearities $-|u|^\frac{2α}{d-α}u$ and $-(|x|^{-2α} * |u|^2) u$. By adopting Kenig-Tsutsumi \cite{mets}, Kenig-Merle \cite{keme} and Killip-Visan \cite{kv} arguments, we show the energy concentration of radial solutions near the maximal existence time. For this purpose we use Sobolev inequalities for radial functions and establish strong energy decoupling of profiles. And we also show that when the kinetic energy is confined the maximal existence time is finite for some large class of initial data satisfying the initial energy $E(φ)$ is less than energy of ground state $E(W_α)$ but $\||\nabla|^\frac\alpha2 φ\|_{L^2} \ge \||\nabla|^\frac\alpha2 W_α\|_{L^2}$.

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

On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity

We study the Cauchy problem for the fractional Schrödinger equation $$ i\partial_tu = (m^2-Δ)^\frac\alpha2 u + F(u) in \mathbb{R}^{1+n}, $$ where $ n \ge 1$, $m \ge 0$, $1 < α< 2$, and $F$ stands for the nonlinearity of Hartree type: $$F(u) = λ(\frac{ψ(\cdot)}{|\cdot|^γ} * |u|^2)u$$ with $λ= \pm1, 0 <γ< n$, and $0 \le ψ\in L^\infty(\mathbb R^n)$. We prove the existence and uniqueness of local and global solutions for certain $α$, $γ$, $λ$, $ψ$. We also remark on finite time blowup of solutions when $λ= -1$.

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↗