Recent Trends in Nonlinear Partial Differential Equations and Related Problems
Org:
Yakine Bahri (University of Victoria),
Slim Ibrahim (University of Victoria) and
Hiroaki Kikuchi (Meiji University)
[
PDF]
 YAKINE BAHRI, UVic
Transverse stability of line soliton for wave guide Schrödinger equations. [PDF]

The transverse stability has been studied for 2D NLS equations in the spatial cylinder $\mathbb{R} \times \mathbb{T}$. It consists of studying the stability of the standing waves for 1D NLS under the 2D NLS flow.
In this talk, we consider the waveguide Schrödinger equations
$$ i \partial_{t} \psi + \partial_{xx} \psi  D_{y} \psi +\psi^{p1} \psi = 0, \qquad \mbox{in $\mathbb{R} \times \mathbb{R}
\times \mathbb{T}$},$$
where $1 < p< 5,$ $D_{y} := \sqrt{ \partial_{yy}}$ and $\mathbb{T} =
\mathbb{R}/2\pi\mathbb{Z}$. This equation was introduced by Xu who showed a
modified scattering result.
We will show that the transverse stability depends on the frequencies
$\omega$. We will classify the result with respect to a critical frequency $\omega_{p}$ i.e. we will discuss the cases $0<\omega <
\omega_{p}$, $\omega > \omega_{p}$ and $\omega = \omega_{p}$.
This is a joint work with Hiroaki Kikuchi and Slim Ibrahim.
 JUNHO CHOI, UNIST
On an Interior Layer for the Burgers Equations in $\mathbb{R}$ [PDF]

As a simplified model derived from the NavierStokes equations, we consider the viscous Burgers equations in $\mathbb{R}$,
\begin{align}
\begin{split}
&u^\epsilon_t\epsilon u_{xx}+\frac{(u^\epsilon)^2}{2}=f(x,t),\quad x\in \mathbb{R},\quad t\geq 0\&u^\epsilon(x,0)=u_0(x),\quad x\in\mathbb{R},\&u^\epsilon\rightarrow g~as~x\rightarrow \infty ,~u^\epsilon\rightarrow h~as~x\rightarrow\infty~and~g>0>h,\quad \forall t\geq 0 .
\end{split}
\end{align}
We investigate the singular behaviors of their solutions $u^\epsilon$ as the viscosity parameter $\epsilon$ gets smaller. Indeed, when $\epsilon$ gets smaller, $u^\epsilon_x$ has viscous shocks whose slopes are proportial to $1/\epsilon$. So controlling the sharp layer is one of the most important parts in this research.
The idea is constructing the asymptotic expansions in the order of the $\epsilon$ and validating the convergence of the expansions to the solutions $u^\epsilon$ as $\epsilon\rightarrow 0$ in $L^2(0,T;H^1(\mathbb{R})))$ space. In this article, we consider the case where a single viscous shock occurs, i.e. interior layers, and we fully analyse the convergence at any order of $\epsilon$ using the socalled interior layer correctors
 YANXIA DENG, University of Victoria
Global Existence and Singularity of the Nbody Problem. [PDF]

Unstable dispersive Hamiltonian evolution equations, such as semilinear KleinGordon and Schrodinger equations, exhibit "soliton"like solutions. Amongst those one singles out the ground state, which has the lowest energy of all solitons. When the solutions are restricted to have energies at most slightly above that of the ground state, one obtains a trichotomy in forward time for this regime of energies. Recently, we found that this idea could be used to characterize solutions in the $N$body problem. The $N$body problem is an initialvalue problem for ordinary differential equations. The appropriate candidates of the ground states in the $N$body problem are the relative equilibria, which are solutions whose configuration remains an isometry of the initial configuration. The characterization of solutions for the twobody problem in this new perspective is completely solved, and it resembles the results in PDE nicely. For $N>2$, the question becomes quite complicated, and I will give some partial results.
 STEPHEN GUSTAFSON, University of British Columbia
Chiral magnetic skyrmion solutions of 2D LandauLifshitz equations: existence. [PDF]

LandauLifshitz equations are the basic dynamical equations in a micromagnetic description of a ferromagnet. They are naturally viewed as geometric evolution PDE of dispersive (``Schr\"odinger map”) or mixed dispersivediffusive type, which scale critically with respect to the physical energy in two dimensions. We describe results on existence and properties of important topological soliton solutions known as ``chiral magnetic skyrmions". Joint work with Li Wang.
 SLIM IBRAHIM, University of Victoria
Relativistic VlasovMaxwell for strongly magnetized Plasmas [PDF]

An important challenge in plasma physics is to determine whether ionized gases can be confined by strong magnetic fields. After modelling, this question leads to a penalized version of the Relativistic Vlasov Maxwell system, marked by the role of a singular term coming from the inverse of a cyclotron frequency. In this talk, I will go over the wellposedness of the corresponding Cauchy problem. This is a joint work with C. Cheverry.
 MASAHIRO IKEDA, RIKEN AIP Center
Wellposedness of fourthorder Schr\"odinger equation with derivative nonlinearities [PDF]

In this talk, we study wellposedness of the Cauchy problem to the forthorder nonlinear Schr\"odinger equations with $\gamma\in \{1,2,3\}$times derivative nonlinearities in Sobolev space $H^s(\mathbb{R})$:
\begin{equation}
\begin{cases}
i\partial_t u+\partial_x^4u=G\left((\partial_x^{k}u)_{k\le \gamma},(\partial_x^{k}\bar{u})_{k\le \gamma})\right),
& (t,x)\in I\times \mathbb{R},
\\ \ \ u_{t=0}=u_0\in H^s(\mathbb{R}),
\end{cases}
\end{equation}
where $u:I\times \mathbb{R}\rightarrow\mathbb{C}$ is an unknown function, $I:=[T,T]$ denotes the existence time interval of the function $u$, $u_0\in H^s(\mathbb{R})$ is a prescribed function, and for $s\in \mathbb{R}$, $H^s(\mathbb{R})$ denotes $L^2(\mathbb{R})$based Sobolev space. For $m\in \mathbb{N}$ with $m\ge 3$, we mainly consider the $m$th order nonlinearity $G$ of the form
\[
G(z)=G(z_1,\cdots,z_{2(\gamma+1)})
:=\sum_{\alpha=m}C_{\alpha}z^{\alpha},
\]
where $C_{\alpha}\in \mathbb{C}$ with $\alpha\in (\mathbb{N}\cup \{0\})^{2(\gamma+1)}$ are constants. The purpose of this talk is to improve the previous results obtained by several Mathematicians, that is, to treat more general nonlinearity and to prove local wellposedness of the problem in lower order Sobolev space $H^s(\mathbb{R})$.
Our proof of the wellposedness result is based on the contraction argument on a suitable function space, via the Strichartz estimates, Katotype smoothing estimates, KenigRuiz estimates, Maximal function estimates, a linear estimate for inhomogeneous term, the bilinear Strichartz type estimate and the LittlewoodPaley theory.
 TAKAHISA INUI, Osaka University
Strichartz estimates for the damped wave equation and its application to the nonlinear problem [PDF]

In this talk, we consider the damped wave equation (DW). The $L^p$$L^q$ type estimate was firstly obtained by Matsumura in 1976. After his work, many researchers have obtained such type estimates. However, there are less results for the spacetime estimates, which are so called Strichartz estimates. Recently, Watanabe (2017) proved the Strichartz estimates for DW in the low spacedimensional case. In this talk, I show the Strichartz estimates in the higher dimensional case. Moreover, we consider the energy critical nonlinear damped wave equation (NLDW). Precisely, we discuss the local wellposedness, the decay property, and the finite time blowup of the solutions to NLDW.
 ISAO KATO, Department of Mathematics, Kyoto University
Local wellposedness for the Cauchy problem of the Zakharov type system [PDF]

In this talk, we consider the Cauchy problem of the Zakharov type system. The system has no dispersion in some direction in the usual Zakharov system, so we call it the degenerated Zakharov system. The linear part of the degenerated Zakharov system is more complicated than that of the Zakharov system, so it is difficult to apply directly the local wellposedness result by GinibreTsutsumiVelo(1997). There are few wellposedness results for this system. The latest result is given by BarrosLinares(2015) for the three dimensional case, and they applied the linear estimate such as the maximal function estimate and the Strichartz estimate. To obtain the local wellposedness result with lower regularity initial data, we apply the Fourier restriction norm method. In this method, the norm of the function space is reflected in the linear part of the equation. Thus, the method is a very powerful tool to recover the derivative loss and we can obtain wellposedness with low regularity initial data. We treat the nonresonant part and the resonant part more carefully than in the case of the Zakharov system because of the lack of dispersion, then we obtain the system is locally wellposed in some anisotropic Sobolev space with low regularity.
 HIROAKI KIKUCHI, Tsuda University
Minimization problem associated with ground states to combined powertype nonlinear Schr\"odinger equations [PDF]

In this talk, we consider a minimization problem
associated to ground state solutions to the following scalar field equation
\begin{equation}
 \Delta u + \omega u  u^{p1}u 
u^{4}u = 0
\qquad \mbox{in $\mathbb{R}^{3}$},
\end{equation}
where $\omega>0$ and $1 < p <5$.
\par
we shall show that the minimization problem has no minimizer when $\omega \gg 1$ and $1 < p \leq 3$. In addition, we determine an explicit frequency threshold that classifies the existence and nonexistence.
\par
To prove our results, we employ a resolvent expansion as in
Coles and Gustafson (preprint). The resolvent expansion has a singularity because of a resonance
in three space dimensions. The frequency threshold appears thanks to the presence of this singularity.
\par
This talk is based on a joint work with T. Akahori(Shizuoka), S. Ibrahim (Victoria) and H. Nawa (Meiji)
 NOBU KISHIMOTO, Kyoto University
Illposedness of the periodic nonlinear Schr\"odinger equation with thirdorder dispersion and Raman scattering term [PDF]

We consider the nonlinear Schr\"odinger equation with thirdorder dispersion and derivative nonlinearity on the onedimensional torus:
$$ i\partial _tu + \partial _x^2u + i\partial _x^3u = c_1 u^2u +ic_2 \partial _x(u^2u) + i\gamma \partial _x(u^2)u, \qquad t\in \mathbb{R},\quad x\in \mathbb{R}/2\pi \mathbb{Z} ,$$
where $c_1,c_2$ are real constants and $\gamma$ is a complex constant.
This equation is regarded as a mathematical model for the photonic crystal fiber oscillator, and the last term, with the coefficient $\gamma$ having nonzero imaginary part, is related to the intrapulse Raman scattering effect, which is not negligible for ultrashort optical pulses.
Without the Raman scattering term (i.e., $\mathrm{Im} \gamma =0$), or for the nonperiodic problem with any complex $\gamma$, the associated Cauchy problem is known to be locally wellposed in Sobolev spaces.
We show that in the periodic setting the Raman scattering term causes illposedness (more precisely, nonexistence of localintime solutions) of the Cauchy problem in Sobolev and Gevrey spaces.
This talk is based on a joint work with Yoshio Tsutsumi (Kyoto University).
 SHENGYI SHEN, University of Victoria
Analysis about a Two fluid mode and its comparison with MHD system [PDF]

We study a two fluid system which describes the motion of two charged particles in a strict neutral incompressible plasma. We study the wellposdness of the system in both space dimensions two and three. Regardless of the size of the initial data, we prove the global wellposedness of the Cauchy problem when the space dimension is two. In space dimension three, we construct global weaksolutions, and we prove the local wellposedness of Katotype solutions. These solutions turn out to be global when the initial data are sufficiently small. We also study the stability of the solution around zero given that the initial data is small and has sufficient regularity. It turns out that our system is a system of regularityloss and the $ L^2 $ norm of lower derivatives of the solution decays. At last, this two fluid system can derive the classic MHD at least formally. Arsenio, Ibrahim and Masmoudi (2015) proved that the two fluid system converges to MHD under some constrain. We showed numerically that the two fluid system converges to MHD with no such constrain.
 IKKEI SHIMIZU, Kyoto University
Remarks on local theory for Schr\"odinger maps near harmonic maps [PDF]

In this talk, we consider the Schr\"odinger map equation (Landau–Lifshitz equation). In the study of the equation, one of the major method is to reduce it to some nonlinear Schr\"odinger equation, which is called the modified Schr\"odinger map equation. This talk will focus on the case of equivariant solutions under the Coulomb gauge condition. We show that the derivation of modified equation can be justified even from rough solutions. This result leads to an improvement on the uniqueness of solutions near harmonic maps.
 KEISUKE TAKASAO, Department of Mathematics, Kyoto University
Global existence of weak solution for volume preserving mean curvature flow via phase field method [PDF]

In this talk, we consider the phase field method for the volume preserving mean
curvature flow. Given an $C^1$ hypersurface, we prove the global existence of the weak
solution for the volume preserving mean curvature flow via the reaction diffusion equation
with a nonlocal term. In particular, we show the $L^2$boundedness of the mean curvature which is the key estimate to prove the main theorem. In addition, we prove the monotonicity formula for the reaction diffusion equation.
 TAIPENG TSAI, University of British Columbia
Existence, uniqueness, and regularity results for elliptic equations with drift terms in critical weak spaces [PDF]

We consider Dirichlet problems for linear elliptic equations of second order in divergence form
on a bounded or exterior smooth domain $\Omega$ in $\mathbb{R}^n$, $n \ge 3$, with drifts
$\mathbf{b}$ in the critical weak $L^n$space $L^{n,\infty}(\Omega ; \mathbb{R}^n )$, and
div\,$\mathbf{b}\ge0$ in $L^{n/2, \infty }(\Omega )$. We first establish existence and
uniqueness of weak solutions in $W^{1,p}(\Omega )$ or $D^{1,p}(\Omega )$ for any $p$ with $n' =
n/(n1)< p < n$. By duality, a similar result also holds for the dual problem. Next, we prove
$W^{1,n+\epsilon}$ or $W^{2, n/2+\delta}$regularity of weak solutions of the dual problem for
some $\epsilon, \delta >0$ when the domain $\Omega$ is bounded. By duality, these results enable
us to obtain a quite general uniqueness result as well as an existence result for weak solutions
belonging to $\bigcap_{p< n' }W^{1,p}(\Omega )$. Finally, we prove a uniqueness result for
exterior problems, which implies in particular that (very weak) solutions are unique in both
$L^{n/(n2),\infty}(\Omega )$ and $L^{n,\infty}(\Omega )$. This is a joint work with Hyunseok Kim.
 LI WANG, University of British Columbia
Chiral magnetic skyrmion solutions of 2D LandauLifshitz equations: stability. [PDF]

LandauLifshitz equations are the basic dynamical equations in a micromagnetic description of a ferromagnet. They are naturally viewed as geometric evolution PDE of dispersive (``Schr\"odinger map”) or mixed dispersivediffusive type, which scale critically with respect to the physical energy in two dimensions. We describe results on the stability of important topological soliton solutions known as ``chiral magnetic skyrmions". Joint work with Stephen Gustafson.
 XINWEI YU, University of Alberta
A New Type of Regularity Criterion for the 3D NavierStokes Equations [PDF]

We present a new kind of regularity criterion for the global wellposedness problem of the three dimensional NavierStokes equations in the whole space. The main novelty of this new criterion is that it involves the shape of the magnitude of the velocity. More specifically, we prove that if for every fixed time in $(0,T)$, the region of high velocity, appropriately defined with a parameter $q$, shrinks fast enough as $q\nearrow \infty$, then the solution stays regular beyond $T$. We further argue that reasonable flows satisfy our criterion, and singularity in NavierStokes is highly unlikely.
This is joint work with Prof. Chuong V. Tran of the University of St. Andrews, United Kingdom.