Goldman–Turaev formality from the Knizhnik–Zamolodchikov connection

For an oriented 2-dimensional manifold Σ of genus g with n boundary components, the space $\mathbb{C}{\pi }_1(\mathrm{\Sigma })/[\mathbb{C}{\pi }_1(\mathrm{\Sigma }),\mathbb{C}{\pi }_1(\mathrm{\Sigma })]$ carries the Goldman–Turaev Lie bialgebra structure defined in terms of intersections and self-intersections of curves. Its associated graded Lie bialgebra (under the natural filtration) is described by cyclic words in $H_1(\mathrm{\Sigma })$ and carries the structure of a necklace Schedler Lie bialgebra. The isomorphism between these two structures in genus zero has been established in [13] using Kontsevich integrals and in [2] using solutions of the Kashiwara–Vergne problem.In this note, we give an elementary proof of this isomorphism over $\mathbb{C}$. It uses the Knizhnik–Zamolodchikov connection on $\mathbb{C}\backslash \{z_1,\dots z_n\}$. We show that the isomorphism naturally depends on the complex structure on the surface. The proof of the isomorphism for Lie brackets is a version of the classical result by Hitchin [9]. Surprisingly, it turns out that a similar proof applies to cobrackets.Furthermore, we show that the formality isomorphism constructed in this note coincides with the one defined in [2] if one uses the solution of the Kashiwara–Vergne problem corresponding to the Knizhnik–Zamolodchikov associator.     

Global,decaying solutions of a focusing energy-critical heat equation in R4

We study solutions of the focusing energy-critical nonlinear heat equation $u_t=\mathrm{\Delta }u?|u{|}^{2}u$ in ${\mathbb{R}}^4$. We show that solutions emanating from initial data with energy and ${\dot{H}}^1$-norm below those of the stationary solution W are global and decay to zero, via the “concentration-compactness plus rigidity” strategy of Kenig–Merle [33], [34]. First, global such solutions are shown to dissipate to zero, using a refinement of the small data theory and the $L^2$-dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza–Seregin–Sverak [17], [18] in an argument similar to that of Kenig–Koch [32] for the Navier–Stokes equations.     

The Cauchy problem for the generalized Zakharov–Kuznetsov equation on modulation spaces

We consider the Cauchy problem for the generalized Zakharov–Kuznetsov equation ${?}_{t}u+{?}_{x_1}\mathrm{\Delta }u={?}_{x_1}(u^{m+1})$ on three and higher dimensions. We mainly study the local well-posedness and the small data global well-posedness in the modulation space $M_{2,1}^0({\mathbb{R}}^n)$ for $m\ge 4$ and $n\ge 3$. We also investigate the quartic case, i.e., $m=3$.     

The uniqueness of an indefinite nonlinear diffusion problem in population genetics,part II

We study the following Neumann problem which models the “complete dominance” case of population genetics of two alleles.

$\{\begin{array}{c}{u}_t=d{u}^{″}+g(x)u^2(1?u)\text{in}(0,1)\times (0,\infty ),\hfill \\ 0\le u\le 1\text{in}(0,1)\times (0,\infty ),\hfill \\ u^{\prime }(0,t)=u^{\prime }(1,t)=0\text{in}(0,\infty ),\hfill \end{array}$

where g changes sign in $(0,1)$. It is known that this equation has a nontrivial steady state $u_d$ for d sufficiently small [5]. It has been conjectured by Nagylaki and Lou [2] that $u_d$ is a unique nontrivial steady state if ${\int }_{\mathrm{\Omega }}g(x)dx\ge 0$. This was proved in [6] if g changes sign only once. In this paper under additional condition on $g(x)$ we treat the case when g has multiple zeros.     

Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant

This paper deals with a two-competing-species chemotaxis system with consumption of chemoattractant

$\{\begin{array}{cc}{u}_t=d_1\mathrm{\Delta }u?\mathrm{?}?(u{\chi }_1(w)\mathrm{?}w)+{\mu }_{1}u(1?u?a_{1}v),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=d_2\mathrm{\Delta }v?\mathrm{?}?(v{\chi }_2(w)\mathrm{?}w)+{\mu }_{2}v(1?a_{2}u?v),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ w_t=d_3\mathrm{\Delta }w?(\alpha u+\beta v)w,\hfill & x\in \mathrm{\Omega },t>0\hfill \end{array}$

under homogeneous Neumann boundary conditions in a bounded domain $\mathrm{\Omega }?{\mathbb{R}}^n$ ($n\ge 1$) with smooth boundary, where the initial data $(u_0,v_0)\in {(C^0(\stackrel{￣}{\mathrm{\Omega }}))}^2$ and $w_0\in W^{1,\infty }(\mathrm{\Omega })$ are non-negative and the parameters $d_1,d_2,d_3>0$, ${\mu }_1,{\mu }_2>0$, $a_1,a_2>0$ and $\alpha ,\beta >0$. The chemotactic function ${\chi }_i(w)$ ($i=1,2$) is smooth and satisfying some conditions. It is proved that the corresponding initial–boundary value problem possesses a unique global bounded classical solution if one of the following cases hold: for $i=1,2$,(i) ${\chi }_i(w)={\chi }_{0,i}>0$ and

${6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}<\frac{\pi \sqrt{d_i{d}_3}}{\sqrt{n+1}{\chi }_{0,i}}?\frac{2\sqrt{d_i{d}_3}}{\sqrt{n+1}{\chi }_{0,i}}\arctan ?\frac{d_i?d_3}{2}\sqrt{\frac{n+1}{d_i{d}_3}};$

(ii) $0<{6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}\le \frac{d_3}{3(n+1){6{\chi }_{i}6}_{L^{\infty }[0,{6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}]}}\min ?\{\frac{2d_i}{d_i+d_3},1\}$.Moreover, we prove asymptotic stabilization of solutions in the sense that:? If $a_1,a_2\in (0,1)$ and $u_0\ne 0\ne v_0$, then any global bounded solution exponentially converge to $(\frac{1?a_1}{1?a_1{a}_2},\frac{1?a_2}{1?a_1{a}_2},0)$ as $t\to \infty$;? If $a_1>1>a_2>0$ and $v_0\ne 0$, then any global bounded solution exponentially converge to $(0,1,0)$ as $t\to \infty$;? If $a_1=1>a_2>0$ and $v_0\ne 0$, then any global bounded solution algebraically converge to $(0,1,0)$ as $t\to \infty$.     

Global well-posedness of partially periodic KP-I equation in the energy space and application

In this article, we address the Cauchy problem for the KP-I equation

${?}_{t}u+{?}_x^{3}u?{?}_x^{?1}{?}_y^{2}u+u{?}_{x}u=0$

for functions periodic in y. We prove global well-posedness of this problem for any data in the energy space $\mathbf{E}=\{u\in L^2(\mathbb{R}\times \mathbb{T}),{?}_{x}u\in L^2(\mathbb{R}\times \mathbb{T}),{?}_x^{?1}{?}_{y}u\in L^2(\mathbb{R}\times \mathbb{T})\}$. We then prove that the KdV line soliton, seen as a special solution of KP-I equation, is orbitally stable under this flow, as long as its speed is small enough.     

Approximation of maximum of Gaussian random fields

This contribution is concerned with Gumbel limiting results for supremum ${\mathbb{M}}_{\mathit{n}}={\sup }_{\mathit{t}\in [\mathbf{0},{\mathit{T}}_{\mathit{n}}]}?|X_{\mathit{n}}(\mathit{t})|$ with $X_{\mathit{n}},\mathit{n}\in {\mathbb{N}}^2$ centered Gaussian random fields with continuous trajectories. We show first the convergence of a related point process to a Poisson point process thereby extending previous results obtained in [8] for Gaussian processes. Furthermore, we derive Gumbel limit results for ${\mathbb{M}}_{\mathit{n}}$ as $\mathit{n}\to \infty$ and show a second-order approximation for $\mathbb{E}{\{{\mathbb{M}}_{\mathit{n}}^p\}}^{1/p}$ for any $p\ge 1$.     

Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity

In a previous work, it was shown how the linearized strain tensor field $\mathbf{e}:=\frac{1}{2}(\mathbf{?}{\mathit{u}}^T+\mathbf{?}\mathit{u})\in {\mathbb{L}}^2(\Omega )$ can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain $\Omega ?{\mathbb{R}}^3$, instead of the displacement vector field $\mathit{u}\in {\mathit{H}}^1(\Omega )$ in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition $\mathit{u}=\mathbf{0}$ on a portion ${\Gamma }_0$ of the boundary of Ω can be recast, again as boundary conditions on ${\Gamma }_0$, but this time expressed only in terms of the new unknown $\mathbf{e}\in {\mathbb{L}}^2(\Omega )$.     

On maximizing the fundamental frequency of the complement of an obstacle

Let $\mathrm{\Omega }?{\mathbb{R}}^n$ be a bounded domain satisfying a Hayman-type asymmetry condition, and let D be an arbitrary bounded domain referred to as an “obstacle”. We are interested in the behavior of the first Dirichlet eigenvalue ${\lambda }_1(\mathrm{\Omega }?(x+D))$.First, we prove an upper bound on ${\lambda }_1(\mathrm{\Omega }?(x+D))$ in terms of the distance of the set $x+D$ to the set of maximum points $x_0$ of the first Dirichlet ground state ${?}_{{\lambda }_1}>0$ of Ω. In short, a direct corollary is that if

(1)${\mu }_{\mathrm{\Omega }}:=\underset{x}{\max }?{\lambda }_1(\mathrm{\Omega }?(x+D))$

is large enough in terms of ${\lambda }_1(\mathrm{\Omega })$, then all maximizer sets $x+D$ of ${\mu }_{\mathrm{\Omega }}$ are close to each maximum point $x_0$ of ${?}_{{\lambda }_1}$.Second, we discuss the distribution of ${?}_{{\lambda }_1(\mathrm{\Omega })}$ and the possibility to inscribe wavelength balls at a given point in Ω.Finally, we specify our observations to convex obstacles D and show that if ${\mu }_{\mathrm{\Omega }}$ is sufficiently large with respect to ${\lambda }_1(\mathrm{\Omega })$, then all maximizers $x+D$ of ${\mu }_{\mathrm{\Omega }}$ contain all maximum points $x_0$ of ${?}_{{\lambda }_1(\mathrm{\Omega })}$.