Aram L. Karakhanyan
In this paper we prove Bernstein type theorems for a class of stationary points of the Alt-Caffarelli functional in $\mathbb R^2$ and $\mathbb R^3$.
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This paper is aimed to establish well-posedness in several settings for the Cauchy problem associated to a model arising in the study of capillary-gravity flows. More precisely, we determinate local well-posedness conclusions in classical Sobolev spaces and some spaces adapted to the energy of the equation. A key ingredient is a commutator estimate involving the Hilbert transform and fractional derivatives. We also study local well-posedness for the associated periodic initial value problem. Additionally, by determining well-posedness in anisotropic weighted Sobolev spaces as well as some uniq…
In the present paper we study a singular perturbation problem for a Navier-Stokes-Korteweg model with Coriolis force. Namely, we perform the incompressible and fast rotation asymptotics simultaneously, while we keep the capillarity coefficient constant in order to capture surface tension effects in the limit. We consider here the case of variable rotation axis: we prove the convergence to a linear parabolic-type equation with variable coefficients. The proof of the result relies on compensated compactness arguments. Besides, we look for minimal regularity assumptions on the variations of the a…
Alexandrov-type theorem for singular capillary CMC hypersurfaces in the half-space
本文研究了半空间中相对等周型问题临界点的分类，通过引入新型移位距离函数，在特定正则性假设下证明了奇异毛细管CMC超曲面的Alexandrov型定理。
Some remarks on singular capillary cones with free boundary
本文研究奇异极小毛细锥的平坦性，提出稳定性判据，证明在特定维度与曲率条件下锥为平坦，并改进图形化超曲面的正则性。
We consider the two-dimensional capillary-gravity water waves problem where the free surface $Γ_t$ intersects the bottom $Γ_b$ at two contact points. In our previous works \cite{MW2, MW3}, the local well-posedness for this problem has been proved with the contact angles less than $π/16$. In this paper, we study the case where the contact angles belong to $(0, π/2)$. It involves much worse singularities generated from corresponding elliptic systems, which have a strong influence on the regularities for the free surface and the velocity field. Combining the theory of singularity decompositions f…
Improved regularity for minimizing capillary hypersurfaces
本文研究了极小毛细超曲面的奇点集正则性，证明其奇点集余维数至少为4，并在毛细角接近0、π/2或π时基于与单相Bernoulli问题的联系给出改进估计。
The objective of the present work is to provide a well-posedness result for a capillary driven thin film equation with insoluble surfactant. The resulting parabolic system of evolution equations is not only strongly coupled and degenerated, but also of mixed orders. To the best of our knowledge the only well-posedness result for a capillary driven thin film with surfactant is provided in [4] by the same author, where a severe smallness condition on the surfactant concentration is assumed to prove the result. Thus, in spite of an intensive analytical study of thin film equations with surfactant…
Local Well-Posedness for Compressible Capillary-Gravity Water Waves with Acute Contact Angles
本文研究具有锐角接触角的可压缩毛细重力水波的局部适定性，通过建立几何结构、奇异性分析与先验能量估计，得到了首个含接触点的可压缩欧拉方程自由边值问题的局部适定性结果。
In this Letter we consider long capillary-gravity waves described by a fully nonlinear weakly dispersive model. First, using the phase space analysis methods we describe all possible types of localized travelling waves. Then, we especially focus on the critical regime, where the surface tension is exactly balanced by the gravity force. We show that our long wave model with a critical Bond number admits stable travelling wave solutions with a singular crest. These solutions are usually referred to in the literature as peakons or peaked solitary waves. They satisfy the usual speed-amplitude rela…
Equivariant critical point theory and bifurcation of $3d$ gravity-capillary Stokes waves
本文通过变分Lyapunov-Schmidt约简与等变Morse-Conley理论，研究了三维重力-毛细斯托克斯波的存在性和分岔现象，揭示了斯托克斯波的拓扑分组特性。
Bifurcation analysis for axisymmetric capillary water waves with vorticity and swirl
该研究针对带一般涡量、旋度及表面张力的定常轴对称水波，通过变量替换克服坐标奇异性，将问题转化为适用于Rabinowitz全局分岔定理的形式，构造出从平坦层流分岔的局部与全局解曲线。
In this paper, we study the forced mean curvature flows and the prescribed mean curvature equations of both graphs and level-sets with capillary-type boundary conditions on a $C^3$ bounded domain, which is not necessarily convex. We prove a priori gradient estimates locally Lipschitz in time. Under an assumption on the forcing term, we prove that the gradient estimates are globally Lipschitz in time. As a consequence, we obtain the existence theorem of solutions. In our formulation, we recover the known results of the gradient estimates on a strictly convex $C^3$ bounded domain. Next, we study…
On the Viscosity Solutions of Parabolic p-Laplacian Equations with Capillary-Type Boundary Conditions
本文研究了具有毛细型边界条件的奇异/退化抛物p-拉普拉斯方程粘性解的适定性及大时间渐进行为，通过最大原理和近似过程结合椭圆特征值问题，证明了解的存在性、唯一性和渐进行为，同时讨论了非凸域的扩展。
Existence of pure capillary solitary waves in constant vorticity flows
本文研究了恒定涡度流中纯表面张力刚性机制的破坏，通过构造有限深度的纯表面张力孤立波，验证了长波标度下的KdV方程及homoclinic轨道的存在性。
Two-dimensional capillary liquid drop: Craig-Sulem formulation on $\mathbb{T}^1$ and bifurcations from multiple eigenvalues of rotating waves
该研究针对二维近圆形毛细液滴自由边界问题，推导了圆周及一维平坦环面上的Craig-Sulem公式，证明其哈密顿结构与运动常数，通过多重特征值分岔得到角动量参数化的唯一对称旋转波解轨道。
New minimal surfaces in the sphere from capillary minimal cones
该研究通过双倍自由边界极小锥构造了球面上的新极小曲面，并展示了毛细功能在单相自由边界与极小曲面之间的联系。
In this paper, we discuss singular Neumann boundary problem for a class of nonlinear parabolic equations in one space dimension. Our boundary problem describes motion of a planar curve sliding along the boundary with a zero contact angle, which can be viewed as a limiting model for the capillary phenomenon. We study the uniqueness and existence of solutions by using the viscosity solution theory. We also show the convergence of the solution to a traveling wave as time proceeds to infinity when the initial value is assumed to be convex.
We study the partial differential equation max{Lu - f, H(Du)}=0 where u is the unknown function, L is a second-order elliptic operator, f is a given smooth function and H is a convex function. This is a model equation for Hamilton-Jacobi-Bellman equations arising in stochastic singular control. We establish the existence of a unique viscosity solution of the Dirichlet problem that has a Holder continuous gradient. We also show that if H is uniformly convex, the gradient of this solution is Lipschitz continuous.
Singular limit phenomenon in a nonlinear elliptic model arising in electrochemistry
研究电化学腐蚀中带符号指数型非线性Neumann边值条件的奇异摄动调和问题，证明低电导率下解迹在分数阶Sobolev空间中的收敛性，并得到光滑边界结的尖锐对数能量展开。
Topological singularities arising from fractional-gradient energies
该研究证明平面正则区域上经标度的Ginzburg-Landau型泛函相对于雅可比平坦收敛Γ收敛于涡旋型能量，通过与含Riesz位势的标准泛函比较得紧性与Γ-下极限，用连接截断类涡旋函数的直接论证得Γ-上极限。
We study the low Mach low Freude numbers limit in the compressible Navier-Stokes equations and the transport equation for evolution of an entropy variable -- the potential temperature $Θ$. We consider the case of well-prepared initial data on "flat" tours and Reynolds number tending to infinity, and the case of ill-prepared data on an infinite slab. In both cases, we show that the weak solutions to the primitive system converge to the solution to the anelastic Navier-Stokes system and the transport equation for the second order variation of $Θ$
We consider singular perturbations of elliptic systems depending on a parameter ? such that, for ? = 0 the boundary conditions are not adapted to the equation (they do not satisfy the Shapiro - Lopatinskii condition). The limit holds only in very abstract spaces out of distribu- tion theory involving complexi?cation and non-local phenomena. This system appears in the thin shell theory when the middle surface is el- liptic and the shell is fixed on a part of the boundary and free on the rest. We use a heuristic reasoning applying some simplifications which allow to reduce the original problem i…
In this work, we prove the existence of a family of solutions of the Allen-Cahn equation with nonlinear Neumann boundary condition under some constraints, whose nodal sets concentrate asymptotically to a given volume nondegenerate capillary hypersurface in a compact Riemannian manifold. Our construction is inspired by the works of Pacard and of Pacard and Ritoré.
We are interested in the general Choquard equation \begin{multline*} \sqrt{\strut -Δ+ m^2} \ u - mu + V(x)u - \fracμ{|x|} u = \left( \int_{\mathbb{R}^N} \frac{F(y,u(y))}{|x-y|^{N-α}} \, dy \right) f(x,u) - K (x) |u|^{q-2}u \end{multline*} under suitable assumptions on the bounded potential \(V\) and on the nonlinearity \(f\). Our analysis extends recent results by the second and third author on the problem with $μ= 0$ and pure-power nonlinearity $f(x,u)=|u|^{p-2}u$. We show that, under appropriate assumptions on the potential, whether the ground state does exist or not. Finally, we study the a…
The Brio system is a two-by-two system of conservation laws arising as a simplified model in ideal magnetohydrodynamics (MHD). The system has the form \begin{align*} \partial_t u+\partial_x \Big({\textstyle \frac{u^2+v^2}{2}}\Big)=0,\\ \partial_t v+\partial_x \big(v(u-1)\big)=0. \end{align*} It was found in previous works that the standard theory of hyperbolic conservation laws does not apply to this system since the characteristic fields are not genuinely nonlinear on the set $v=0$. As a consequence, certain Riemann problems have no weak solutions in the traditional class of functions of boun…
This paper is concerned with the following singularly perturbed non-local semi-linear problem \begin{equation} \label{h} \tag{$\ast$} \begin{cases} \varepsilon^2 Δu=\frac{m}{\int_Ωe^{u}{\mathrm{d}x}}u e^u\quad &\mathrm{in}~Ω,\\ u= u_0~&\mathrm{on}~\partialΩ, \end{cases} \end{equation} which corresponds to the stationary problem of a chemotaxis system describing the aerobic bacterial movement, where $Ω$ is a smooth bounded domain in $\mathbb{R}^N (N\geq 1)$, $\varepsilon, m$ and $u_0$ are positive constants. We show that the problem \eqref{h} admits a unique classical solution which is of bound…
Linear stability analysis for a system of singular amplitude equations arising in biomorphology
本文研究伴随守恒律的对流 Turing 分岔中奇异振幅方程组指数周期解的线性稳定性，通过双参数矩阵摄动分析证明已有必要条件对 Schneider 扩散稳定性亦充分，揭示其沿非正规双曲慢流形及横向均值模式的稳定性机制。
We consider a class of generalised stochastic porous media equations with multiplicative Lipschitz continuous noise. These equations can be related to physical models exhibiting self-organised criticality. We show that these SPDEs have unique SVI solutions which depend continuously on the initial value. In order to formulate this notion of solution and to prove uniqueness in the case of a slowly growing nonlinearity, the arising energy functional is analysed in detail.
On a singular perturbation problem arising in the theory of Evolutionary Distributions
针对各向异性演化分布系统的奇异摄动问题，本文建立弱解存在性理论，提出数值求解算法，并通过种群竞争模型实验展示其显著特征。
We consider positive singular solutions of PDEs arising from double phase functionals. Exploiting a rather new version of the moving plane method originally developed by Sciunzi, we prove symmetry and monotonicity properties of such solutions.
Moore-Gibson-Thompson (MGT) equations, which describe acoustic waves in a heterogeneous medium, are considered. These are the third order in time evolutions of a predominantly hyperbolic type. MGT models account for a finite speed propagation due to the appearance of thermal relaxation coefficient τ {>} {0} in front of the third order time derivative. Since the values of τ are relatively small and often negligible, it is important to understand the asymptotic behavior and characteristics of the model when τ {\to} {0}. This is a particularly delicate issue since the τ- dynamics is governed by a…
Fredholm properties of singular elliptic operators arising in the study of point defects
本文研究平面斑图系统点缺陷动力学中奇异椭圆算子的Fredholm性质，通过双重加权Sobolev空间刻画无穷远与原点附近的代数衰减/增长，并给出算子单射、满射或同构的权重条件及核与值域的精确描述。
This article describes the use of algebraic methods in a phase plane analysis of ordinary differential equations. The method is illustrated by the study of capillary-gravity steady surface waves propagating in shallow water. We consider the (fully nonlinear, weakly dispersive) Serre-Green-Naghdi equations with surface tension, because it provides a tractable model that, in the same time, is not too simple so the interest of the method can be emphasised. In particular, we analyse a special class of solutions, the solitary waves, which play an important role in many fields of Physics. In capilla…
We continue the study of positive singular solutions of PDEs arising from double phase functionals started in [6]. In particular, we consider the case $p<q < 2$, and we relax the assumption on the capacity of the singular set using an intrinsic notion of capacity.
We show that singular sets of free boundaries arising in codimension one anisotropic geometric variational problems are $\mathcal H ^{n-3}$-negligible, where $n$ is the ambient space dimension. In particular our results apply to capillarity type problems, and establish everywhere regularity in the three-dimensional case.
We consider a singular Liouville equation on a compact surface, arising from the study of Chern-Simons vortices in a self dual regime. Using new improved versions of the Moser-Trudinger inequalities (whose main feature is to be scaling invariant) and a variational scheme, we prove new existence results.
Motivated by an equation arising in magnetohydrodynamics, we prove that Holder continuous weak solutions of a nonlinear parabolic equation with singular drift velocity are classical solutions. The result is proved using the space-time Besov spaces introduced by Chemin and Lerner, combined with energy estimates, without any minimality assumption on the Holder exponent of the weak solutions.
A singular infinite dimensional Hamilton-Jacobi-Bellman equation arising from a storage problem
本文推导了描述含无穷多非原子主体存储模型经济均衡的无穷维 mean field game master equation，并深入研究了其导出的奇异且定义于 Hilbert 空间上的 Hamilton-Jacobi-Bellman 方程的数学性质。
Attractors for Singular-Degenerate Porous Medium Type Equations Arising in Models for Biofilm Growth
本文研究了生物膜生长模型中出现的奇异退化孔隙介质类型方程的吸引子存在性及性质。
Solvability of a doubly singular boundary value problem arising in front propagation for reaction-diffusion equations
针对 $p$-Laplacian 型反应-扩散-对流方程行波研究中出现的含 $z^{-α}$ 奇性与零边值的双重奇异 ODE，本文在给定参数及连续系数条件下探讨其正解的可解性。
We give an exhaustive characterization of singular weak solutions for ordinary differential equations of the form $\ddot{u}\,u + \frac{1}{2}\dot{u}^2 + F'(u) =0$, where $F$ is an analytic function. Our motivation stems from the fact that in the context of hydrodynamics several prominent equations are reducible to an equation of this form upon passing to a moving frame. We construct peaked and cusped waves, fronts with finite-time decay and compact solitary waves. We prove that one cannot obtain peaked and compactly supported traveling waves for the same equation. In particular, a peaked travel…
A finite element approach for minimizing line and surface energies arising in the study of singularities in liquid crystals
本文提出了一种基于有限元的算法，用于最小化液晶中奇异性的线和表面能量，通过引入障碍和表面能量，揭示了最小化配置的复杂结构。
We study scaling limits of the weakly driven Zhang and the Bak-Tang-Wiesenfeld (BTW) model for self-organized criticality. We show that the weakly driven Zhang model converges to a stochastic partial differential equation (PDE) with singular-degenerate diffusion. In addition, the deterministic BTW model is shown to converge to a singular-degenerate PDE. Alternatively, the proof of the scaling limit can be understood as a convergence proof of a finite-difference discretization for singular-degenerate stochastic PDEs. This extends recent work on finite difference approximation of (deterministic)…
On a countable sequence of homoclinic orbits arising near a saddle-center point
针对含五阶色散项的修正Korteweg-de Vries方程行波约化相关的特定四阶方程，严格证明了鞍中心点附近极限同宿轨道复解析延拓中存在四重奇点时，可数小参数值集合上振荡消失且存在同宿轨道的猜想。
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