Francisco Gancedo
We consider a family of contour dynamics equations depending on a parameter $\al$ with $0<α\leq 1$. The vortex patch problem of the 2-D Euler equation is obtained taking $α\to 0$, and the case $α=1$ corresponds to a sharp front of the QG equation. We prove local-in-time existence for the family of equations in Sobolev spaces.
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KPP transition fronts in a one-dimensional two-patch habitat
本文针对带交界面条件的一维KPP型双斑块模型，通过构造上下解结合极限论证得到整体解，首次证明了跨斑块移动的过渡波前的存在性。
Guided by numerical simulations, we present the proof of two results concerning the behaviour of SQG sharp fronts and $α$-patches. We establish that ellipses are not rotational solutions and we prove that initially convex interfaces may lose this property in finite time.
This paper is devoted to the study of spatial propagation dynamics of species in locally spatially inhomogeneous patchy environments or media. For a lattice differential equation with monostable nonlinearity in a discrete homogeneous media, it is well-known that there exists a minimal wave speed such that a traveling front exists if and only if the wave speed is not slower than this minimal wave speed. We shall show that strongly localized spatial inhomogeneous patchy environments may prevent the existence of transition fronts (generalized traveling fronts). Transition fronts may exist in weak…
Patch solutions for the surface quasigeostrophic (SQG) equation model sharp temperature fronts in atmospheric and oceanic flows. We establish local well-posedness for SQG sharp fronts of low Sobolev regularity, $H^{2+s}$ for arbitrarily small $s>0$, allowing for fronts with unbounded curvature.
This paper is about the evolution of a temperature front governed by the Surface quasi-geostrophic equation. The existence part of that program within the scale of Sobolev spaces was obtained by one of the authors [10]. Here we revisit that proof introducing some new tools and points of view which allow us to conclude the also needed uniqueness result.
Unified theory for regularity persistence of vortex patch boundaries
针对含径向卷积核的二维有源标量方程，建立涡斑边界 Sobolev 正则性保持的统一局部理论，证明特定条件下轮廓动力学方程的局部解存在唯一性，丰富了这类方程的正则性研究成果。
We prove local well-posedness in the Sobolev spaces $\dot H^s(\mathbb{T})$, with $s>7/2$, for an initial value problem for a nonlocal, cubically nonlinear, dispersive equation that provides an approximate description of the evolution of surface quasi-geostrophic (SQG) fronts with small slopes.
This paper considers the temperature patch problem for the incompressible Boussinesq system with no diffusion and viscosity in the whole space $\mathbb{R}^2$. We prove that for initial patches with $W^{2,\infty}$ boundary the curvature remains bounded for all time. The proof explores new cancellations that allow us to bound $\nabla^2u$, even for those components given by time dependent singular integrals with kernels with nonzero mean on circles. In addition, we give a different proof of the $C^{1+γ}$ regularity result in [23], $0<γ<1$, using the scale of Sobolev spaces for the velocity. Furth…
Spreading properties of the Fisher--KPP equation when the intrinsic growth rate is maximal in a moving patch of bounded size
本文研究一维时空异质 Fisher-KPP 方程在中央快速生长斑块移动时的传播性质，揭示了非局部牵引和锁定等现象，并在斑块恒速恒大小情形下给出了传播速度的完整显式公式。
In this paper, we give characterizations of usual wave front set and Sobolev type wave front set in terms of wave packet transform without any restriction on basic wave packet.
Quadratic lifespan for the sublinear $α$-SQG sharp front problem
研究次线性 $\alpha$-SQG 方程中接近涡盘的涡补丁稳定性，证明其解在 $O(\varepsilon^{-2})$ 时间内保持 $\varepsilon$-接近，方法为仿微分 Birkhoff 正规形约化。
We consider a one-dimensional nonlocal nonlinear equation of the form: $\partial_t u = (Λ^{-α} u)\partial_x u - νΛ^βu$ where $Λ=(-\partial_{xx})^{\frac 12}$ is the fractional Laplacian and $ν\ge 0$ is the viscosity coefficient. We consider primarily the regime $0<α<1$ and $0\le β\le 2$ for which the model has nonlocal drift, fractional dissipation, and captures essential features of the 2D $α$-patch models. In the critical and subcritical range $1-α\le β\le 2$, we prove global wellposedness for arbitrarily large initial data in Sobolev spaces. In the full supercritical range $0 \le β<1-α$, we …
In this paper we address the temperature patch problem of the 2D viscous Boussinesq system without heat diffusion term. The temperature satisfies the transport equation and the initial data of temperature is given in the form of non-constant patch, usually called the temperature front initial data. Introducing a good unknown and applying the method of striated estimates, we prove that our partially viscous Boussinesq system admits a unique global regular solution and the initial $C^{k,γ}$ and $W^{2,\infty}$ regularity of the temperature front boundary with $k\in \mathbb{Z}^+ = \{1,2,\cdots\}$ …
We study the temperature front problem for the 3D viscous Boussinesq equation. We prove that the $C^{k,γ}$ ($k\geq 1$, $0<γ< 1$) and $W^{2,\infty}$ regularity of a temperature front is locally preserved along the evolution as well as globally preserved under a smallness condition in a critical space. In particular, beside giving another proof of the main result in \cite{GGJ20}, we also extend it to a more general class of regular patch.
Spaces of distributions with Sobolev wave front in a fixed conic set: compactness, pullback by smooth maps and the compensated compactness theorem
本文为具有固定锥形集中Sobolev波前集的分布空间引入局部凸拓扑，推广拉回、紧性、伪微分算子连续性与Rellich引理，并应用于微局域缺陷测度与补偿紧性定理。
Algebraic rates of stability for front-type modulated waves in Ginzburg Landau equations
该研究针对复金兹堡-朗道方程中双稳区域的波前型调制波，运用指数三分理论分析空间模，证明了代数衰减初始扰动下带渐近相位的代数速率收敛结果。
The $α$-SQG patch problem is illposed in $C^{2,β}$ and $W^{2,p}$
本文证明对于 $0<α<1/2$，$α$-SQG 斑块问题在所有 $β<1$ 的 $C^{2,β}$ 空间及除 $p=2$ 外的 $W^{2,p}$ 空间中强不适定，与 Euler 斑块的全局适定性形成鲜明对比。
We study the propagation of singularities for semilinear Schrodinger equations with quadratic Hamiltonians, in particular for the semilinear harmonic oscillator. We show that the propagation still occurs along the flow the Hamiltonian flow, but for Sobolev regularities in a certain range and provided the notion of Sobolev-wave front set is conveniently modified. The proof makes use of a weighted version of the paradifferential calculus, adapted to our situation. The results can be regarded as the Schrodinger counterpart of those known for semilinear hyperbolic equations, which hold with the us…
We prove that the steady state of a class of multidimensional reaction-diffusion systems is asymptotically stable at the intersection of unweighted space and exponentially weighted Sobolev spaces, and pay particular attention to a special case, namely, systems of equations that arise in combustion theory. The steady-state solutions considered here are the end states of the traveling fronts associated with the systems, and thus the present results complement recent papers \cite{GLS1, GLS2, GLS3, GLSR, GLY} that study the stability of traveling fronts.
Singularities of solutions to nonlinear Schrödinger equations
该研究借助波包变换分析非线性薛定谔方程初值问题解的波前集，提出了判定解在特定时刻特定方向属于s阶索伯列夫空间的充分条件，推进了相关方程解的正则性研究。
Paralinearization and extended lifespan for solutions of the $ α$-SQG sharp front equation
本文对接近圆形涡旋的α-SQG锋面方程进行抛物线性化，利用抛物微分Birkhoff正规型约简及方程的哈密顿性质，在最奇异情形下证明了初始涡旋块的二次寿命结果，首次给出锋面正规型的长时间存在性证明。
Propagation of anisotropic Gabor singularities for Schrödinger type equations
本文研究了一类Schrödinger型演化方程解的各向异性Gabor波前集传播，证明了传播子在Shubin--Sobolev空间上的连续性，并揭示了波前集传播遵循主符号的Hamilton流。
We illustrate the composition properties for an extended family of SG Fourier integral operators. We prove continuity results for operators in this class with respect to $L^2$ and weighted modulation spaces, and discuss continuity on $\mathscr{S}$, $\mathscr{S}^\prime$ and on weighted Sobolev spaces. We study mapping properties of global wave-front sets under the action of these Fourier integral operators. We extend classical results to more general situations. For example, there are no requirements of homogeneity for the phase functions. Finally, we apply our results to the study of of the pr…
In this paper, we consider the linearized Vlasov-Poisson equation around an homogeneous Maxwellian equilibrium in a weakly collisional regime: there is a parameter $\eps$ in front of the collision operator which will tend to $0$. Moreover, we study two cases of collision operators, linear Boltzmann and Fokker-Planck. We prove a result of Landau damping for those equations in Sobolev spaces uniformly with respect to the collision parameter $\eps$ as it goes to $0$.
Geometric Integration by Parts and Sobolev Spaces on Vector Bundles: A Unified Global Approach
本文提出一种统一的全局框架，用于研究Riemann流形上的向量丛Sobolev空间理论，通过显式高阶几何积分求解公式，推导出协变导数的正式伴随，并在任意Riemann流形上建立Sobolev嵌入和Rellich-Kondrachov紧致性定理的自包含证明。
In this paper, KdV-type equations with time- and space-dependent coefficients are considered. Assuming that the dispersion coefficient in front of $u_{xxx}$ is positive and uniformly bounded away from the origin and that a primitive function of the ratio between the anti-dissipation and the dispersion coefficients is bounded from below, we prove the existence and uniqueness of a solution $u$ such that $h u$ belongs to a classical Sobolev space, where $h$ is a function related to this ratio. The LWP in $H^s(\mathbb{R})$, $s>1/2$, in the classical (Hadamard) sense is also proven under an assumpt…
We study the cubic ab-family of equations, which includes both the Fokas-Olver-Rosenau-Qiao (FORQ) and the Novikov (NE) equations. For $a\neq0$, it is proved that there exist initial data in the Sobolev space $H^s$, $s<3/2$, with non-unique solutions. Multiple solutions are constructed by studying the collision of 2-peakon solutions. Furthermore, we prove the novel phenomenon that for some members of the family, collision between 2-peakons can occur even if the "faster" peakon is in front of the "slower" peakon.
We are concerned with supersonic vortex sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients, in [10] the authors have derived a pseudo-differential equation which describes the time evolution of the discontinuity front of the vortex sheet. In agreement with the classical stability analysis, the problem is weakly stable if $|[v\cdotτ]|>2\sqrt{2}\,c$, and the well-posedness was obtained in standard weighted Sobolev spaces. The aim of the present paper is to improve the result of [10], by showing the existence of the s…
We are concerned with supersonic vortex sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients we derive an evolution equation for the discontinuity front of the vortex sheet. This is a pseudo-differential equation of order two. In agreement with the classical stability analysis, if the jump of the tangential component of the velocity satisfies $|[v\cdotτ]|<2\sqrt{2}\,c$ (here $c$ denotes the sound speed) the symbol is elliptic and the problem is ill-posed. On the contrary, if $|[v\cdotτ]|>2\sqrt{2}\,c$, then the probl…
This paper deals with the stability of semi-wavefronts to the following delay non-local monostable equation: $\dot{v}(t,x) = Δv(t,x) - v(t,x) + \int_{\R^d}K(y)g(v(t-h,x-y))dy, x \in \R^d,\ t >0;$ where $h>0$ and $d\in\Z_+$. We give two general results for $d\geq1$: on the global stability of semi-wavefronts in $L^p$-spaces with unbounded weights and the local stability of planar wavefronts in $L^p$-spaces with bounded weights. We also give a global stability result for $d=1$ which includes the global stability on Sobolev spaces. Here $g$ is not assumed to be monotone and the kernel $K$ is not …
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