Igor Kukavica, Vlad Vicol
We consider the Euler equations in a three-dimensional Gevrey-class bounded domain. Using Lagrangian coordinates we obtain the Gevrey-class persistence of the solution, up to the boundary, with an explicit estimate on the rate of decay of the Gevrey-class regularity radius.
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In this paper, we mainly consider the Gevrey regularity and analyticity of the solution to a generalized two-component shallow water wave system with higher-order inertia operators, namely, $m=(1-\partial_x^2)^su$ with $s>1$. Firstly, we obtain the Gevrey regularity and analyticity for a short time. Secondly, we show the continuity of the data-to-solution map. Finally, we prove the global Gevrey regularity and analyticity in time.
On the radius of analyticity and Gevrey regularity for the Boltzmann equation
本文研究了Boltzmann方程在硬势场中的分析半径和Gevrey正则性，通过宏微观分解和次椭圆估计获得短时和全局时间的正则性估计。
In this paper we mainly investigate the Cauchy problem of some Camassa-Holm type systems. By constructing a new auxiliary function, we present a generalized Ovsyannikov theorem. By using this theorem and the basic properties of Sobolev-Gevrey spaces, we prove the Gevrey regularity and analyticity of these systems. Moreover, we obtain a lower bound of the lifespan and the continuity of the data-to-solution map.
We are concerned with an isothermal model of viscous and capillary compressible fluids derived by J. E. Dunn and J. Serrin (1985), which can be used as a phase transition model. Compared with the classical compressible Navier-Stokes equations, there is a smoothing effect on the density that comes from the capillary terms. First, we prove that the global solutions with critical regularity that have been constructed in [11] by the second author and B. Desjardins (2001), are Gevrey analytic. Second, we extend that result to a more general critical L p framework. As a consequence, we obtain algebr…
We address the global persistence of analyticity and Gevrey-class regularity of solutions to the two and three-dimensional visco-elastic second-grade fluid equations. We obtain an explicit novel lower bound on the radius of analyticity of the solutions to the second-grade fluid equations that does not vanish as $t\to \infty$. Applications to the damped Euler equations are given.
We study the persistence of the Gevrey class regularity of solutions to nonlinear wave equations with real analytic nonlinearity. Specifically, it is proven that the solution remains in a Gevrey class, with respect to some of its spatial variables, during its whole life-span, provided the initial data is from the same Gevrey class with respect to these spatial variables. In addition, for the special Gevrey class of analytic functions, we find a lower bound for the radius of the spatial analyticity of the solution that might shrink either algebraically or exponentially, in time, depending on th…
We establish analyticity of the subcritical and critical quasi-geostrophic equations in critical Besov spaces. The main method is so-called Gevrey estimates, which is motivated by the work of Foias and Temam. We show that mild solutions θ(t), are Gevrey regular, i.e. they satisfy the estimate \sup_{t>0}\|e^{αt^{1/γ}Λ_1}θ(t)\|_{\cap L}<\infty. for a suitably chosen α>0 and a scaling invariant Besov space {\cap L}.
We show that solutions to a large class of inviscid equations, in Eulerian variables, extend as holomorphic functions of time, with values in a Gevrey class (thus space-analytic), and are solutions of complexified versions of the said equations. The class of equations we consider includes those of fluid dynamics such as the Euler, surface quasi-geostrophic, Boussinesq and magnetohydrodynamic equations, as well as other equations with analytic nonlinearities. The initial data are assumed to belong to a $\mathit{Gevrey\:class}$, i.e., analytic in the space variable. Our technique follows that of…
It has been thought for a while that the Prandtl system is only well-posed under the Oleinik monotonicity assumption or under an analyticity assumption. We show that the Prandtl system is actually locally well-posed for data that belong to the Gevrey class 7/4 in the horizontal variable x. Our result improves the classical local well-posedness result for data that are analytic in x (that is Gevrey class 1). The proof uses new estimates, based on non-quadratic energy functionals.
We address the problem of analyticity up to the boundary of solutions to the Euler equations in the half space. We characterize the rate of decay of the real-analyticity radius of the solution $u(t)$ in terms of $\exp{\int_{0}^{t} \Vert \nabla u(s) \Vert_{L^\infty} ds}$, improving the previously known results. We also prove the persistence of the sub-analytic Gevrey-class regularity for the Euler equations in a half space, and obtain an explicit rate of decay of the radius of Gevrey-class regularity.
In this work we study the Cauchy problem in Gevrey spaces for a generalized class of equations that contains the case $b=0$ of the $b$-equation. For the generalized equation, we prove that it is locally well-posed for initial data in Gevrey spaces. Moreover, as we move to global well-posedness, we show that for a particular choice of the parameter in the equation the local solution is global analytic in both time and spatial variables.
In this paper we study the weighted Gevrey class regularity of Euler equation in the whole space R 3. We first establish the local existence of Euler equation in weighted Sobolev space, then obtain the weighted Gevrey regularity of Euler equation. We will use the weighted Sobolev-Gevrey space method to obtain the results of Gevrey regularity of Euler equation, and the use of the property of singular operator in the estimate of the pressure term is the improvement of our work.
Global well-posedness for the nonlinear wave equation in analytic Gevrey spaces
本文研究了非线性波动方程在解析Gevrey空间中的整体适定性，获得了其解的空间解析性半径的渐近衰减率。
In this paper, we study the Cauchy problem for an integrable multi-component (2N-component) peakon system which is involved in an arbitrary polynomial function. Based on a generalized Ovsyannikov type theorem, we first prove the existence and uniqueness of solutions for the system in the Gevrey-Sobolev spaces with the lower bound of the lifespan. Then we show the continuity of the data-to-solution map for the system. Furthermore, by introducing a family of continuous diffeomorphisms of a line and utilizing the fine structure of the system, we demonstrate the system exhibits unique continuation…
In this paper, we study the Cauchy problem for the generalized Keller-Segel system with the cell diffusion being ruled by fractional diffusion: \begin{equation*} \begin{cases} \partial_{t}u+Λ^αu-\nabla\cdot(u\nabla ψ)=0\quad &\mbox{in}\ \ \mathbb{R}^n\times(0,\infty), -Δψ=u\quad &\mbox{in}\ \ \mathbb{R}^n\times(0,\infty), u(x,0)=u_0(x), \ \ &\mbox{in}\ \ \mathbb{R}^n. \end{cases} \end{equation*} In the case that $1<α\leq 2$, we prove local well-posedness for any initial data and global well-posedness for small initial data in critical Besov spaces $\dot{B}^{-α+\frac{n}{p}}_{p,q}(\mathbb{R}^{n}…
Gevrey regularity and analyticity for the solutions of the Vlasov-Navier-Stokes system
本文采用傅里叶空间方法，证明了Vlasov-Navier-Stokes系统解的$\frac{1}{s}$-Gevrey正则性与解析性的传播性，给出定量估计并推导得出对应解的存在性及全局Gevrey解的存在条件。
We study a Euler-Bernoulli beam equation with localized discontinuous structural damping. As our main result, we prove that the associated $C_0$-semigroup $(S(t))_{t\geq0}$ is of Gevrey class $δ>24$ for $t>0$, hence immediately differentiable. Moreover, we show that $(S(t))_{t\geq0}$ is exponentially stable.
We show that the solution to the Cauchy problem of the 3D nematic liquid crystal flows, with initial data belongs to a critical Besov space, belongs to a Gevrey class. More precisely, it is proved that for any $(u_{0},d_{0} - \overline{d}_{0}) \in \dot{B}^{\frac{3}{p}-1}_{p,1} (\mathbb{R}^{3}) \times \dot{B}^{\frac{3}{q}}_{q,1} (\mathbb{R}^{3})$ with some suitable conditions imposed on $p, q\in(1,\infty)$, there exists $T^{*}>0$ depending only on initial data, such that the nematic liquid crystal flows admits a unique solution $(u,d)$ on $\mathbb{R}^{3} \times (0,T^{*})$, and satisfies \begin{…
In this work we prove the weighted Gevrey regularity of solutions to the incompressible Euler equation with initial data decaying polynomially at infinity. This is motivated by the well-posedness problem of vertical boundary layer equation for fast rotating fluid. The method presented here is based on the basic weighted $L^2$- estimate, and the main difficulty arises from the estimate on the pressure term due to the appearance of weight function.
We prove real analyticity of all the streamlines, including the free surface, of a steady stratified flow of water over a flat bed in the absence of stagnation points, with a Hölder continuous Bernoulli function and a Hölder continuously differentiable density function. Furthermore, we show that if the Bernoulli function and the density function possess some Gevrey regularity of index s, then the stream function admits the same Gevrey regularity throughout the fluid domain; in particular if the Gevrey index s equals to 1, then we obtain analyticity of the stream function. The regularity result…
This work is concerned with the Gevrey regularity and analyticity of the solution to a weakly dissipative Camassa-Holm system. We first demonstrate the local Gevery regularity and analyticity of this equation. Then, we disscuss the continuity of the data-to-solution map. Finally, we obtain the global Gevery regularity of this system in Gevery class $G_σ$ with $σ\geq 1$ in time.
We prove real analyticity of all the streamlines, including the free surface, of a gravity- or capillary-gravity-driven steady flow of water over a flat bed, with a Hölder continuous vorticity function, provided that the propagating speed of the wave on the free surface exceeds the horizontal fluid velocity throughout the flow. Furthermore, if the vorticity possesses some Gevrey regularity of index $s$, then the stream function admits the same Gevrey regularity throughout the fluid domain; in particular if the Gevrey index $s$ equals to 1, then we obtain analyticity of the stream function. The…
We prove that for a certain class of kernels $K(y)$ that viscosity solutions of the integro-differential equation $$ \int_{\mathbb R^n} (u(x+y) - 2 u(x) + u(x-y)) K(y) dy = f(x,u(x)) $$ are locally analytic if $f$ is an analytic function. This extends the result of Albanese, Fiscella, Valdinoci that such solutions belong to certain Gevrey classes.
A Gevrey class semigroup, exponential decay and Lack of analyticity for a system formed by a Kirchhoff-Love plate equation and the equation of a membrane-like electric network with indirect fractional damping
本文研究 Kirchhoff-Love 板与膜状电网耦合系统的半群性质，证明其在 $θ\in[0,1[$ 非解析、$θ=1$ 解析，指数衰减，且 $θ\in]0,1[$ 时为 Gevrey 类 $s>1/θ$。
The Gevrey class of the Euler-Bernoulli beam model with singularities
本文研究具有奇点的欧拉-伯努利梁模型，证明其半群属于盖尔雷类4，并显示模型具有指数稳定性、线性稳定性及平滑效应。
We consider the inhomogeneous heat equation on the half-space $\mathbb R_{+}^{d}$ with Neumann boundary conditions. We prove a space-time Gevrey regularity of the solution, with a radius of analyticity uniform up to the boundary of the half-space. We also address the case of homogeneous Robin boundary conditions. Our results generalize the case of homogeneous Dirichlet boundary conditions established by Kukavica and Vicol in [10].
We discuss gain of analyticity phenomenon of solutions to the initial value problem for semilinear Schroedinger equations with gauge invariant nonlinearity. We prove that if the initial data decays exponentially, then the solution becomes real-analytic in the space variable and a Gevrey function of order 2 in the time variable except in the initial plane. Our proof is based on the energy estimates developed in our previous work and on fine summation formulae concerned with a matrix norm.
Spatial and temporal analyticity of solutions to semi-linear parabolic systems with analytic nonlinearities
本文研究具解析非线性项的半线性抛物方程组解的空间与时间解析性，证明其在空间变量的Gevrey类中关于时间解析。
Space-time analyticity and refined analyticity radius of the Navier-Stokes equations in the critical Besov spaces
本文证明了临界Besov空间中小初始数据的不可压缩Navier-Stokes方程全局解的时空解析性，结合Gevrey类估计与迭代导数技术推导了高阶导数衰减率与解析半径下界，并获得了任意初始数据下空间解析半径下界的对数改进。
We consider the Cauchy problem for an equation of Korteweg-de Vries-Kawahara type with initial data in the analytic Gevrey spaces. By using linear, bilinear and trilinear estimates in analytic Bourgain spaces, we establish the local well-posedness for this problem. By using an approximate conservation law, we extend this to a global result in such a way that the radius of analyticity of solutions is uniformly bounded below by a fixed positive number for all time.
The radius of spatial analyticity for solutions of the KdV equation is studied. It is shown that the analyticity radius does not decay faster than $t^{-1/4}$ as time $t$ goes to infinity. This improves the works [Selberg, da Silva, Lower bounds on the radius of spatial analyticity for the KdV equation, Annales Henri Poincaré, 2017, 18(3): 1009-1023] and [Tesfahun, Asymptotic lower bound for the radius of spatial analtyicity to solutions of KdV equation, arXiv preprint arXiv:1707.07810, 2017]. Our strategy mainly relies on a higher order almost conservation law in Gevrey spaces, which is inspir…
We address the analyticity and large time decay rates for strong solutions of the Hall-MHD equations. By Gevrey estimates, we show that the strong solution with small initial date in $H^r(\mathbb{R}^3)$ with $r>\f 52$ becomes analytic immediately after $t>0$, and the radius of analyticity will grow like $\sqrt{t}$ in time. Upper and lower bounds on the decay of higher order derivatives are also obtained, which extends the previous work by Chae and Schonbek (J. Differential Equations 255 (2013), 3971--3982).
Lagrangian time-analyticity for the Euler equations in a Sobolev domain
本文在仅假设区域具有Sobolev正则性的条件下，证明了Euler方程解的拉格朗日轨迹具有时间解析性。
In this paper, similar to the incompressible Euler equation, we prove the propagation of the Gevrey regularity of solutions to the three-dimensional incompressible ideal magnetohydrodynamics (MHD) equations. We also obtain an uniform estimate of Gevery radius for the solution of MHD equation.
Global existence and analyticity of $L^p$ solutions to the compressible fluid model of Korteweg type
针对零声速Korteweg型可压缩流体，利用线性化系统的纯抛物结构，在$L^p$型混合Besov空间证明强解全局存在与Gevrey解析性。
We consider the incompressible Euler equations on ${\mathbb R}^d$, where $d \in \{ 2,3 \}$. We prove that: (a) In Lagrangian coordinates the equations are locally well-posed in spaces with fixed real-analyticity radius (more generally, a fixed Gevrey-class radius). (b) In Lagrangian coordinates the equations are well-posed in highly anisotropic spaces, e.g.~Gevrey-class regularity in the label $a_1$ and Sobolev regularity in the labels $a_2,...,a_d$. (c) In Eulerian coordinates both results (a) and (b) above are false.
We present a technique for derivation of a priori bounds for Gevrey-Sobolev norms of space-periodic three-dimensional solutions to evolutionary partial differential equations of hydrodynamic type. It involves a transformation of the flow velocity in the Fourier space, which introduces a feedback between the index of the norm and the norm of the transformed solution, and results in emergence of a mildly dissipative term. To illustrate the technique, we derive finite-time bounds for Gevrey-Sobolev norms of solutions to the Euler and inviscid Burgers equations, and global in time bounds for the V…
In this paper we consider the inviscid limit for the periodic solutions to Navier-Stokes equation in the framework of Gevrey class. It is shown that the lifespan for the solutions to Navier-Stokes equation is independent of viscosity, and that the solutions of the Navier-Stokes equation converge to that of Euler equation in Gevrey class as the viscosity tends to zero. Moreover the convergence rate in Gevrey class is presented.
This paper is concerned with the cubic Szegő equation $$ i\partial_t u=Π(|u|^2 u), $$ defined on the $L^2$ Hardy space on the one-dimensional torus $\mathbb T$, where $Π: L^2(\mathbb T)\rightarrow L^2_+(\mathbb T)$ is the Szegő projector onto the non-negative frequencies. For analytic initial data, it is shown that the solution remains spatial analytic for all time $t\in (-\infty,\infty)$. In addition, we find a lower bound for the radius of analyticity of the solution. Our method involves energy-like estimates of the special Gevrey class of analytic functions based on the $\ell^1$ norm of Fou…
Gevrey instability in the inviscid inflow-outflow problem
研究周期通道上二维不可压缩欧拉方程的流入-流出问题，通过设定边界条件，证明在热夫雷空间中该问题全局适定性，指出流入-流出速度边界条件与解的唯一性及解析性的关系。
On propagation of regularities and evolution of radius of analyticity in the solution of the fifth order KdV-BBM model
该研究针对描述单向水波的五阶KdV-BBM模型初值问题，证明了解的正则性传播特性、Gevrey类中的局部适定性，并给出解析半径演化的显式上下界公式。
We prove linear inviscid damping near a general class of monotone shear flows in a finite channel, in Gevrey spaces. It is an essential step towards proving nonlinear inviscid damping for general shear flows that are not close to the Couette flow, which is a major open problem in 2d Euler equations.
We consider the Nernst-Planck equations describing the nonlinear time evolution of multiple ionic concentrations in a two-dimensional incompressible fluid. The velocity of the fluid evolves according to either the Euler or Darcy's equations, both forced nonlinearly by the electric forces generated by the presence of charged ions. We address the global well-posedness and Gevrey regularity of the resulting electrodiffusion models in the periodic setting.
We address the Mach limit problem for the Euler equations in the analytic spaces. We prove that, given analytic data, the solutions to the compressible Euler equations are uniformly bounded in a suitable analytic norm and then show that the convergence toward the incompressible Euler solution holds in the analytic norm. We also show that the same results hold more generally for Gevrey data with the convergence in the Gevrey norms.
Improved lower bound for the radius of analyticity for the modified KdV equation
本文针对散焦 mKdV 方程，在带 cosh 权的 Gevrey 空间中构造新的几乎守恒律，证明解全局存在，并将空间解析半径下界改进为 cT^{-1/2}。
We give a proof of the analiticity in time for the particle trajectories associated with the solutions of some transport equations when the initial datum is a patch. These results are obtained from a precise study of the Beurling transform, which provides estimates for the solutions of some new equations satisfied by the lagrangian flow.
Through Borel summation methods, we analyze the Boussinesq equations for coupled fluid velocity and temperature fields. We prove that an equivalent system of integral equations in the Borel variable p dual to 1/t has a unique solution in a class of exponentially bounded functions, implying the existence of a classical solution to the Boussinesq equations in a complex t-region that includes a real positive time axis segment. For analytic initial data and forcing, it is shown that the solution is Borel summable, implying that that formal series in powers of t is Gevrey-1 asymptotic, and within t…
The set of nonzero external forces for which the zero function is in the global attractor of the 2D Navier-Stokes equations is shown to be meagre in a Fréchet topology. A criterion in terms of a Taylor expansion in complex time is used to characterize the forces in this set. This leads to several relations between certain Gevrey subclasses of $C^{\infty}$ and a new upper bound for a Gevrey norm of solutions in the attractor, valid in the strip of analyticity in time.
Through Borel summation methods, we analyze two different variations of the Navier-Stokes equation --the Boussinesq equation and the magnetic Benard equation. This method has previously been applied to the Navier-Stokes equation. We prove that an equivalent system of integral equations in each case has a unique solution, which is exponentially bounded for p in R^{+}, p being the Laplace dual variable of 1/t. This implies the local existence of a classical solution in a complex t-region that includes a real positive time (t)-axis segment. In this formalism, global existence of PDE solutions bec…
We prove propagation of $\frac{1}{s}$-Gevrey regularity $(s\in(0,1))$ for the Vlasov-Poisson system on $\mathbb{T}^d$ using a Fourier space method in analogy to the results proved for the 2D Euler system in \cite{KV} and \cite{LO}. More precisely, we give a quantitative estimate for the growth in time of the $\frac{1}{s}$-Gevrey norm for the solution of the system in terms of the force field and the gradient in the velocity variable of the distribution of matter. As an application, we show global existence of $\frac{1}{s}$-Gevrey solutions ($s\in (0,1)$) for the Vlasov-Poisson system in $\math…
Persistence of conormal analyticity for the Euler equations in the half-space
该研究证明了三维不可压缩欧拉方程在半空间中的余法解析性通过Hʳ范数保持持久性，确立了相关解析性半径下界，还证明了两类余法空间的范数等价性。
We study partial analyticity of solutions to elliptic systems and analyticity of level sets of solutions to nonlinear elliptic systems. We consider several applications, including analyticity of flow lines for bounded stationary solutions to the 2-d Euler equation, and analyticity of water waves with and without surface tension.
Certain systems of inviscid fluid dynamics have the property that for solutions that are only slightly better than differentiable in Eulerian variables, the corresponding Lagrangian trajectories are analytic in time. We elucidate the mechanisms in fluid dynamics systems that give rise to this automatic Lagrangian analyticity, as well as mechanisms in some particular fluids systems which prevent it from occurring. We give a conceptual argument for a general fluids model which shows that the fulfillment of a basic set of criteria results in the analyticity of the trajectory maps in time. We then…
Gevrey regularity solution for initial data in Triebel-Lizorkin-Lorentz spaces via single norm defined by nonlinearity of frequency
本文通过引入单一范数迭代空间并利用 Fefferman-Stein 不等式，在临界 Triebel-Lizorkin-Lorentz 空间中建立了 Navier-Stokes 方程的全局适定性与 mild 解的 Gevrey 正则性，推广了已有的 Besov 空间结果。
We give a new, simpler, proof of nonlinear Landau damping on T^d in Gevrey-1/s regularity (s > 1/3) which matches the regularity requirement predicted by the formal analysis of Mouhot and Villani in the original proof of Landau damping [Acta Mathematica 2011]. Our proof combines in a novel way ideas from the original proof of Landau damping and the proof of inviscid damping in 2D Euler [arXiv:1306.5028]. As in the work on 2D Euler, we use paraproduct decompositions and controlled regularity loss to replace the Newton iteration scheme employed in the original proof. We perform time-response est…
We consider the Euler equations on $\mathbb{T}^d$ with analytic data and prove lower bounds for the radius of spatial analyticity $ε(t)$ of the solution using a new method based on inductive estimates in standard Sobolev spaces. Our results are consistent with similar previous results proved by Kukavica and Vicol, but give a more precise dependence of $ε(t)$ on the radius of analyticity of the initial datum.
Half-space problem on the Boltzmann equation with zero Mach number at infinity
研究Boltzmann方程在半空间中的长时间动力学，采用扩散反射边界条件和小扰动，证明了低正则性解和Gevrey正则性传播，克服了边界效应和非线性相互作用的挑战。
Global analytic solutions of a pseudospherical Novikov equation
本文研究描述伪球面的Novikov方程，通过Sobolev空间全局适定性、Gevrey空间初值的新估计及Kato-Masuda定理得到解析半径下界，证明解在两变量上全局解析，并赋予对应条带伪球面相关的全局解析度量。
Regularity and stability of two coupled Euler-Bernoulli equations with a localized singular structural damping
本文研究了带有局部奇异阻尼的双耦合欧拉-伯努利方程组的正则性与稳定性，推导了均匀稳定性结果并描述了系统长期行为。
On the local analyticity for the Euler equations
本文研究了 Euler 方程在边界附近解析、远处 Sobolev 正则的初值解，提出 diamond-analyticity 框架结合非标准迭代方案，证明了解析半径商不受限，深化了对流体方程解析性与边界效应的理解。
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