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 $\mathbb{T}^3$. Furthermore, the propagation of Gevrey regularity can be easily modified to obtain the same result in $\mathbb{R}^d$. In particular, this implies global existence of analytic $(s=1)$ and $\frac{1}{s}$-Gevrey solutions ($s\in (0,1)$) for the Vlasov-Poisson system in $\mathbb{R}^3$.
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.
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.
The micro-local Gevrey regularity of a class of "sums of squares" with real analytic coefficients is studied in detail. Some partial regularity result is also given.
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.
In this paper, we study the Gevrey class regularity for solutions of the spatially homogeneous Landau equations in the hard potential case and the Maxwellian molecules case.
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.
By using the energy-type inequality, we obtain, in this paper, the result on propagation of Gevrey regularity for the solution of the spatially homogeneous Landau equation in the cases of Maxwellian molecules and hard potential.
In this paper, we establish the Gevrey regularity of solutions for a class of degenerate Monge-Ampère equations in the plane, under the assumption that one principle entry of the Hessian is strictly positive and an appropriately finite type degeneracy.
In this paper, following the techniques of Foias and Temam, we establish suitable Gevrey class regularity of solutions to the supercritical quasi-geostrophic equations in the whole space, with initial data in "critical" Sobolev spaces. Moreover, the Gevrey class that we obtain is "near optimal" and as a corollary, we obtain temporal decay rates of higher order Sobolev norms of the solutions. Unlike the Navier-Stokes or the subcritical quasi-geostrophic equations, the low dissipation poses a difficulty in establishing Gevrey regularity. A new commutator estimate in Gevrey classes, involving the…
The Navier--Stokes system is considered in a compact Riemannian manifold. Gevrey class regularity is proven under Lions boundary conditions: in 2D for the Rectangle, Cylinder, and Hemisphere, and in 3D for the Rectangle. The cases of the 2D Sphere and 2D and 3D Torus are also revisited.
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.
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.
In this paper we study the action of pseudo-differential operators acting on Gevrey spaces. We introduce classes of classical symbols with spatial Gevrey regularity. As the spatial Gevrey regularity of a symbol $p(\cdot,ξ)$ may depend on the frequency $ξ$, the action of the associated pseudo-differential operator $op(p)$ may induce a loss of regularity. The proof is based on a para-product decomposition.
We prove via FBI-transform a result concerning the microlocal Gevrey regularity of analytic vectors for operators sums of squares of vector fields with real-valued real analytic coefficients of Hörmander type, thus providing a microlocal version, in the analytic category, of a result due to M. Derridj in "Local estimates for Hörmander's operators of first kind with analytic Gevrey coefficients and application to the regularity of their Gevrey vectors", concerning the problem of the local regularity for the Gevrey vectors for sums of squares of vector fields with real-valued real analytic/Gevre…
We prove for some singular kernels $K(x,y)$ that viscosity solutions of the integro-differential equation $\int_{\mathbb{R}^n} \left[u(x+y)+u(x-y)-2u(x)\right]\,K(x,y)dy=f(x)$ locally belong to some Gevrey class if so does $f$. The fractional Laplacian equation is included in this framework as a special case.
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.
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.
In this paper we consider the non-cutoff Boltzmann equation in spatially inhomogeneous case. We prove the propagation of Gevrey regularity for the so-called smooth Maxwellian decay solutions to the Cauchy problem of spatially inhomogeneous Boltzmann equation, and obtain Gevrey regularity of order $1/(2s)$ in the velocity variable $v$ and order 1 in the space variable $x$. The strategy relies on our recent results for spatially homogeneous case (J. Diff. Equ. 253(4) (2012), 1172-1190. DOI: 10.1016/j.jde.2012.04.023). Rather, we need much more intricate analysis additionally in order to handle w…
We study the regularity of Gevrey vectors for Hörmander operators $$ P = \sum_{j=1}^m X_j^2 + X_0 + c$$ where the $X_j$ are real vector fields and $c(x)$ is a smooth function, all in Gevrey class $G^{s}.$ The principal hypothesis is that $P$ satisfies the subelliptic estimate: for some $\varepsilon >0, \; \exists \,C$ such that $$\|v\|_\varepsilon^2 \leq C\left(|(Pv, v)| + \|v\|_0^2\right) \qquad \forall v\in C_0^\infty.$$ We prove directly (without the now familiar use of adding a variable $t$ and proving suitable hypoellipticity for $Q=-D_t^2-P$ and then, using the hypothesis on the iterates…
This Note derives regularity bounds for a Gevrey criterion when the Cauchy problem of elliptic equations is solved by regularization. When utilizing the regularization, one knows that checking such criterion is basically problematic, albeit its importance to engineering circumstances. Therefore, coping with that impediment helps us improve the use of some regularization methods in real-world applications. This work also consider the presence of the power-law nonlinearities.
In this paper, we study the Gevrey regularity of spatially homogeneous Boltzmann equation without angular cutoff. We prove the propagation of Gevrey regularity for $C^\infty$ solutions with the Maxwellian decay to the Cauchy problem of spatially homogeneous Boltzmann equation. The idea we use here is based on the framework of Morimoto's recent paper (See Morimoto: J. Pseudo-Differ. Oper. Appl. (2010) 1: 139-159, DOI:10.1007/s11868-010-0008-z), but we extend the range of the index $γ$ satisfying $γ+ 2s \in (-1,1)$, $s\in (0,1/2)$ and in this case we consider the kinetic factor in the form of $Φ…
The sharp Gevrey hypoellipticity is provided for the following generalization of the Métivier operator, "Non-hypoellipticité analytique pour $D_{x}^{2}+\left( x^{2} + y^{2}\right)D_{y}^{2}$" by G. Métivier, \begin{align*} D_{x}^{2}+\left(x^{2n+1}D_{y}\right)^{2}+\left(x^{n}y^{m}D_{y}\right)^{2}, \end{align*} in $Ω$ open neighborhood of the origin in $\mathbb{R}^{2}$, where $n$ and $m$ are positive integers.
We consider a bi-dimensional viscous incompressible fluid in interaction with a beam located at its boundary. We show the existence of strong solutions for this fluid-structure interaction system, extending a previous result where we supposed that the initial deformation of the beam was small. The main point of the proof consists in the study of the linearized system and in particular in proving that the corresponding semigroup is of Gevrey class.
This is an addendum to a previous article, which aims to provide the proofs of some results in that paper (Theorem 7.5 and Proposition 9.15) which were removed from its final version. The reason for such omission is that these proofs follow quite closely others already present in the literature, with minor modifications. I make them publicly available for the sake of completeness.
Motivated by \cite{DG19}, we prove the global existence and large time behavior of small solutions to 2-D Prandtl system for data with Gevrey 2 regularity in the $x$ variable and Sobolev regularity in the $y$ variable. In particular, we extend the global well-posedness result in \cite{PZ5} for 2-D Prandtl system with analytic data to data with optimal Gevery regularity in the sense of \cite{Ger1}.
For $ q $, $ a $ integers such that $ a \geq 1 $, $ 1 < q $, $ (x, y) \in U $, $ U $ a neighborhood of the origin in $ \mathbb{R}^{2} $, we consider the operator $$ D_{x}^{2} + x^{2(q-1)} D_{y}^{2} + y^{2a} D_{y}^{2} . $$ Slightly modifying the method of proof of \cite{monom} we can see that it is Gevrey $ s_{0} $ hypoelliptic, where $ s_{0}^{-1} = 1 - a^{-1} (q - 1) q^{-1} $. Here we show that this value is optimal, i.e. that there are solutions to $ P u = f $ with $ f $ more regular than $ G^{s_{0}} $ that are not better than Gevrey $ s_{0} $. The above operator reduces to the Métivier opera…
In this paper, we establish Gevrey class regularity of solutions to a class of dissipative equations with an analytic nonlinearity in the whole space. This generalizes the results of Ferrari and Titi in the periodic space case with initial data in $L^2-$based Sobolev spaces to the $L^p$ setting and in the whole space. Our generalization also includes considering rougher initial data, in negative Sobolev spaces in some cases including the Navier-Stokes and the subcritical quasi-geostrophic equations, and allowing the dissipation operator to be a fractional Laplacian. Moreover, we derive global …
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…
We use sequences which depend on two parameters to define families of ultradifferentiable functions which contain Gevrey classes. It is shown that such families are closed under superposition, and therefore inverse closed as well. Furthermore, we study partial differential operators whose coefficients satisfy the extended Gevrey regularity. To that aim we introduce appropriate wave front sets and derive a theorem on propagation of singularities. This extends related known results in the sense that weaker assumptions on the regularity of the coefficients are imposed.
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 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…
In this short note we capitalize on and complete our previous results on the regularity of the homogenized coefficients for Bernoulli perturbations by addressing the case of the Poisson point process, for which the crucial uniform local finiteness assumption fails. In particular, we strengthen the qualitative regularity result first obtained in this setting by the first author to Gevrey regularity of order~2. The new ingredient is the independence of Poisson point processes, in a form recently used by Giunti, Gu, Mourrat, and Nitzschner.
In this paper, we study the Gevrey regularity of weak solutions for a class of linear and semi-linear kinetic equations, which are the linear model of spatially inhomogeneous Boltzmann equations without an angular cutoff.
In \cite{JB1}, Benameur proved a blow-up result of the non regular solution of $(NSE)$ in the Sobolev-Gevrey spaces. In this paper we improve this result, precisely we give an exponential type explosion in Sobolev-Gevrey spaces with less regularity on the initial condition. Fourier analysis is used.
In this paper, we study the Gevrey regularity of weak solutions for a class of linear and semi-linear kinetic equations, which are the linear model of spatially inhomogeneous Boltzmann equations without an angular cutoff.
In this paper we study the Gevrey regularity for the weak solutions to the Cauchy problem of the non-cutoff spatially homogeneous Botlzmann equation for the Maxwellian molecules model with the singularity exponent $s\in (0,1)$. We establish that any weak solution belongs to the Gevrey spaces for any positive time.
In this paper, we provide a complete regularity analysis for an abstract system of coupled hyperbolic and parabolic equations in a complex Hilbert space. We are able to decompose the unit square of the parameters into three parts where the semigroup associated with the system is analytic, of specific order Gevrey classes, and non-smoothing, respectively. Moreover, we will show that the orders of Gevrey class are sharp, under proper conditions.
This work is concerned with the broad question of propagation of regularity for smooth solutions to non-linear Vlasov equations. For a class of equations (that includes Vlasov-Poisson and relativistic Vlasov-Maxwell), we prove that higher regularity in space is propagated, locally in time, into higher regularity for the moments in velocity of the solution. This in turn can be translated into some anisotropic Sobolev higher regularity for the solution itself, which can be interpreted as a kind of weak propagation of space regularity. To this end, we adapt the methods introduced in the context o…
We show that the linearized Vlasov-Poisson equations around self-similar non-homogeneous states near zero contain the full plasma echo mechanism, yielding Gevrey 3 as a critical stability class. Moreover, here Landau damping may persist despite blow-up: We construct a critical Gevrey regularity class in which the force field converges in $L^2$. Thus, on the one hand, the physical phenomenon of Landau damping holds. On the other hand, the density diverges to infinity in Sobolev regularity. Hence, ``strong damping'' cannot hold.
