In this article we study the generalized dispersion version of the Kadomtsev-Petviashvili II equation, on $\T \times \R$ and $\T \times \R^2$. We start by proving bilinear Strichartz type estimates, dependent only on the dimension of the domain but not on the dispersion. Their analogues in terms of Bourgain spaces are then used as the main tool for the proof of bilinear estimates of the nonlinear terms of the equation and consequently of local well-posedness for the Cauchy problem.
We consider the wave equation with Dirichlet boundary conditions in the exterior of a cylinder in R 3 and we construct a global in time parametrix to derive sharp dispersion estimates for all frequencies (low and high) and, as a corollary, Strichartz estimates, all matching the R 3 case.
We establish quantitative asymptotic behavior of positive solutions of a family of nonlinear elliptic equations on the half cylinder near the end. This unifies the study of isolated singularities of some semilinear elliptic equations, such as the Yamabe equation, Hardy-Hénon equation etc.
We establish well-posedness for the Cauchy problem associated to the dispersion generalized Zakharov-Kutnesov equation in the cylinder. Our main ingredient is a localized Strichartz estimate and an argument of compactness
We prove unique existence of local-in-time smooth solutions of the Navier-Stokes equations for initial data in $L^{p}$ and $p \in [3, \infty)$ in an infinite cylinder, subject to the Neumann boundary condition.
Weighted Dirichlet-type inequalities for the decreasing rearrangement in cylinders
Martin-Luther-University of Halle（哈勒马丁路德大学）；Università di Napoli Federico II（那不勒斯费德里科二世大学）；Università degli studi della Campania “L. Vanvitelli”（坎帕尼亚卢多维维特利大学）
Here we define Rarita-Schwinger operators on cylinders and construct their fundamental solutions. Further the fundamental solutions to the cylindrical Rarita-Schwinger type operators are achieved by applying translation groups. In turn, a Borel-Pompeiu Formula, Cauchy Integral Formula and a Cauchy Transform are presented for the cylinders. Moreover we show a construction of a number of conformally inequivalent spinor bundles on these cylinders. Again we construct Rarita-Schwinger operators and their fundamental solutions in this setting. Finally we study the remaining Rarita-Schwinger type ope…
We study the existence and uniqueness of a solution to a linear stationary convection-diffusion equation stated in an infinite cylinder, Neumann boundary condition being imposed on the boundary. We assume that the cylinder is a junction of two semi-infinite cylinders with two different periodic regimes. Depending on the direction of the effective convection in the two semi-infinite cylinders, we either get a unique solution, or one-parameter family of solutions, or even non-existence in the general case. In the latter case we provide necessary and sufficient conditions for the existence of a s…
We construct nontrivial entire solutions for a bistable reaction-diffusion equation in a class of domains that are unbounded in one direction. The motivation comes from recent results of Berestycki, Bouhours, and Chapuisat concerning propagation and blocking phenomena in infinite domains. A key assumption in their study was the "cylinder-like" assumption: their domains are supposed to be straight cylinders in a half space. The purpose of this paper is to consider domains that tend to a straight cylinder in one direction. We also prove the existence of an entire solution for a one-dimensional p…
Global regular axially symmetric solutions with large swirl in a cylinder with periodic conditions on the top and on the bottom are proved. On the lateral part of its boundary the boundary slip conditions are assumed. The proof is obtained step by step extending a local solution and showing that the initial data are not increasing.
We consider the incompressible 2D Euler equation in an infinite cylinder $\mathbb{R}\times \mathbb{T}$ in the case when the initial vorticity is non-negative, bounded, and compactly supported. We study $d(t)$, the diameter of the support of vorticity, and prove that it allows the following bound: $d(t)\leq Ct^{1/3}\log^2 t$ when $t\rightarrow\infty$.
We prove the existence of positive and of nodal solutions for $-Δu = |u|^{p-2}u+μ|u|^{q-2}u$, $u\in {\rm H_0^1}(Ω)$, where $μ>0$ and $2<q<p=2N(N-2)$, for a class of open subsets $Ω$ of $\mathbb{R}^N$ lying between two infinite cylinders.
We consider the initial boundary value problem of non-homogeneous stochastic heat equation. The derivative of the solution with respect to time receives heavy random perturbation. The space boundary is Lipschitz and we impose non-zero cylinder condition. We prove a regularity result after finding suitable spaces for the solution and the pre-assigned datum in the problem. The tools from potential theory, harmonic analysis and probability are used. Some Lemmas are as important as the main Theorem.
We address the uniqueness of the nonzero stationary state for a reaction-diffusion system of Fisher-KPP type that does not satisfy the comparison principle. Although the uniqueness is false in general, it turns out to be true under biologically natural assumptions on the parameters. This Liouville-type result is then used to characterize the wake of traveling waves. All results are extended to an analogous nonlocal reaction-diffusion equation that contains as a particular case the cane toads equation with bounded traits.
An explicit saturating set consisting of eigenfunctions of Stokes operator in general 3D Cylinders is proposed. The existence of saturating sets implies the approximate controllability for Navier--Stokes equations in $\rm 3D$ Cylinders under Lions boundary conditions.
Asymptotic behavior and Liouville-type theorems for axisymmetric stationary Navier-Stokes equations outside of an infinite cylinder with a periodic boundary condition
We consider the incompressible Navier-Stokes equations in the cylinder $\R \times \T$, with no exterior forcing, and we investigate the long-time behavior of solutions arising from merely bounded initial data. Although we do not know if such solutions stay uniformly bounded for all times, we prove that they converge in an appropriate sense to the family of spatially homogeneous equilibria as $t \to \infty$. Convergence is uniform on compact subdomains, and holds for all times except on a sparse subset of the positive real axis. We also improve the known upper bound on the $L^\infty$ norm of th…
We study the incompressible Navier-Stokes equations in the two-dimensional strip $\mathbb{R} \times [0,L]$, with periodic boundary conditions and no exterior forcing. If the initial velocity is bounded, we prove that the solution remains uniformly bounded for all times, and that the vorticity distribution converges to zero as $t \to \infty$. We deduce that, after a transient period, a laminar regime emerges in which the solution rapidly converges to a shear flow governed by the one-dimensional heat equation. Our approach is constructive and gives explicit estimates on the size of the solution …
This paper concernes with the existence of heteroclinic solutions for the following class of elliptic equations $$ -Δ{u}+A(εx, y)V'(u)=0, \quad \mbox{in} \quad Ω, $$ where $ε>0$, $Ω=\R \times \D$ is an infinite cylinder of $\mathbb{R}^N$ with $N \geq 2$. Here, we have considered a large class of potential $V$ that includes the Ginzburg-Landau potential $V(t)=(t^{2}-1)^{2}$ and two geometric conditions on the function $A$. In the first condition we assume that $A$ is asymptotic at infinity to a periodic function, while in the second one $A$ satisfies $$ 0<A_0=A(0,y)=\inf_{(x,y) \in Ω}A(x,y) < \…
Let $s \in {\mathbb N}$, $T_1,T_2 \in {\mathbb R}$, $T_1<T_2$, and let $Ω, ω$ be bounded domains in ${\mathbb R}^n$, $n \geq 1$ such that $ω\subset Ω$ and the complement $Ω\setminus ω$ have no non-empty compact components in $Ω$. We investigate the problem of approximation of solutions to parabolic Lamé type system from the Lebesgue class $L^2(ω\times (T_1,T_2))$ in a cylinder domain $ω\times (T_1,T_2) \subset {\mathbb R}^{n+1}$ by more regular solutions in a bigger domain $Ω\times (T_1,T_2)$. As an application of the obtained approximation theorems we construct Carleman's formulas for recover…
We consider in this paper a reaction-diffusion system in presence of a flow and under a KPP hypothesis. While the case of a single-equation has been extensively studied since the pioneering Kolmogorov-Petrovski-Piskunov paper, the study of the corresponding system with a Lewis number not equal to 1 is still quite open. Here, we will prove some results about the existence of travelling fronts and generalized travelling fronts solutions of such a system with the presence of a non-linear spacedependent loss term inside the domain. In particular, we will point out the existence of a minimal speed,…
We introduce a geometrical extension of the FitzHugh-Nagumo equations describing propagation of electrical impulses in nerve axons. In this extension, the axon is modeled as a warped cylinder, rather than a straight line, as is usually done, while pulses propagate on its surface, as is the case with real axons. We prove the stability of electrical impulses for a standard cylinder and existence and stability of pulse-like solutions for warped cylinders whose radii are small and vary slowly along their lengths.
In this note we consider the action functional \[ \int_{\mathbb{R} \times ω} \left( 1 - \sqrt{ 1 - |\nabla u|^2 } + W(u) \right) \, \mathrm{d}t, \] where $W$ is a double well potential and $ω$ is a bounded domain of $\mathbb{R}^{N-1}$. We prove existence, one-dimensionality and uniqueness (up to translation) of a smooth minimizing phase transition between the two stable states $u=1$ and $u=-1$. The question of existence of at least one minimal heteroclinic connection for the non autonomous model \[ \int_{\mathbb{R}} \left( 1 - \sqrt{1-|u'|^2} + a(t) W(u) \right) \, \mathrm{d}t \] is also addre…
We construct a maximal function associated with a family of skewed cylinders. These cylinders, which are defined as tubular neighborhoods of trajectories of a mollified flow, appear in the study of fluid equations such as the Navier-Stokes equations and the Euler equations. We define a maximal function subordinate to these cylinders, and show it is of weak type $(1, 1)$ and strong type $(p, p)$ by a covering lemma. As an application, we give an alternative proof for the higher derivatives estimate of smooth solutions to the three-dimensional Navier-Stokes equations.
In this paper we consider a reaction-diffusion equation of Fisher-KPP type inside an infinite cylindrical domain in $\mathbb{R}^{N+1}$, coupled with a reaction-diffusion equation on the boundary of the domain, where potentially fast diffusion is allowed. We will study the existence of an asymptotic speed of propagation for solutions of the Cauchy problem associated with such system, as well as the dependence of this speed on the diffusivity at the boundary and the amplitude of the cylinder. When $N=1$ the domain reduces to a strip between two straight lines. This models the effect of two roads…
In this paper, a new formulation for the three dimensional Euler equations is derived. Since the Euler system is hyperbolic-elliptic coupled in a subsonic region, so an effective decoupling of the hyperbolic and elliptic modes is essential for any development of the theory. The key idea in our formulation is to use the Bernoulli's law to reduce the dimension of the velocity field by defining new variables $(1,β_2=\frac{u_2}{u_1},β_3=\frac{u_3}{u_1})$ and replacing $u_1$ by the Bernoulli's function $B$ through $u_1^2=\frac{2(B-h(ρ))}{1+β_2^2+β_3^2}$. We find a conserved quantity for flows with …
In this paper, for a family of second-order parabolic equation with rapidly oscillating and time-dependent periodic coefficients, we are interested in an approximate two-sphere one-cylinder inequality for these solutions in parabolic periodic homogenization, which implies an approximate quantitative propagation of smallness. The proof relies on the asymptotic behavior of fundamental solutions and the Lagrange interpolation technique.
A regularity criterion for the angular component of velocity in the norm $L_\infty(0,T;L_p(Ω)),\;\frac 3 p <1$ in axisymmetric Navier Stokes equations in a cylinder
Let $u$ be a nonnegative, local, weak solution to the porous medium equation for $m\ge2$ in a space-time cylinder $Ω_T$. Fix a point $(x_o,t_o)\inΩ_T$: if the average \[ a{\buildrel\mbox{def}\over{=}}\frac1{|B_r(x_o)|}\int_{B_r(x_o)}u(x,t_o)\,dx>0, \] then the quantity $|\nabla u^{m-1}|$ is locally bounded in a proper cylinder, whose center lies at time $t_o+a^{1-m}r^2$. This implies that in the same cylinder the solution $u$ is Hölder continuous with exponent $α=\frac1{m-1}$, which is known to be optimal. Moreover, $u$ presents a sort of instantaneous regularisation, which we quantify.
We study multiplicity of the supercritical traveling front solutions for scalar reaction-diffusion equations in infinite cylinders which invade a linearly unstable equilibrium. These equations are known to possess traveling wave solutions connecting an unstable equilibrium to the closest stable equilibrium for all speeds exceeding a critical value. We show that these are, in fact, the only traveling front solutions in the considered problems for sufficiently large speeds. In addition, we show that other traveling fronts connecting to the unstable equilibrium may exist in a certain range of the…
Global axisymmetric solutions and incompressible limit for the 3D isentropic compressible Navier-Stokes equations in annular cylinders with swirl and large initial data
School of Mathematics and Statistics, Southwest University（西南大学数学与统计学院）；School of Mathematics and Statistics, Xiamen University of Technology（厦门理工学院数学与统计学院）
A fundamental solution of Laplace's equation in three dimensions is expanded in harmonic functions that are separated in parabolic or elliptic cylinder coordinates. There are two expansions in each case which reduce to expansions of the Bessel functions $J_0(kr)$ or $K_0(kr)$, $r^2=(x-x_0)^2+(y-y_0)^2$, in parabolic and elliptic cylinder harmonics. Advantage is taken of the fact that $K_0(kr)$ is a fundamental solution and $J_0(kr)$ is the Riemann function of partial differential equations on the Euclidean plane.
The time evolution of a collisionless plasma is modeled by the relativistic Vlasov-Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. In this work, the setting is two and one-half dimensional, that is, the distribution functions of the particles species are independent of the third space dimension. We consider the case that the plasma is located in an infinitely long cylinder and is influenced by an external magnetic field. We prove existence of stationary solutions and give conditions on the external magnetic field under wh…
We prove, under generic assumptions, that the special variational traveling wave that minimizes the exponentially weighted Ginzburg-Landau functional associated with scalar reaction-diffusion equations in infinite cylinders is the long-time attractor for the solutions of the initial value problems with front-like initial data. The convergence to this traveling wave is exponentially fast. The obtained result is mainly a consequence of the gradient flow structure of the considered equation in the exponentially weighted spaces and does not depend on the precise details of the problem. It strength…
We consider the conformal wave equation on the Einstein cylinder with a defocusing cubic non-linearity. Motivated by a method developed by Rostworowski-Maliborski on the existence of time periodic solutions to the spherically symmetric Einstein-Klein-Gordon system, we study perturbations around the zero solution as a formal series expansion and assume that the perturbations bifurcate from $1-$mode. In the center of this work stands a rigorous proof on how one can choose the initial data to cancel out all secular terms in the resonant system. Interestingly, our analysis reveals that the only po…
In this article, we prove the scattering for the quintic defocusing nonlinear Schrödinger equation on cylinder $\mathbb{R} \times \mathbb{T}$ in $H^1$. We establish an abstract linear profile decomposition in $L^2_x h^α$, $0 < α\le 1$, motivated by the linear profile decomposition of the mass-critical Schrödinger equation in $L^2(\mathbb{R}^d )$, $d\ge 1$. Then by using the solution of the one-discrete-component quintic resonant nonlinear Schrödinger system, whose scattering can be proved by using the techniques in $1d$ mass critical NLS problem by B. Dodson, to approximate the nonlinear profi…
We prove the existence of a countable family of Delaunay type domains Ω_j in M^n x R, where M^n is the Riemannian manifold S^n or H^n and n is at least 2, bifurcating from the cylinder B^n x R (where B^n is a geodesic ball of radius 1 in M^n) for which the first eigenfunction of the Laplace-Beltrami operator with zero Dirichlet boundary condition also has constant Neumann data at the boundary. The domains Ω_j are rotationally symmetric and periodic with respect to the R-axis of the cylinder and as j converges to 0 the domain Ω_j converges to the cylinder B^n x R.
