Weighted Decay Estimates for the Wave Equation with Radially Symmetric Data
Paschalis Karageorgis
研究四维或高维空间中具有径向对称数据的齐次波动方程。利用Riemann算子的一些新的整表示，我们建立了解的加权衰减估计。
We study the homogeneous wave equation with radially symmetric data in four or higher space dimensions. Using some new integral representations for the Riemann operator, we establish weighted decay estimates for the solution.
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We prove dilation invariant inequalities involving radial functions, poliharmonic operators and weights that are powers of the distance from the origin. Then we discuss the existence of extremals and in some cases we compute the best constants.
Unconditional stability of radially symmetric steady sates of compressible viscous fluids with inflow/outflow boundary conditions
本文证明了带流入/流出边界条件的可压缩粘性流体径向对称稳态具有无条件稳定性，即相关演化问题的任意解最终都会收敛至单一径向对称稳态。
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Radially symmetric solutions to a Lane-Emden type system
本文研究一类Lane-Emden型系统的径向对称解存在性，回答了da Silva和do O（2024）提出的问题，并讨论了非齐次版本及若干开放问题。
Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system
有限时间爆破所示发生的径向对称解的临界准线性Smoluchowski-Poisson系统提供的初始条件的质量超过一个明确的阈值。在超临界的情况下，爆破显示发生任何积极的质量。这个证明依赖于一个新的维里类型的恒等式。
We consider the problem \begin{equation}(1)\;\;\; \begin{cases} S_k(D^2u)= λ|x|^σ (1-u)^q &\mbox{in }\;\; B,\\ u <0 & \mbox{in }\;\; B,\\ u=0 &\mbox{on }\partial B, \end{cases} \end{equation} where $B$ denotes the unit ball in $\mathbb{R}^n$, $n>2k$ ($k\in \mathbb{N}$), $λ>0$, $q > k$ and $σ\geq 0$. We study the existence, uniqueness and multiplicity of negative bounded radially symmetric solutions of (1). The methodology to obtain our results is based on a dynamical system approach. For this, we introduce a new transformation which reduces problem (1) to an autonomous two dimensional generali…
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针对单位球中带齐次Dirichlet边值的径向对称程函方程|Du|=V，研究消失粘性方法，构造粘性解与极限解显式公式，证明收敛性并推导出ε|log ε|型局部收敛速率，还讨论了解的唯一性及给出例子。
In this article we derive C^1-a priori estimates on the Riemann invariants of the Euler compressible equations in the case of cylindrical or spherical symmetry. These estimates allow then to construct shock waves with a time of existence proportional to the distance to the origin at the initial time.
Existence of radially symmetric stationary solutions for viscous and Heat-conductive ideal Gas
本文在球体外部无界区域上证明了可压缩粘性导热多方理想气体径向对称定常解在远场状态小邻域内的唯一存在性，并给出了向远场状态的代数衰减速率估计。
In this paper a nonlinear coupled Schrodinger system in the presence of mixed cubic and superlinear power laws is considered. Focus are made on the steady state solutions of the continuous system for existence and uniqueness by minimizing problem on some associated Nehari manifold.
Weighted HLS inequalities for radial functions and Strichartz estimates for wave and Schroedinger equations
本文利用加权Hardy-Littlewood-Sobolev（HLS）不等式，从Morawetz型估计出发，导出了自由波方程径向对称解的整体或局部时间上的Lahartz估计.用同样的方法，我们也得到了自由Schroedinger方程径向对称解的加权端点带导数增益的Schroedinger估计。利用Stein和Weiss的加权不等式和Muckenhoupt的加权Lebesgue空间中的Hardy-Littlewood极大不等式证明了径向对称函数的加权HLS不等式.在径向对称条件下，我们得到了比通常的HLS不等式和Hohartz估计更好的结果。
This paper is devoted to the generic transversality of radially symmetric stationary solutions of nonlinear parabolic systems of the form \[ \partial_t w(x,t) = -\nabla V\bigl(w((x,t))\bigr) + Δ_x w(x,t) \,, \] where the space variable $x$ is multidimensional and unbounded. It is proved that, generically with respect to the potential $V$, radially symmetric stationary solutions that are stable at infinity (in other words, that approach a minimum point of $V$ at infinity in space) are transverse; as a consequence, the set of such solutions is discrete. This result can be viewed as the extension…
Nonlinear Schroedinger equations with radially symmetric data of critical regularity
研究了一类具有临界正则性的径向对称纯幂非线性Schroedinger方程小解的整体存在性。在径向对称条件下，我们将注意力集中在非线性项的幂比伪共形项的幂稍小且初始数据属于尺度不变的齐次索伯列夫空间的情形.尽管初值是负阶可微的，但只要初值具有小范数，非线性Schroedinger方程就有整体解。在证明这一结果的关键成分是一个有效的使用全球加权平滑估计特定的径向对称的解决方案。
Radially symmetric shadow wave solutions to the system of multidimensional pressureless gas dynamics are introduced, which allow one to capture concentration of mass. The transformation to a one-dimensional system with source terms is performed and physically meaningful boundary conditions at the origin are determined. Entropy conditions are derived and applied to single out physical (nonnegative mass) and dissipative (entropic) solutions. A complete solution to the pseudo-Riemann problem with initial data exhibiting a single shock on a sphere is obtained.
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We consider the semilinear curl-curl wave equation $s(x) \partial_t^2 U +\nabla\times\nabla\times U + q(x) U \pm V(x) |U|^{p-1} U = 0 \mbox{ for } (x,t)\in \mathbb{R}^3\times\mathbb{R}$. For any $p>1$ we prove the existence of time-periodic spatially localized real-valued solutions (breathers) both for the $+$ and the $-$ case under slightly different hypotheses. Our solutions are classical solutions that are radially symmetric in space and decay exponentially to $0$ as $|x|\to \infty$. Our method is based on the fact that gradient fields of radially symmetric functions are annihilated by the …
In this paper, we study the Cauchy problem for radially symmetric homogeneous non-cutoff Boltzmann equation with Maxwellian molecules, the initial datum belongs to Shubin space of the negative index which can be characterized by spectral decomposition of the harmonic oscillators. The Shubin space of the negative index contains the measure functions. Based on this spectral decomposition, we construct the weak solution with Shubin class initial datum, we also prove that the Cauchy problem enjoys Gelfand-Shilov smoothing effect, meaning that the smoothing properties are the same as the Cauchy pro…
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Time-Asymptotic Stability of the Stationary Solution to the Impermeable Wall Problem for the radially Symmetric Navier-Stokes-Korteweg Equations
该研究针对外部域上的径向对称NSK方程不可渗透壁问题，证明当初值为定态解的小扰动且边界数据足够小时，存在全局时间强解并渐近收敛至定态解，所用方法结合了基本能量估计与精心设计的能量泛函。
The aim of the paper is to derive spectral estimates on the eigenvalue moments of the magnetic Schroedinger operators defined on the two-dimensional disk with a radially symmetric magnetic field and radially symmetric electric potential.
We prove nonlinear stability of compactly supported expanding star-solutions of the mass-critical gravitational Euler-Poisson system. These special solutions were discovered by Goldreich and Weber in 1980. The expanding rate of such solutions can be either self-similar or non-self-similar (linear), and we treat both types. An important outcome of our stability results is the existence of a new class of global-in-time radially symmetric solutions, which are not homogeneous and therefore not encompassed by the existing works. Using Lagrangian coordinates we reformulate the associated free-bounda…
Hardy-Sobolev inequalities involving mixed radially and cylindrically symmetric weights
本文研究了函数在R^d空间上的加权Hardy-Sobolev不等式，探讨了不同权重下的有效性条件及极值函数的存在性。
We consider the quadratic nonlinear Schrödinger system \begin{align*} \begin{cases} i\partial_t u +Δu =v \overline{u},\\ i\partial_t v +κΔv =u^2, \end{cases} \text{ on } I \times \mathbb{R}^d, \end{align*} where $1\leq d \leq 6$ and $κ>0$. In the lower dimensional case $d=1,2,3$, it is known that the $H^1$-solution is global in time. On the other hand, there are finite time blow-up solutions when $d=4,5,6$ and $κ=1/2$. The condition of $κ=1/2$ is called mass-resonance. In this paper, we prove finite time blow-up under radially symmetric assumption when $d=5,6$ and $κ\neq 1/2$ and we show blow-…
Subsonic time-periodic solutions to radially symmetric spiral flows in an annulus
研究环形区域中具有非零角速度的径向对称非等熵欧拉系统的亚声速时间周期解，在特定边界条件下建立其存在性与稳定性，克服导数损失等困难，关键在于重写熵的径向导数及控制扰动积累。
We investigate the following eigenvalue problem \begin{align*} \begin{cases} -\operatorname{div}\left( L(x) |\nabla u| ^{p-2}\nabla u\right)=λK(x)|u|^{p-2}u \quad \text{in } A_{R_1}^{R_2} , u=0\quad \text{on } \partial A_{R_1}^{R_2} , \end{cases} \end{align*} where $A_{R_1}^{R_2}:=\{x\in\mathbb{R}^N: R_1<|x|<R_2\}$ $(0< R_1<R_2\leq\infty)$, $λ>0$ is a parameter, the weights $L$ and $K$ are measurable with $L$ positive a.e. in $A_{R_1}^{R_2}$ and $K$ possibly sign-changing in $A_{R_1}^{R_2}$. We prove the existence of the first eigenpair and discuss the regularity and positiveness of eigenfunct…
Global well-posedness for supercritical SQG with perturbations of radially symmetric data
本文针对超临界耗散SQG方程大初始数据整体适定性难题，利用径向解的增强衰减，证明大径向数据的小扰动存在唯一整体解。
In this paper, we verify the Glassey conjecture in the radial case for all spatial dimensions, which states that, for the nonlinear wave equations of the form $\Box u=|\nabla u|^p$, the critical exponent to admit global small solutions is given by $p_c=1+\frac{2}{n-1}$. Moreover, we are able to prove the existence results with low regularity assumption on the initial data and extend the solutions to the sharp lifespan. The main idea is to exploit the trace estimates and KSS type estimates.
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Global large strong solutions to the radially symmetric compressible Navier-Stokes equations in 2D solid balls
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Global behaviour of radially symmetric solutions stable at infinity for gradient systems
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This paper is concerned with weighted energy estimates and diffusion phenomena for the initial-boundary problem of the wave equation with space-dependent damping term in an exterior domain. In this analysis, an elliptic problem was introduced by Todorova and Yordanov. This attempt was quite useful when the coefficient of the damping term is radially symmetric. In this paper, by modifying their elliptic problem, we establish weighted energy estimates and diffusion phenomena even when the coefficient of the damping term is not radially symmetric.
Inviscid Limit Problem of radially symmetric stationary solutions for compressible Navier-Stokes equation
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Blow-up of radially symmetric solutions for a cubic NLS type system in dimension 4
针对四维能量临界的三次非线性 Schrödinger 系统，本文利用凸性方法证明低初始能量、高动能的径向对称解有限时间爆破，并通过集中紧性与变分方法建立基态及最佳 Sobolev 常数。
We consider two critical Rellich inequalities with singularities at both the origin and the boundary in the higher order critical radial Sobolev spaces $W_{0, {\rm rad}}^{k, p}$, where $1< p = \frac{N}{k}$. We give the explicit values of the optimal constants of two critical Rellich inequalities for radially symmetric functions in $W_{0, {\rm rad}}^{k, \frac{N}{k}}$. Furthermore the (non-)attainability of the optimal constants are also discussed.
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In this paper we prove that the equation $ -( r^αϕ(|u'(r)|)u'(r))' = λr^γf(u(r)), ~0<r<R$, where $α, γ, {\bf{R}}$ are given real numbers, $ϕ: (0, \infty) \to (0, \infty)$ is a suitable twice differentiable function, $λ> 0$ is a real parameter and $f:{\bf{R}}\to{\bf{R}}$ is continuous, admits an infinite sequence of sign-changing solutions satisfying $u'(0) =u(R) =0$. The function $f$ is required to satisfy $tf(t)>0$ for $ t\neq 0$. Our technique explores fixed point arguments applied to suitable integral equations and shooting arguments. Our main result extends earlier ones in the case $ϕ$ is …
Optimal Decay Estimates for the Radially Symmetric Compressible Navier-Stokes Equations
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The aim of the paper is to derive spectral estimates on the eigenvalue moments of the magnetic Dirichlet Laplacian defined on the two-dimensional disk with a radially symmetric magnetic field.
We consider radial solutions to the cubic Schr{ö}dinger equation on the Heisenberg group$$i\partial_t u - Δ_{\mathbb{H}^1} u = |u|^2u, \quadΔ_{\mathbb{H}^1} = \frac{1}{4}(\partial_x^2+\partial_y^2) + (x^2+y^2)\partial_s^2, \quad(t,x,y,s) \in \mathbb{R}\times\mathbb{H}^1.$$This equation is a model for totally non-dispersive evolution equations. We show existence of ground state traveling waves with speed $β\in (-1,1)$. When the speed $β$ is sufficiently close to $1$, we prove their uniqueness up to symmetries and their smoothness along the parameter $β$. The main ingredient is the emergence of …
Here, we study radial solutions for first- and second-order stationary Mean-Field Games (MFG) with congestion on $\mathbb{R}^d$. MFGs with congestion model problems where the agents' motion is hampered in high-density regions. The radial case, which is one of the simplest non one-dimensional MFG, is relatively tractable. As we observe in this paper, the Fokker-Planck equation is integrable with respect to one of the unknowns. Consequently, we obtain a single equation substituting this solution into the Hamilton-Jacobi equation. For the first-order case, we derive explicit formulas; for the ell…
Global Smooth Radially Symmetric Solutions to a Multidimensional Radiation Hydrodynamics Model
本文研究三维有界同心环形区域内辐射 Euler 方程的初边值问题，证明了在初始数据足够小的条件下存在唯一的全局光滑径向对称解。
Given two probability measures $μ, ν$ on $\mathbb{R}^d$, in subharmonic order, we describe optimal stopping times $τ$ that maximize/minimize the cost functional $\mathbb{E} |B_0 - B_τ|^α$, $α> 0$, where $(B_t)_t$ is Brownian motion with initial law $μ$ and with final distribution --once stopped at $τ$-- equal to $ν$. Under the assumption of radial symmetry on $μ$ and $ν$, we show that in dimension $d \geq 3$ and $α\neq 2$, there exists a unique optimal solution given by a non-randomized stopping time characterized as the hitting time to a suitably symmetric barrier. We also relate this problem…
We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules enjoys the same Gelfand-Shilov regularizing effect as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator.
A Sharp Mass Threshold for Boundedness in a Critical Quasilinear Chemotaxis Model With Indirect Signal Production: The Radially Symmetric Case
针对空间维数大于2的具有间接信号产生的临界拟线性趋化模型，在径向对称情形下确立了尖锐质量阈值M_c，明确了不同质量与能量条件下解的全局有界性或无穷时间爆破行为，并关联了其与相关不等式最优常数的关系。
Higher-order derivatives of radially symmetric functions
该研究推导了径向对称函数n阶偏导数张量Frobenius范数的简洁逐点公式，规避高阶径向导数的技术困难，刻画了径向对称函数在高阶Sobolev空间中的子空间，并推广到Banach格上的Sobolev型空间。
Global strong solutions to the radially symmetric compressible MHD equations in 2D solid balls with arbitrary large initial data
本文证明了二维实心球上具有任意大初始数据的径向对称可压缩MHD方程全局强解的存在性，并给出了密度一致有界的条件，推广了已有结果。
Solutions of the 2D Radially Symmetric Vlasov-Maxwell System with an Initial Focusing Phase
本文研究二维径向对称Vlasov-Maxwell系统，构造了初始电荷密度和电场$C^k$范数任意小但后期$L^\infty$范数任意大的解，揭示了聚焦阶段导致的增长现象。
Global well-posedness of radially symmetric strong solutions to two-dimensional compressible liquid crystal flows with large data and vacuum
该研究针对具大初值和真空的二维可压缩液晶流径向对称强解，在β>1条件下建立其整体适定性，改进了相关研究结果，关键利用director径向对称性的刚性机制阻止集中现象。
This paper studies the transport of a mass $μ$ in $\real^d, d \geq 2,$ by a flow field $v= -\nabla K*μ$. We focus on kernels $K=|x|^α/ α$ for $2-d\leq α<2$ for which the smooth densities are known to develop singularities in finite time. For this range This paper studies the transport of a mass $μ$ in $\real^d, d \geq 2,$ by a flow field $v= -\nabla K*μ$. We focus on kernels $K=|x|^α/ α$ for $2-d\leq α<2$ for which the smooth densities are known to develop singularities in finite time. For this range we prove the existence for all time of radially symmetric measure solutions that are monotone …
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