Samer Israwi
We consider here the $1D $ Green-Naghdi equations that are commonly used in coastal oceanography to describe the propagation of large amplitude surface waves. We show that the solution of the Green-Naghdi equations can be constructed by a standard Picard iterative scheme so that there is no loss of regularity of the solution with respect to the initial condition.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper we consider the 1D Green-Naghdi system. This system describes the evolution of water waves over a flat bottom in the shallow water regime in terms of the surface height $h$ and the horizontal velocity $u$. We give a Lagrangian formulation of the 1D Green-Naghdi system on a Sobolev type diffeomorphism group. As an application of this formulation we prove local well-posedness for $(h,u)$ in the Sobolev space $(1+H^s(\R)) \times H^{s+1}(\R),\; s > 1/2$. This improves the local well-posedness range for the 1D Green-Naghdi system.
We derive here a variant of the 2D Green-Naghdi equations that model the propagation of two-directional, nonlinear dispersive waves in shallow water. This new model has the same accuracy as the standard $2D $ Green-Naghdi equations. Its mathematical interest is that it allows a control of the rotational part of the (vertically averaged) horizontal velocity, which is not the case for the usual Green-Naghdi equations. Using this property, we show that the solution of these new equations can be constructed by a standard Picard iterative scheme so that there is no loss of regularity of the solutio…
On the blow-up scenario for some modified Serre-Green-Naghdi equations
针对 Clamond 等人提出的改进色散关系的 mSGN 系统，本文刻画其精确爆破情形，并证明一类解会在有限时间内产生奇性，结果亦适用于弱表面张力 SGN 系统。
Oscillatory and regularized shock waves for a modified Serre-Green-Naghdi system
本文提出含人工色散与耗散项的修正 Serre-Green-Naghdi 系统，证明其行波解在参数趋零时收敛到浅水熵激波，以解释色散激波及扩散对涌波的作用。
The (Serre-)Green-Naghdi system is a non-hydrostatic model for the propagation of surface gravity waves in the shallow-water regime. Recently , Favrie and Gavrilyuk proposed in [Nonlinearity, 30(7) (2017)] an efficient way of numerically computing approximate solutions to the Green-Naghdi system. The approximate solutions are obtained through solutions of an augmented quasilinear system of balance laws, depending on a parameter. In this work, we provide quantitative estimates showing that any regular solution of the Green-Naghdi system is the limit of solutions to the Favrie-Gavrilyuk system a…
We provide the existence and asymptotic description of solitary wave solutions to a class of modified Green-Naghdi systems, modeling the propagation of long surface or internal waves. This class was recently proposed by Duch{ê}ne, Israwi and Talhouk (Stud. Appl. Math.,137 (2016)) in order to improve the frequency dispersion of the original Green-Naghdi system while maintaining the same precision. The solitary waves are constructed from the solutions of a constrained minimization problem. The main difficulties stem from the fact that the functional at stake involves low order non-local operator…
We introduce a new class of Green-Naghdi type models for the propagation of internal waves between two (1+1)-dimensional layers of homogeneous, immiscible, ideal, incompressible, irrotational fluids, vertically delimited by a flat bottom and a rigid lid. These models are tailored to improve the frequency dispersion of the original bi-layer Green-Naghdi model, and in particular to manage high-frequency Kelvin-Helmholtz instabilities, while maintaining its precision in the sense of consistency. Our models preserve the Hamiltonian structure, symmetry groups and conserved quantities of the origina…
In this paper we consider the 1D Green-Naghdi system over a nonflat bottom. This system describes the evolution of water waves over an uneven bottom in the shallow water regime in terms of the water depth $h$ and the horizontal velocity $u$. Using a Lagrangian formulation of this system on a Sobolev type diffeomorphism group we prove local well-posedness for $(h,u)$ in the Sobolev space $(1+H^s(\mathbb R)) \times H^{s+1}(\mathbb R),\; s > 1/2$. This improves the local well-posedness range.
In this paper we introduce a new reformulation of the Green-Naghdi model in the Camassa-Holm regime for the propagation of internal waves over a flat topography derived by Duchêne, Israwi and Talhouk. These new Green-Naghdi systems are adapted to improve the frequency dispersion of the original model, they share the same order of precision as the standard one but have an appropriate structure which makes them much more suitable for the numerical resolution. We develop a second order splitting scheme where the hyperbolic part of the system is treated with a high-order finite volume scheme and t…
Global weak solutions of the Serre-Green-Naghdi equations with surface tension
针对带表面张力的 Serre–Green–Naghdi 方程，本文利用其 H^1 等价能量结构，证明小能量初值下全局弱耗散解的存在性，并给出 Riemann 不变量的单侧 Oleinik 估计。
We study the well-posedness of the initial value problem for a wide class of singular evolution equations. We prove a general well-posedness theorem under three assumptions easy to check: the first controls the singular part of the equation, the second the behavior of the nonlinearities, and the third one assumes that an energy estimate can be found for the linearized system. We allow losses of derivatives in this energy estimate and therefore construct a solution by a Nash-Moser iterative scheme. As an application to this general theorem, we prove the well-posedness of the Serre and Green-Nag…
In this paper we address the Cauchy problem for two systems modeling the propagation of long gravity waves in a layer of homogeneous, incompressible and inviscid fluid delimited above by a free surface, and below by a non-necessarily flat rigid bottom. Concerning the Green-Naghdi system, we improve the result of Alvarez-Samaniego and Lannes (Invent. Math., 2008) in the sense that much less regular data are allowed, and no loss of derivatives is involved. Concerning the Boussinesq-Peregrine system, we improve the lower bound on the time of existence provided by M{é}sognon-Gireau (Adv. Different…
Long time well-posedness and full justification of a Whitham-Green-Naghdi system
本文证明了模拟浅水表面重力波的 Whitham-Green-Naghdi 系统与水波方程在 $O(μ^2(\varepsilon+β))$ 精度上一致，并建立了其在长时间尺度上的适定性和解的收敛性。
The Green-Naghdi equations are a nonlinear dispersive perturbation of the nonlinear shallow water equations, more precise by one order of approximation. These equations are commonly used for the simulation of coastal flows, and in particular in regions where the water depth vanishes (the shoreline). The local well-posedness of the Green-Naghdi equations (and their justification as an asymptotic model for the water waves equations) has been extensively studied, but always under the assumption that the water depth is bounded from below by a positive constant. The aim of this article is to remove…
We consider an asymptotic 1D (in space) rotation-Camassa-Holm (R-CH) model, which could be used to describe the propagation of long-crested shallow-water waves in the equatorial ocean regions with allowance for the weak Coriolis effect due to the Earth's rotation. This model equation has similar wave-breaking phenomena as the Camassa-Holm equation. It is analogous to the rotation-Green-Naghdi (R-GN) equations with the weak Earth's rotation effect, modeling the propagation of wave allowing large amplitude in shallow water. We provide here a rigorous justification showing that solutions of the R…
Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart
本文通过数值实验研究一维 SGN 方程及其全色散 WGN 对应方程，构造 WGN 孤立波解并分析稳定性、调制振荡与爆破现象，提出 Fourier 谱方法结合 GMRES 和四阶 Runge-Kutta 的高效数值方案。
In this work we prove that the solution of the Serre-Green-Naghdi equation cannot be globally defined when the interface reaches the impervious bottom tangentially. As a consequence, our result complements the paper \emph{Camassa, R., Falqui, G., Ortenzi, G., Pedroni, M., \& Thomson, C. Hydrodynamic models and confinement effects by horizontal boundaries. Journal of Nonlinear Science, 29(4), 1445-1498, 2019.} Furthermore, we also prove that the solution to the $abcd-$Boussinesq system can change sign in finite time. Finally, we provide with a proof of a scenario of finite time singularity for …
In this short note, we present a multi-symplectic structure of the Serre-Green-Naghdi (SGN) equations modelling nonlinear long surface waves in shallow water. This multi-symplectic structure allow the use of efficient finite difference or pseudo-spectral numerical schemes preserving exactly the multi-symplectic form at the discrete level.
In this note we present the current status of the derivation of a viscous Serre-Green-Naghdi system. For this goal, the flow domain is separated into two regions. The upper region is governed by inviscid Euler equations, while the bottom region (the so-called boundary layer) is described by Navier-Stokes equations. We consider a particular regime linking the Reynolds number and the shallowness parameter. The computations presented in this note are performed in the fully nonlinear regime. The boundary layer flow reduces to a Prantdl-like equation. Further approximations seem to be needed to obt…
Numerical study of the Serre-Green-Naghdi equations in 2D
本文利用 Fourier 谱方法与 Krylov 子空间技术数值研究二维无旋 SGN 方程，验证线状一维孤立波的横向稳定性，并发现不存在二维局域稳定结构。
For surface gravity waves propagating in shallow water, we propose a variant of the fully nonlinear Serre-Green-Naghdi equations involving a free parameter that can be chosen to improve the dispersion properties. The novelty here consists in the fact that the new model conserves the energy, contrary to other modified Serre's equations found in the literature. Numerical comparisons with the Euler equations show that the new model is substantially more accurate than the classical Serre equations, specially for long time simulations and for large amplitudes.
The equations of motion of lossless compressible nonclassical fluids under the so-called Green--Naghdi theory are considered for two classes of barotropic fluids: (\textit{i}) perfect gases and (\textit{ii}) liquids obeying a quadratic equation of state. An exact reduction in terms of a scalar acoustic potential and the (scalar) thermal displacement is achieved. Properties and simplifications of these model nonlinear acoustic equations for unidirectional flows are noted. Specifically, the requirement that the governing system of equations for such flows remain hyperbolic is shown to lead to re…
This article is concerned with bending plate theory for thermoelastic diffusion materials under Green-Naghdi theory. First, we present the basic equations which characterize the bending of thin thermoelastic diffusion plates for type II and III models. The theory allows for the effect of transverse shear deformation without any shear correction factor, and permits the propagation of waves at a finite speed without energy dissipation for type II model and with energy dissipation for type III model. By the semigroup theory of linear operators, we prove the well-posedness of the both models and t…
Discrete transparent boundary conditions for the linearized Green-Naghdi system of equations
本文提出线性化Green-Naghdi方程组的人工边界条件，采用拉普拉斯变换或Z变换求逆，通过交错网格和集中网格两种空间离散化方法，结合Crank-Nicolson时间离散化，验证了数值方法的准确性及稳定性。
The Serre-Green-Naghdi system is a coupled, fully nonlinear system of dispersive evolution equations which approximates the full water wave problem. The system is an extension of the well known shallow-water system to the situation where the waves are long, but not so long that dispersive effects can be neglected. In the current work, the focus is on deriving mass, momentum and energy densities and fluxes associated with the Serre-Green-Naghdi system. These quantities arise from imposing balance equations of the same asymptotic order as the evolution equations. In the case of an even bed, the …
We are interested in asymptotic models for the propagation of internal waves at the interface between two shallow layers of immiscible fluid, under the rigid-lid assumption. We review and complete existing works in the literature, in order to offer a unified and comprehensive exposition. Anterior models such as the shallow water and Boussinesq systems, as well as unidirectional models of Camassa-Holm type, are shown to descend from a broad Green-Naghdi model, that we introduce and justify in the sense of consistency. Contrarily to earlier works, our Green-Naghdi model allows a non-flat topogra…
We study here the propagation of long waves in the presence of vorticity. In the irrotational framework, the Green-Naghdi equations (also called Serre or fully nonlinear Boussinesq equations) are the standard model for the propagation of such waves. These equations couple the surface elevation to the vertically averaged horizontal velocity and are therefore independent of the vertical variable. In the presence of vorticity, the dependence on the vertical variable cannot be removed from the vorticity equation but it was however shown in [?] that the motion of the waves could be described using …
We rigorously justify in 3D the main asymptotic models used in coastal oceanography, including: shallow-water equations, Boussinesq systems, Kadomtsev-Petviashvili (KP) approximation, Green-Naghdi equations, Serre approximation and full-dispersion model. We first introduce a ``variable'' nondimensionalized version of the water-waves equations which vary from shallow to deep water, and which involves four dimensionless parameters. Using a nonlocal energy adapted to the equations, we can prove a well-posedness theorem, uniformly with respect to all the parameters. Its validity ranges therefore f…
We study here Green-Naghdi type equations (also called fully nonlinear Boussinesq, or Serre equations) modeling the propagation of large amplitude waves in shallow water. The novelty here is that we allow for a general vorticity, hereby allowing complex interactions between surface waves and currents. We show that the a priori 2+1-dimensional dynamics of the vorticity can be reduced to a finite cascade of two-dimensional equations: with a mechanism reminiscent of turbulence theory, vorticity effects contribute to the averaged momentum equation through a Reynolds-like tensor that can be determi…
We study the existence and the regularity of the biharmonic Green kernel in a Brelot biharmonic space whose associated harmonic spaces have Green kernels. We show by some examples that this kernel does not always exist. We then introduce and study the adjoint of the given biharmonic space. This study was initiated by Smyrnelis, however, it seems that several results were incomplete and we clarify them here.
We study the existence of the Green function for an elliptic system in divergence form $-\nabla\cdot a\nabla$ in $\mathbb{R}^d$, with $d>2$. The tensor field $a=a(x)$ is only assumed to be bounded and $λ$-coercive. For almost every point $y \in \mathbb{R}^d$, the existence of a Green's function $G(a; \cdot, y)$ centered in $y$ has been proven in [J. Conlon, A. Giunti and F.Otto, "Green's function for elliptic systems: Delmotte-Deuschel bounds", 2017]. In this paper, we show that the set of points $y \in \mathbb{R}^d$ for which $G(a; \cdot, y)$ does not exist has zero $p$-capacity, for an expon…
In this paper, we use the Green-Naghdi theory of thermomechanics of continua to derive a nonlinear theory of thermoelasticity with microtemperatures of type III. This theory permits propagation of both thermal and microtemperatures waves at finite speeds with dissipation of energy. The equations of the linear theory are also obtained. With the help of the semigroup theory of linear operators we establish that the linear anisotropic problem is well posed and we study the asymptotic behavior of the solutions. Finally, we investigate the impossibility of the localization in time of solutions.
We study ($p$-harmonic) singular functions, defined by means of upper gradients, in bounded domains in metric measure spaces. It is shown that singular functions exist if and only if the complement of the domain has positive capacity, and that they satisfy very precise capacitary identities for superlevel sets. Suitably normalized singular functions are called Green functions. Uniqueness of Green functions is largely an open problem beyond unweighted $\mathbf{R}^n$, but we show that all Green functions (in a given domain and with the same singularity) are comparable. As a consequence, for $p$-…
Optimal result involving the Green's function
研究三维球中临界Sobolev指数非线性椭圆方程正解存在与不存在的边界，通过适当选择测试函数确定最优结果。
We prove that for an open domain $D \subset \mathbb{R}^d $ with $d \geq 2 $ , for every (measurable) uniformly elliptic tensor field $a$ and for almost every point $y \in D$ , there exists a unique Green's function centred in $ y $ associated to the vectorial operator $ -\nabla \cdot a\nabla $ in D. In particular, when $d > 2$ this result also implies the existence of the fundamental solution for elliptic systems, i.e. the Green function for $ -\nabla \cdot a\nabla $ in $ \mathbb{R}^d $. Moreover, introducing an ensemble $\langle\cdot \rangle$ over the set of uniformly elliptic tensor fields, …
We prove the existence and pointwise bounds of the Green functions for stationary Stokes systems with measurable coefficients in two dimensional domains. We also establish pointwise bounds of the derivatives of the Green functions under a regularity assumption on the $L_1$-mean oscillations of the coefficients.
We present a new method for the existence and pointwise estimates of a Green's function of non-divergence form elliptic operator with Dini mean oscillation coefficients. We also present a sharp comparison with the corresponding Green's function for constant coefficients equations.
Green functions of mixed boundary value problems for stationary Stokes systems in two dimensions
本文在二维Reifenberg平坦区域中，针对带可测系数的平稳Stokes系统混合Dirichlet-共法向导数问题，证明了Green函数的存在性、唯一性并给出多种估计。
We consider in the plane the problem of reconstructing a domain from the normal derivative of its Green's function with pole at a fixed point in the domain. By means of the theory of conformal mappings, we obtain existence, uniqueness, (non-spherical) symmetry results, and a formula relating the curvature of the boundary of the domain to the normal derivative of its Green's function.
We establish existence and various estimates of fundamental matrices and Green's matrices for divergence form, second order strongly parabolic systems in arbitrary cylindrical domains under the assumption that solutions of the systems satisfy an interior Hölder continuity estimate. We present a unified approach valid for both the scalar and the vectorial cases.
In this paper, we establish existence, uniqueness, and scale-invariant estimates for fundamental solutions of non-homogeneous second order elliptic systems with bounded measurable coefficients in $\mathbb{R}^n$ and for the corresponding Green functions in arbitrary open sets. We impose certain non-homogeneous versions of de Giorgi-Nash-Moser bounds on the weak solutions and investigate in detail the assumptions on the lower order terms sufficient to guarantee such conditions.
Green's function and Large time behavior for the 1-D compressible Euler-Maxwell system
本文研究了带松弛的一维可压缩 Euler-Maxwell 系统的 Green 函数与大时间行为，通过构造线性化系统的 Green 函数并结合能量估计，证明了非线性系统解的整体存在性与最优时间衰减率。
We establish existence and pointwise estimates of fundamental solutions and Green's matrices for divergence form, second order strongly elliptic systems in a domain $Ω\subseteq \mathbb{R}^n$, $n \geq 3$, under the assumption that solutions of the system satisfy De Giorgi-Nash type local Hölder continuity estimates. In particular, our results apply to perturbations of diagonal systems, and thus especially to complex perturbations of a single real equation.
Let $M$ be a complete non-compact Riemannian manifold and let $σ$ be a Radon measure on $M$. We study the problem of existence or non-existence of positive solutions to a semilinear elliptic inequaliy \begin{equation*} -Δu\geq σu^{q}\quad \text{in}\,\,M, \end{equation*} where $q>1$. We obtain necessary and sufficent criteria for existence of positive solutions in terms of Green function of $Δ$. In particular, explicit necessary and sufficient conditions are given when $M$ has nonnegative Ricci curvature everywhere in $M$, or more generally when Green's function satisfies the 3G-inequality.
Existence of pulse solutions in a bulk--surface reaction--diffusion system for cell polarization
本文针对二维区域中用于细胞极化的体-表面反应扩散系统，采用奇异摄动方法证明了其稳态脉冲解的存在性，其分析可为其他体-表面型系统应用该方法提供指导。
In three-dimensional case, we consider two classical operators: Schrodinger operator and an operator in the divergence form. For slowly-decaying oscillating potentials, we establish spatial asymptotics of the Green's function. The main term in this asymptotics involves vector-valued analytic function whose behavior is studied away from the spectrum. The absolute continuity of the spectrum is established as a corollary. For the operator in the divergence form, we consider the wave equation and establish existence of wave operators.
The Green's function of polyharmonic operators with diverging coefficients: Construction and sharp asymptotics
该研究针对闭黎曼流形上带大正参数α的多调和算子(Δ_g + α)^k，证明了其格林函数的存在唯一性与正性，得到显式依赖α的逐点估计，并发现大α下远离对角线区域的指数衰减特性。
This paper deals with global existence and uniqueness for the lake equations with a bottom topography vanishing on the shore. Our result generalizes previous studies that assumed the depth to be nondegenerate. Elliptic estimates for degenerate equations has to be established studying the behavior of the associated Green function.
For a given finite subset $S$ of a compact Riemannian manifold $(M,g)$ whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric on $M \setminus S$ such that each point of $S$ corresponds to an asymptotically flat end and that the Schouten tensor of the conformal metric belongs to the boundary of the given cone. As a by-product, we define a purely local notion of Ricci lower bounds for continuous metrics which are conformal to smooth metrics and prove a corresponding volume comparison theore…
Cas d'existence de solutions d'EDP
本文研究了几何偏微分方程解的存在性，并讨论了相关不等式和原理。
Green function in metric measure spaces
本文研究度量测度空间中Cheeger Q-Laplacian格林函数的存在性与唯一性，在相对紧致域和全局情形下证明唯一性，并补充了无界空间中全局格林函数的存在性结果。
We consider the problem of wave propagation for a 2-D rectilinear optical waveguide which presents some perturbation. We construct a mathematical framework to study such a problem and prove the existence of a solution for the case of small imperfections. Our results are based on the knowledge of a Green's function for the rectilinear case.
Green's Function and Solution Representation for a Boundary Value Problem Involving the Prabhakar Fractional Derivative
本文研究了一个涉及普拉哈拉德分数导数的时间二阶偏微分方程的第一类边值问题，通过利用普拉哈拉德核的结构性质和广义 Mittag-Leffler 函数，将问题转化为 Volterra 型积分方程，并据此显式构造了对应的格林函数，进而推导出解的闭式积分表示，并证明了其存在性和唯一性。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。