The momentum ray transform $I_m^k$ integrates a rank $m$ symmetric tensor field $f$ on ${\mathbb R}^n$ over lines with the weight $t^k$, $I_m^kf(x,ξ)=\int_{-\infty}^\infty t^k\langle f(x+tξ),ξ^m\rangle\,\mathrm{d}t$. Let $N^k_m=(I^k_m)^*I^k_m$ be the normal operator of $I_m^k$. To what extent is a symmetric $m$-tensor field $f$ determined by the data $(N_m^0f,\dots,N_m^rf)$ given for some $0\le r\le m$? The Saint Venant operator $W^r_m$ is a linear differential operator of order $m-r$ with constant coefficients on the space of symmetric $m$-tensor fields. We derive an explicit formula expressing $W^r_mf$ in terms of $(N_m^0f,\dots,N_m^rf)$. The tensor field $W^r_mf$ represents the full local information on $f$ that can be extracted from the data $(N_m^0f,\dots,N_m^rf)$.
In this article, we work with a generalized Saint Venant operator introduced by Vladimir Sharafutdinov to describe the kernel of the integral moment transforms over symmetric m-tensor fields in n-dimensional Euclidean space. We also provide an equivalence between the injectivity question for the integral moment transforms and the generalized Saint Venant operator over symmetric tensor fields of the Schwartz class.
We address three related problems in the theory of elasticity, formulated in the framework of double forms: the Saint-Venant compatibility condition, the existence and uniqueness of solutions for equations arising in incompatible elasticity, and the existence of stress potentials. The scope of this work is for manifolds with boundary of arbitrary dimension, having constant sectional curvature. The central analytical machinery is the regular ellipticity of a boundary-value problem for a bilaplacian operator, and its consequences, which were developed in [KL21]. One of the novelties of this work…
In this paper we investigate the deformation of cylindrical linearly elastic shells using the Koiter model. We formulate and solve the relaxed Saint-Venant's problem for thin cylindrical tubes made of isotropic and homogeneous elastic materials. To this aim, we adapt a method established previously in the three-dimensional theory of elasticity. We present a general solution procedure to determine closed-form solutions for the extension, bending, torsion and flexure problems. We remark the analogy and formal resemblance of these solutions to the classical Saint-Venant's solutions for solid cyli…
A nonlinear Korn inequality based on the Green-Saint Venant strain tensor is proved, whenever the displacement is in the Sobolev space $W^{1,p}$, $p\geq 2$, under Dirichlet conditions on a part of the boundary. The inequality can be useful in proving the coercivity of a nonlinear elastic energy.
Exponential stability of the linearized viscous Saint-Venant equations using a quadratic Lyapunov function
Ecole Nationale des Ponts et Chaussées, Institut Polytechnique de Paris（国立路桥学校，巴黎理工学院）；Korean Institute for Advanced Study（韩国高等研究院）；University of California, Berkeley（加州大学伯克利分校）
We study the exponential stability in the $H^{2}$ norm of the nonlinear Saint-Venant (or shallow water) equations with arbitrary friction and slope using a single Proportional-Integral (PI) control at one end of the channel. Using a good but simple Lyapunov function we find a simple and explicit condition on the gain the PI control to ensure the exponential stability of any steady-states. This condition is independent of the slope, the friction coefficient, the length of the river, the inflow disturbance and, more surprisingly, can be made independent of the steady-state considered. When the i…
The standard multilayer Saint-Venant system consists in introducing fluid layers that are advected by the interfacial velocities. As a consequence there is no mass exchanges between these layers and each layer is described by its height and its average velocity. Here we introduce another multilayer system with mass exchanges between the neighborhing layers where the unknowns are a total height of water and an average velocity per layer. We derive it from Navier-Stokes system with an hydrostatic pressure and prove energy and hyperbolicity properties of the model. We also give a kinetic interpre…
On global classical and weak solutions with arbitrary large initial data to the multi-dimensional viscous Saint-Venant system and compressible Navier-Stokes equations subject to the BD entropy condition under spherical symmetry
We initiate the study of the finiteness condition $\int_Ωu(x)^{-β}\,dx\leq C(Ω,β)<+\infty$ where $Ω\subseteq{\mathbb{R}}^n$ is an open set and $u$ is the solution of the Saint Venant problem $Δu=-1$ in $Ω$, $u=0$ on $\partialΩ$. The central issue which we address is that of determining the range of values of the parameter $β>0$ for which the aforementioned condition holds under various hypotheses on the smoothness of $Ω$ and demands on the nature of the constant $C(Ω,β)$. Classes of domains for which our analysis applies include bounded piecewise $C^1$ domains in ${\mathbb{R}}^n$, $n\geq 2$, w…
Global-in-time Well-posedness of Classical Solutions to the Vacuum Free Boundary Problem for the 1-D Viscous Saint-Venant System with Large Data
The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong（香港中文大学数学科学研究所及数学系）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；School of Mathematical Sciences, CMA-Shanghai and MOE-LSC, Shanghai Jiao Tong University（上海交通大学数学科学学院、上海数学中心与教育部拉格朗日计算科学中心）
Let $I_{m}$ denote the Euclidean ray transform acting on compactly supported symmetric $m$-tensor field distributions $f$, and $I_{m}^{*}$ be its formal $L^2$ adjoint. We study a unique continuation result for the normal operator $N_{m}=I_{m}^{*}I_{m}$. More precisely, we show that if $N_{m}$ vanishes to infinite order at a point $x_0$ and if the Saint-Venant operator $W$ acting on $f$ vanishes on an open set containing $x_0$, then $f$ is a potential tensor field. This generalizes two recent works of Ilmavirta and Mönkkönen who proved such unique continuation results for the ray transform of f…
In this study, we propose an improved version of the nonlinear shallow water (or Saint-Venant) equations. This new model is designed to take into account the effects resulting from the large spacial and/or temporal variations of the seabed. The model is derived from a variational principle by choosing the appropriate shallow water ansatz and imposing suitable constraints. Thus, the derivation procedure does not explicitly involve any small parameter.
Global spherically symmetric classical solutions for arbitrary large initial data of the multi-dimensional non-isentropic compressible Navier-Stokes equations
