In the works of K.A. Pericak-Spector and S. Spector \cite{ps88, ps98} a class of self-similar solutions are constructed for the equations of radial isotropic elastodynamics that describe cavitating solutions. Cavitating solutions decrease the total mechanical energy and provide a striking example of non-uniqueness of entropy weak solutions (for polyconvex energies) due to point-singularities at the cavity. To resolve this paradox, we introduce the concept of singular limiting induced from continuum solution (or slic-solution), according to which a discontinuous motion is a slic-solution if its averages form a family of smooth approximate solutions to the problem. It turns out that there is an energetic cost for creating the cavity, which is captured by the notion of slic-solution but neglected by the usual entropic weak solutions. Once this cost is accounted for, the total mechanical energy of the cavitating solution is in fact larger than that of the homogeneously deformed state. We also apply the notion of slic-solutions to a one-dimensional example describing the onset of fracture, and to gas dynamics in Langrangean coordinates with Riemann data inducing vacuum in the wave fan.
The notion of singular limiting induced from continuum solutions (slic-solutions) is applied to the problem of cavitation in nonlinear elasticity, in order to re-assess an example of non-uniqueness of entropic weak solutions (with polyconvex energy) due to a forming cavity.
In this paper we give an explicit sufficient condition for the affine map $u_λ(x):=λx$ to be the global energy minimizer of a general class of elastic stored-energy functionals $I(u)=\int_Ω W(\nabla u)\,dx$ in three space dimensions, where $W$ is a polyconvex function of $3 \times 3$ matrices. The function space setting is such that cavitating (i.e., discontinuous) deformations are admissible. In the language of the calculus of variations, the condition ensures the quasiconvexity of $I(\cdot)$ at $λ\mathbf{1}$, where $\mathbf{1}$ is the $3 \times 3$ identity matrix. Our approach relies on argu…
We consider the problem of dynamic cavity formation in isotropic compressible nonlinear elastic media. For the equations of radial elasticity we construct self-similar weak solutions that describe a cavity emanating from a state of uniform deformation. For dimensions $d =2, 3$ we show that cavity formation is necessarily associated with a unique precursor shock. We also study the bifurcation diagram and do a detailed analysis of the singular asymptotics associated to cavity initiation as a function of the cavity speed of the self-similar profiles. We show that for stress free cavities the crit…
We study the following Helmholtz equation $$ (\nabla +iA(x))^{2} u+ V_{1}(x) u + V_{2}(x) u + λu = f(x) $$ in $\mathbb{R}^d$ with magnetic and electric potentials that are singular at the origin and decay at infinity. We prove the existence of a unique solution satisfying a suitable Sommerfeld radiation condition, together with some a priori estimates. We use the limiting absorption method and a multiplier technique of Morawetz type.
In 2006, Janakiraman [10] showed that if $Ω$ with mean value zero on $S^{n-1}$ satisfies the condition \[ \sup_{|ξ|=1}\int_{S^{n-1}}|Ω(θ)-Ω(θ+δξ)|dσ(θ)\leq Cnδ\int_{S^{n-1}}|Ω(θ)|dσ(θ),\quad 0<δ<\frac{1}{n},\ (\ast) \] then for the singular integral operator $T_Ω$ with homogeneous kernel, the following limiting behavior holds: \[\lim\limits_{λ\rightarrow 0}λm(\{x\in\mathbb{R}^n:|T_Ωf(x)|>λ\})= \frac{1}{n}\|Ω\|_{1}\|f\|_{1},\quad \text{for}\ f\in L^1(\mathbb{R}^n)\ \text{with}\ f\geq 0.\ (\ast\ast)\] In the present paper, we prove that if replacing the condition $(\ast)$ by more general conditi…
This work is devoted to study of a class of elliptic singular perturbed systems and their singular limit to a phase segregating system. We prove existence and uniqueness and study the asymptotic behaviour with convergence to a limiting problem as the interaction rate tends to infinity. The limiting problem is a free boundary problem such that at each point in the domain at least one of the components is zero which implies simultaneously all components can not coexist. We present a novel method, which provides an explicit solution of limiting problem for special choice of parameters. Moreover, …
We study fully nonlinear singularly perturbed parabolic equations and their limits. We show that solutions are uniformly Lipschitz continuous in space and Hölder continuous in time. For the limiting free boundary problem, we analyse the behaviour of solutions near the free boundary. We show, in particular, that, at each time level, the free boundary is a porous set and, consequently, is of Lebesgue measure zero. For rotationally invariant operators, we also derive the limiting free boundary condition.
We study the limiting behavior of the two-dimensional singular stochastic stochastic cubic nonlinear complex Ginzburg-Landau with Gibbs measure initial data. We show that in the appropriate small viscosity and small noise regimes, the limiting dynamics is given by the deterministic cubic nonlinear Schrödinger equation at Gibbs equilibrium. In order to obtain this convergence, our approach combines both heat and Schrödinger analysis, within the framework of the Fourier restriction norm method of Bourgain (1993).
We study global minimizers of the Landau-de Gennes (LdG) energy functional for nematic liquid crystals, on arbitrary three-dimensional simply connected geometries with topologically non-trivial and physically relevant Dirichlet boundary conditions. Our results are specific to the low-temperature limit. We prove (i) that (re-scaled) global LdG minimizers converge uniformly to a (minimizing) limiting harmonic map, away from the singular set of the limiting map; (ii) there exist both a point of maximal biaxiality and a nonempty Lebesgue-null set of uniaxial points near each singular point of the …
In this paper, we discuss singular Neumann boundary problem for a class of nonlinear parabolic equations in one space dimension. Our boundary problem describes motion of a planar curve sliding along the boundary with a zero contact angle, which can be viewed as a limiting model for the capillary phenomenon. We study the uniqueness and existence of solutions by using the viscosity solution theory. We also show the convergence of the solution to a traveling wave as time proceeds to infinity when the initial value is assumed to be convex.
At present, the fundamental solutions of the multidimensional elliptic equation with the several singular coefficients are known and they are expressed in terms of the Lauricella hypergeometric function of many variables. In this paper we study the Neumann problem for a multidimensional elliptic equation with several singular coefficients in the infinite domain. Using the method of the integral energy the uniqueness of solution has been proved. In the course of proving the existence of the explicit solution of the Neumann problem, a differentiation formula, some adjacent and limiting relations…
We study the phenomenon of cavitation for the displacement boundary value problem of radial, isotropic compressible elasticity for a class of stored energy functions of the form $W(F) + h(\det F)$, where $W$ grows like $||F||^n$, and $n$ is the space dimension. In this case it follows (from a result of Vodopyanov, Goldshtein and Reshetnyak) that discontinuous deformations must have infinite energy. After characterizing the rate at which this energy blows up, we introduce a modified energy functional which differs from the original by a null lagrangian, and for which cavitating energy minimizer…
Limiting absorption principle for time-harmonic elastic scattering of plane waves from diffraction gratings
Three Gorges Mathematical Research Center, College of Mathematics and Physics, China Three Gorges University（三峡大学数学与物理学院）；School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院）
We study the limiting dynamics of the Euler Alignment system with a smooth, heavy-tailed interaction kernel $ϕ$ and unidirectional velocity $\mathbf{u} = (u, 0, \ldots, 0)$. We demonstrate a striking correspondence between the entropy function $e_0 = \partial_1 u_0 + ϕ*ρ_0$ and the limiting 'concentration set', i.e., the support of the singular part of the limiting density measure. In a typical scenario, a flock experiences aggregation toward a union of $C^1$ hypersurfaces: the image of the zero set of $e_0$ under the limiting flow map. This correspondence also allows us to make statements abo…
We consider the mean-field limit of systems of particles with singular interactions of the type $-\log|x|$ or $|x|^{-s}$, with $0< s<d-2$, and with an additive noise in dimensions $d \geq 3$. We use a modulated-energy approach to prove a quantitative convergence rate to the solution of the corresponding limiting PDE. When $s>0$, the convergence is global in time, and it is the first such result valid for both conservative and gradient flows in a singular setting on $\mathbb{R}^d$. The proof relies on an adaptation of an argument of Carlen-Loss to show a decay rate of the solution to the limiti…
