Ball Br Research Articles

Global well-posedness for two-phase fluid motion in the Oberbeck–Boussinesq approximation

This paper focuses on the global well-posedness of the Oberbeck–Boussinesq approximation for the unsteady motion of a drop in another bounded fluid separated by a closed interface with surface tension. We assume that the initial state of the drop is close to a ball BR with the same volume as the drop, and that the boundary of the drop is a small perturbation of the boundary of BR. To begin, we introduce the Hanzawa transformation with an added barycenter point to obtain the linearized Oberbeck–Boussinesq approximation in a fixed domain. From there, we establish time-weighted estimates of solutions for the shifted equation using maximal Lp–Lq regularities for the two-phase fluid motion of the linearized system, as obtained by Hao and Zhang [J. Differ. Equations 322, 101–134 (2022)]. Using time decay estimates of the semigroup, we then obtain decay time-weighted estimates of solutions for the linearized problem. Additionally, we prove that these estimates are less than the sum of the initial value and its own square and cube by estimating the corresponding non-linear terms. Finally, the existence and uniqueness of solutions in the finite time interval (0, T) was proven by Hao and Zhang [Commun. Pure Appl. Anal. 22(7), 2099–2131 (2023)]. After that, we demonstrate that the solutions can be extended beyond T by analyzing the properties of the roots of algebraic equations.

On the colimits of certain Sobolev spaces

The aim of this paper is to give an identification of the Sobolev spaces Wk,p(Rn) intrinsically in terms of the Sobolev spaces W0k,p(Br(0)) of functions defined in the ball Br(0) of center zero and radius r>0. To provide such identification we make use of the category theoretic concept of colimit. We apply these techniques to certain examples of nonlinear differential equations.

Uniqueness, symmetry and convergence of positive ground state solutions of the Choquard type equation on a ball

This paper is concerned with the qualitative properties of the positive ground state solutions to the nonlocal Choquard type equation on a ball BR. First, we prove the radial symmetry of all the positive ground state solutions by using Talenti's inequality. Next we develop the Newton's Theorem and then resort to the contraction mapping principle to establish the uniqueness of the positive ground state solution. Finally, by constructing cut-off functions and applying energy comparison method, we show the convergence of the unique positive ground state solutions as the radius R→∞. Our results extend and complement the existing ones in the literature.

Reconstruction from the Fourier transform on the ball via prolate spheroidal wave functions

We give new formulas for finding a compactly supported function v on Rd, d≥1, from its Fourier transform Fv given within the ball Br. For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions (PSWF's). In multidimensions, well-known results of the Radon transform theory reduce the problem to the one-dimensional case. Related results on stability and convergence rates are also given.

Radial singular solutions for the N-Laplace equation with exponential nonlinearities

In this paper, we consider radial distributional solutions of the quasilinear equation −ΔNu=f(u) in the punctured open ball BR\\{0}⊂RN, N≥2. We obtain sharp conditions on the nonlinearity f for extending such solutions to the whole domain BR by preserving the regularity. For a certain class of nonlinearity f we obtain the existence of singular solutions and deduce upper and lower estimates on the growth rate near the singularity.

A characteristic of local existence for nonlinear fractional heat equations in Lebesgue spaces

In this paper, we consider the fractional heat equation ut=△α/2u+f(u) with Dirichlet conditions on the ball BR⊂Rd, where △α/2 is the fractional Laplacian, f:[0,∞)→[0,∞) is continuous and non-decreasing. We present the characterisations of f to ensure the equation has a local solution in Lq(BR) provided that the non-negative initial data u0∈Lq(BR). For q>1 and 1<α<2, we show that the equation has a local solution in Lq(BR) if and only if lims→∞sups−(1+αq/d)f(s)=∞; and for q=1 and 1<α<2 if and only if ∫1∞s−(1+α/d)F(s)ds<∞, where F(s)=sup1≤t≤sf(t)/t. When lims→0f(s)/s<∞, the same characterisations holds for the fractional heat equation on the whole space Rd.

Accuracy analysis of the boundary integral method for steady Navier-Stokes equations around a rotating obstacle

This paper deals with the boundary integral method to study the Navier-Stokes equations around a rotating obstacle. The detail of this method is that the exterior domain is truncated into a bounded domain and a new exterior domain by introducing some open ball BR, and the nonlinear problem in the bounded domain and the linearized problem in the new exterior domain are considered and the approximation coupled problem is obtained. We show that the error between the solution u of Navier-Stokes equations around a rotating obstacle and the solution ue of the approximation coupled problem is O(R−1/4) in the H1-seminorm when |ω| does not exceed some constant.

Minimizers of the Rudin-Osher-Fatemi Functionals in Bounded Domains

Abstract We give some explicit formulas for the minimizers of the Rudin-Osher-Fatemi functionals and on BV(Ω), where Ω ⊂ ℝ2 is a bounded domain containing the ball BR(0) and f = χR(x) is the characteristic function of BR(0).

Wavelet optimal estimations for a density with some additive noises

Using wavelet methods, Fan and Koo study optimal estimations for a density with some additive noises over a Besov ball Br,qs(L) (r,q⩾1) and over L2 risk (Fan and Koo, 2002 [13]). The L∞ risk estimations are investigated by Lounici and Nickl (2011) [19]. This paper deals with optimal estimations over Lp (1⩽p⩽∞) risk for moderately ill-posed noises. A lower bound of Lp risk is firstly provided, which generalizes Fan–Koo and Lounici–Nickl's theorems; then we define a linear and non-linear wavelet estimators, motivated by Fan–Koo and Pensky–Vidakovic's work. The linear one is rate optimal for r⩾p, and the non-linear estimator attains suboptimal (optimal up to a logarithmic factor). These results can be considered as an extension of some theorems of Donoho et al. (1996) [10]. In addition, our non-linear wavelet estimator is adaptive to the indices s, r, q and L.

Optimal [formula omitted]-locating–dominating codes in the infinite king grid

Assume that G=(V,E) is an undirected graph with vertex set V and edge set E. The ball Br(v) denotes the vertices within graphical distance r from v. A subset C⊆V is called an (r,≤l)-locating–dominating code of type B if the sets Ir(F)=⋃v∈F(Br(v)∩C) are distinct for all subsets F⊆V∖C with at most l vertices. We give examples of optimal (r,≤3)-locating–dominating codes of type B in the infinite king grid for all r∈N+ and prove optimality. The infinite king grid is the graph with vertex set Z2 and edge set {{(x1,y1),(x2,y2)}∣|x1−x2|≤1,|y1−y2|≤1}.

Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space

We find lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form T(u)=−∫RdK(x,y)(u(y)−u(x))dy. Here we consider a kernel K(x,y)=ψ(y−a(x))+ψ(x−a(y)) where ψ is a bounded, nonnegative function supported in the unit ball and a means a diffeomorphism on Rd. A simple example being a linear function a(x)=Ax. The upper and lower bounds that we obtain are given in terms of the Jacobian of a and the integral of ψ. Indeed, in the linear case a(x)=Ax we obtain an explicit expression for the first eigenvalue in the whole Rd and it is positive when the determinant of the matrix A is different from one. As an application of our results, we observe that, when the first eigenvalue is positive, there is an exponential decay for the solutions to the associated evolution problem. As a tool to obtain the result, we also study the behavior of the principal eigenvalue of the nonlocal Dirichlet problem in the ball BR and prove that it converges to the first eigenvalue in the whole space as R→∞.

Global Non-Small Data Existence of Spherically Symmetric Solutions to Nonlinear Viscoelasticity in a Ball

We consider some initial-boundary value problems for non-linear equations of the three dimensional viscoelasticity. We examine the Dirichlet and the Neumann boundary conditions. We assume that the stress tensor is a nonlinear tensor valued function depending on the strain tensor fulfilling the rules of the continuum mechanics. We consider the initial-boundary value problems in a ball BR with radius R . Since, we are interested in proving global existence the spherically symmetric solutions are considered. Therefore we have to examine the spherically symmetric viscoelasticity system in spherical coordinates. Applying the energy method implies estimates in weighted anisotropic Sobolev spaces, where the weight is a power function of radius. Hence the origin of coordinates becomes a singular point. First the existence of weak solutions is proved. Next having appropriate estimates the weak solutions appear bounded and continuous. We have to emphasize that non-small data problem is considered.

Finite Morse index solutions of an elliptic equation with supercritical exponent

We study the behavior of finite Morse index solutions of the equation−Δu=|x|α|u|p−1uin Ω⊂RN, where p>1, α>−2, and Ω is a bounded or unbounded domain. We show that there is a critical power p=p¯(α) larger than the usual critical exponent N+2N−2 such that this equation with Ω=RN has no nontrivial stable solution for 1<p<p¯(α) but it admits a family of stable positive solutions when p⩾p¯(α). For a positive solution u with finite Morse index, we classify the singularity of u at the origin when Ω is a punctured ball BR(0)\\{0}, and we classify its behavior at infinity when Ω=RN∖BR(0). We show that the behavior depends crucially on whether p is below or above the critical power p¯(α). We also demonstrate how a duality method can be used to obtain sharper results.

Small Surfaces of Willmore Type in Riemannian Manifolds

In this paper we investigate the properties of small surfaces of Willmore type in Riemannian manifolds. By small surfaces we mean topological spheres contained in a geodesic ball of small enough radius. In particular, we show that if there exist such surfaces with positive mean curvature in the geodesic ball Br(p) for arbitrarily small radius r around a point p in the Riemannian manifold, then the scalar curvature must have a critical point at p.

Lorenz-like flows: exponential decay of correlations for the Poincaré map, logarithm law, quantitative recurrence

AbstractIn this paper we prove that the Poincaré map associated to a Lorenz-like flow has exponential decay of correlations with respect to Lipschitz observables. This implies that the hitting time associated to the flow satisfies a logarithm law. The hitting time τr(x,x0) is the time needed for the orbit of a point x to enter a ball Br(x0) centered at x0, with small radius r, for the first time. As the radius of the ball decreases to 0 its asymptotic behavior is a power law whose exponent is related to the local dimension of the physical measure at x0: for each x0 such that the local dimension dμ (x0) exists, holds for μ almost each x. In a similar way, it is possible to consider a quantitative recurrence indicator quantifying the speed of an orbit to come back to its starting point. Similar results hold for this recurrence indicator.

Transformations of measures by Gauss maps

The goal of this paper is to introduce a new class of transformations of measures on R having the form (heuristically) T = φ · ∇φ/|∇φ|, where φ is some function; our construction exhibits an interesting link between two actively developing areas: optimal transportation and geometrical glows (see [1], [2], and [3] concerning these directions). Let A ⊂ R, d ≥ 2, be a compact convex set and let μ = %0 dx be a probability measure on A equivalent to the restriction of Lebesgue measure. Let ν = %1 dx be a probability measure on the ball Br := {x : |x| ≤ r} equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping T such that ν = μ ◦ T−1 and T = φ · n, where φ : A → [0, r] is a continuous potential with convex sub-level sets and n is the Gauss map of the corresponding level sets of φ. Moreover, the mapping T is invertible and essentially unique. Our proof employs the optimal transportation techniques. In the case of smooth φ the level sets of φ are governed by the Gauss curvature flow ẋ(s) = −sd−1 %1(sn) %0(x) K(x) · n(x), where K is the Gauss curvature. Throughout we assume that d ≥ 2 and denote by H the n-dimensional Hausdorff measure. Let IntA denote the interior of a set A. Given a compact convex set V ⊂ R with boundary M = ∂V , for an arbitrary point x ∈M , let us set

Real Bargmann spaces, Fischer decompositions, and sets of uniqueness for polyharmonic functions

In this paper a positive answer is given to the following question of W.K. Hayman: if a polyharmonic entire function of order k vanishes on k distinct ellipsoids in the euclidean space R n then it vanishes everywhere. Moreover a characterization of ellipsoids is given in terms of an extension property of solutions of entire data functions for the Dirichlet problem answering a question of D. Khavinson and H.S. Shapiro. These results are consequences from a more general result in the context of direct sum decom- positions (Fischer decompositions) of polynomials or functions in the algebra A(BR) of all real-analytic functions defined on the ball BR of radius R and center 0 whose Taylor series of homogeneous polynomials converges compactly in BR. The main result states that for a given elliptic polynomial P of degree 2k and suciently

Precise Asymptotic Properties of Solutions to Two-Parameter Elliptic Eigenvalue Problems in a Ball

Abstract We consider the simple pendulum type equation with two-parameters μ, λ &gt; 0 in a ball BR. For a given μ &gt; 0, a solution triple (μ, λ(μ), uμ) ∈ R+ 2 × C2(B̅R) is obtained by a variational method, and uμ develops a boundary layer as μ → ∞. We establish the two-term asymptotic formulas for ∥uμ∥∞, u′μ(R)2 and the variational eigencurve λ(μ) as μ → ∞. It is shown that the coefficients of the second terms of ∥uμ∥∞, u′μ(R)2 and λ(μ) depend on 1/R, the mean curvature of ∂BR. Since the second term of the maximum norm of the solution for the corresponding one parameter problems does not depend on the mean curvature of ∂BR, the results obtained here seem to be new.

On the static and dynamic equilibrium of concentrated loads in linear acoustics and linear elasticity

The displacement field in an unbounded linear elastic fluid subjected to a time-dependent point force is obtained by using integral transform techniques. Differentiation of the displacement field yields the pressure field. It is shown that the pressure on the surface of a spherical ball B r of radius r centered at the point where the force is applied is statically equivalent in the limit as r→0 to only one-third of the force. The remaining two-thirds are carried by the inertia terms. It is also shown, by an independent reasoning, that a point force cannot be carried in static equilibrium by a linear elastic fluid. The displacement field corresponding to an unbounded isotropic linear-elastic solid subjected to a time-dependent point force (the Stokes solution) is also obtained by using integral transform techniques. As is well-known, the tractions of the Stokes solution on the surface of a spherical ball B r are statically equivalent in the limit as r→0 to the force itself; consequently, the inertia terms do not contribute to the dynamic equilibrium of B r. The contrast between the response of a fluid and that of an isotropic solid under the action of a point force is discussed.

Note on propagation of classical waves: II

1. Main results and notations. We consider Cauchy's problem for the classical wave equation in :with initial conditions u(0, x) = uo(x) and ∂tu(0, x) = u1 (x). In (1·1) δ and m2 stand for the Laplacian in and a constant respectively. Note that a second order hyper-bolic differential equation with real constants can be reduced to (1·1) ([1], p. 183). Let uj (j = 0,1) be C∞ funotions on whose support is contained in the closed ball BR = {xεl; |x| ≥R} for some R &gt; 0. In this note we shall show that the solution u(t, x) of (1·1) possesses the following properties.