Example Of Non-uniqueness Research Articles

Static Black Binaries in de Sitter Space.

We construct the first four-dimensional multiple black hole solution of general relativity with a positive cosmological constant. The solution consists of two static black holes whose gravitational attraction is balanced by the cosmic expansion. These static binaries provide the first four-dimensional example of nonuniqueness in general relativity without matter.

Deterministic particle approximation for nonlocal transport equations with nonlinear mobility

We construct a deterministic, Lagrangian many-particle approximation to a class of nonlocal transport PDEs with nonlinear mobility arising in many contexts in biology and social sciences. The approximating particle system is a nonlocal version of the follow-the-leader scheme. We rigorously prove that a suitable discrete piece-wise density reconstructed from the particle scheme converges strongly in Lloc1 towards the unique entropy solution to the target PDE as the number of particles tends to infinity. The proof is based on uniform BV estimates on the approximating sequence and on the verification of an approximated version of the entropy condition for large number of particles. As part of the proof, we also prove uniqueness of entropy solutions. We further provide a specific example of non-uniqueness of weak solutions and discuss the interplay of the entropy condition with the steady states. Finally, we produce numerical simulations supporting the need of a concept of entropy solution in order to get a well-posed semigroup in the continuum limit, and showing the behaviour of solutions for large times.

Time-Dependent Focusing Mean-Field Games: The Sub-critical Case

We consider time-dependent viscous Mean-Field Games systems in the case of local, decreasing and unbounded coupling. These systems arise in mean-field game theory, and describe Nash equilibria of games with a large number of agents aiming at aggregation. We prove the existence of weak solutions that are minimisers of an associated non-convex functional, by rephrasing the problem in a convex framework. Under additional assumptions involving the growth at infinity of the coupling, the Hamiltonian, and the space dimension, we show that such minimisers are indeed classical solutions by a blow-up argument and additional Sobolev regularity for the Fokker-Planck equation. We exhibit an example of non-uniqueness of solutions. Finally, by means of a contraction principle, we observe that classical solutions exist just by local regularity of the coupling if the time horizon is short.

An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions

We consider the weighted Radon transforms RW along hyperplanes in , with strictly positive weights . We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions. In addition, the related weight W is infinitely smooth almost everywhere and is bounded. Our construction is based on the famous example of non-uniqueness of Boman (1993 J. d’Anal. Math. 61 395–401) for the weighted Radon transforms in and on a recent result of Goncharov and Novikov (2016 Eurasian J. Math. Comput. Appl. 4 23–32).

An Example of Non-uniqueness for Radon Transforms with Continuous Positive Rotation Invariant Weights

We consider weighted Radon transforms $$R_W$$ along hyperplanes in $$\mathbb {R}^3$$ with strictly positive weights W. We construct an example of such a transform with non-trivial kernel $$\mathrm {Ker}R_W$$ in the space of infinitely smooth compactly supported functions and with continuous weight. Moreover, in this example the weight W is rotation invariant. In particular, by this result we continue studies of Quinto (J Math Anal Appl 91(2): 510–522, 1983), Markoe and Quinto (SIAM J Math Anal 16(5), 1114–1119, 1985), Boman (J Anal Math 61(1), 395–401, 1993) and Goncharov and Novikov (An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions. arXiv:1709.04194v2 , 2017). We also extend our example to the case of weighted Radon transforms along two-dimensional planes in $$\mathbb {R}^d, \, d\ge 3$$ .

On the uniqueness of solutions to stationary convection–diffusion equations with generalized divergence-free drift

Let A be a skew-symmetric matrix in , — a bounded Lipschitz domain in , . The Dirichlet problem , , has at least one solution obtained by approximating A and passing to the limit. In 2004 V. V. Zhikov constructed an example of nonuniqueness. In the same paper he proved the uniqueness of solutions if the norms of A are o(p) as p goes to infinity. We prove the uniqueness of solutions if for some , which generalizes Zhikov’s theorem.

Strongly Hermitian Einstein–Maxwell solutions on ruled surfaces

This paper produces explicit strongly Hermitian Einstein–Maxwell solutions on the smooth compact 4-manifolds that are $$S^2$$ -bundles over compact Riemann surfaces of any genus. This generalizes the existence results by LeBrun (J Geom Phys 91:163–171, 2015, The Einstein–Maxwell equations and conformally Kaehler geometry. arXiv:1504.06669 , 2016). Moreover, by calculating the (normalized) Einstein–Hilbert functional of our examples, we generalize Theorem E of LeBrun (The Einstein–Maxwell equations and conformally Kaehler geometry. arXiv:1504.06669 , 2016), which speaks to the abundance of Hermitian Einstein–Maxwell solutions on such manifolds. As a bonus, we exhibit certain pairs of strongly Hermitian Einstein–Maxwell solutions, first found in LeBrun (The Einstein–Maxwell equations and conformally Kaehler geometry. arXiv:1504.06669 , 2016), on the first Hirzebruch surface in a form which clearly shows that they are conformal to a common Kahler metric. In particular, this yields a non-trivial example of non-uniqueness of positive constant scalar curvature metrics in a given conformal class.

Liquid toroidal drop in compressional Stokes flow

The deformation of an immiscible toroidal drop embedded in axisymmetric compressional Stokes flow is analysed via the boundary integral formulation in the case of equal viscosity. Numerical simulations are performed for the drop having initially the shape of a torus with circular cross-section. The quasi-stationary dynamic simulations reveal that, when the viscous forces, proportional to the intensity of the flow, are relatively weak compared with the surface tension (the ratio of these forces is characterized by the capillary number,$Ca$), three different scenarios of drop evolution are possible: indefinite expansion of the liquid torus, contraction to the centre and a stationary toroidal shape. When the intensity of the flow is low, the stationary shapes are shown to be close to circular tori. Once the outer flow strengthens, the cross-section of the stationary torus assumes first an elliptic and then an egg-like shape. For the capillary number greater than a critical value,$Ca_{cr}$, toroidal stationary shapes were not found. Remarkably,$Ca_{cr}$is close to the critical capillary number found previously for a simply connected drop flattened in compressional flow. Thus, a new example of non-uniqueness of stationary drop shape in viscous flow is obtained. Approximate stationary solutions in the form of tori with circular and elliptic cross-sections are obtained by minimizing the normal velocity over the drop interface. They are shown to be in good agreement with the stationary shapes from quasi-dynamic simulations for the corresponding intervals of the capillary number.

The problem of dynamic cavitation in nonlinear elasticity

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.

Singular Limiting Induced from Continuum Solutions and the Problem of Dynamic Cavitation

In the works of Pericak-Spector and Spector (Arch Rational Mech Anal. 101:293–317, 1988, Proc. Royal Soc. Edinburgh Sect A 127:837–857, 1997) 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.

An example of non-uniqueness of a simple partial fraction of the best uniform approximation

For arbitrary natural n ≥ 2 we construct an example of a real continuous function, for which there exists more than one simple partial fraction of order ≤ n of the best uniform approximation on a segment of the real axis. We prove that even the Chebyshev alternance consisting of n+1 points does not guarantee the uniqueness of the best approximation fraction. The obtained results are generalizations of known non-uniqueness examples constructed for n = 2, 3 in the case of simple partial fractions of an arbitrary order n.

Examples related to best approximation by simple partial fractions

UDC 517.538 It is shown that an alternance consisting of less than 2n points does not guarantee best approximation by simple partial fractions of degree n. We obtain a sufficient condition for the uniqueness of a simple partial fraction of degree 3 of best approximation and propose a numerical method for verifying this condition. We also construct an example of nonuniqueness of a simple partial fraction of degree 3 of best approximation, as well as some other theoretical examples. Bibliography :4 titles.

Unsteady compressible flow in ducts with varying cross-section: Comparison between the nonconservative Euler system and the axisymmetric flow model

Most of nonconservative hyperbolic systems corresponds to a reduction of an initial three-dimensional problem deriving from a homogenization procedure. Unfortunately, the reduced model gives rise to two new difficulties: the resonance problem corresponding to a splitting or a merging of the genuinely nonlinear waves and the non uniqueness of the Riemann problem solution. The question arises to check whether the two problems correspond and provide similar solutions, at least numerically. In this paper, we propose a comparison between the one-dimensional nonconservative Euler equations modelling the duct with variable cross-sectional area with its original three-dimensional conservative Euler system. Based on the classification of the Riemann problems proposed in [13], we compare the numerical results of the two models for a large series of representative configurations. We also propose a new example of non uniqueness for the Riemann problem involving the resonance phenomena.

A counterexample to uniqueness of generalized characteristics in Hamilton–Jacobi theory

The notion of generalized characteristics plays a pivotal role in the study of propagation of singularities for Hamilton–Jacobi equations. This note gives an example of nonuniqueness of forward generalized characteristics emanating from a given point.

Nonuniqueness of the heat flow of director fields

We give a new example of nonuniqueness of weak solutions of the initial-boundary-value problem for the heat flow of director fields in an infinitely long cylinder in $\mathbb R^{3}$. The example confirms the connection between nonuniqueness of axially symmetric solutions for the harmonic map heat flow and the occurrence of point singularities in the solutions. The result is compared with earlier nonuniqueness results. Traveling wave solutions are used as barrier functions.

ASYMPTOTIC ANALYSIS OF AN ISOTHERMAL GAS FLOW THROUGH A LONG OR THIN PIPE

In this paper we study the compressible stationary isothermal flow through a thin (or long) straight pipe. Starting from the compressible Stokes system, via rigorous asymptotic analysis, as the pipe's thickness tends to zero, we obtain the 1D model describing the effective behavior of the flow. The uniqueness of the solution for such model is proved, if the first component of the function g, appearing on the right-hand side of the system, does not change its sign. If it does change the sign, an example of non-uniqueness of the solution is given.

Anisotropic classes of uniqueness of the solution of the Dirichlet problem for quasi-elliptic equations

We select a class of uniqueness of the solutions of the quasi-elliptic equation with the Dirichlet condition on the boundary of an unbounded domain and show that for domains with irregular behaviour of the boundary this class can be wider than that established in [10] for second-order elliptic equations. For the Laplace equation we construct an example of non-uniqueness of solution of the Dirichlet problem that shows that the class of uniqueness found in this paper cannot be essentially extended.

Viscous two‐fluid flows in perturbed unbounded domains

AbstractTwo stationary plane free boundary value problems for the Navier‐Stokes equations are studied. The first problem models the viscous two‐fluid flow down a perturbed or slightly distorted inclined plane. The second one describes the viscous two‐fluid flow in a perturbed or slightly distorted channel. For sufficiently small data and under certain conditions on parameters the solvability and uniqueness results are proved for both problems. The asymptotic behaviour of the solutions is investigated. For the second problem an example of nonuniqueness is constructed. Computational results of flow problems that are very close to the above problems are presented. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An example of nonuniqueness of the Cauchy problem

Using Mehler kernel, we give an example of nontrivial solution of the homogeneous Cauchy problem of the Hermite heat equation, which is, for each $t$, bounded in the space variables.

Supersymmetric black rings and three-charge supertubes

We present supergravity solutions for $1/8$-supersymmetric black supertubes with three charges and three dipoles. Their reduction to five dimensions yields supersymmetric black rings with regular horizons and two independent angular momenta. The general solution contains seven independent parameters and provides the first example of nonuniqueness of supersymmetric black holes. In ten dimensions, the solutions can be realized as D1-D5-P black supertubes. We also present a worldvolume construction of a supertube that exhibits three dipoles explicitly. This description allows an arbitrary cross section but captures only one of the angular momenta.