Conditions For Uniqueness Of Solutions Research Articles

Existence and uniqueness of stationary solutions in $5$-dimensional minimal supergravity

We study the problem of stationary bi-axially symmetric solutions of the $5$-dimensional minimal supergravity equations. Essentially all possible solutions with nondegenerate horizons are produced, having the allowed horizon cross-sectional topologies of the sphere $S^3$, ring $S^1\times S^2$, and lens $L(p,q)$, as well as the three different types of asymptotics. The solutions are smooth apart from possible conical singularities at the fixed point sets of the axial symmetry. This analysis also includes the solutions known as solitons in which horizons are not present but are rather replaced by nontrivial topology called bubbles which are sustained by dipole fluxes. Uniqueness results are also presented which show that the solutions are completely determined by their angular momenta, electric and dipole charges, and rod structure which fixes the topology. Consequently we are able to identify the finite number of parameters that govern a solution. In addition, a generalization of these results is given where the spacetime is allowed to have orbifold singularities.

Traveling wave solutions of periodic nonlocal Fisher-KPP equations with non-compact asymmetric kernel

<p style='text-indent:20px;'>This paper deals with front propagation for nonlocal monostable equations in spatially periodic habitats. In the authors' earlier works, assuming the existence of principal eigenvalue, it is shown that there are periodic traveling wave solutions to a spatially periodic nonlocal monostable equation with symmetric and compact kernel connecting its unique positive stationary solution and the trivial solution in every direction with all propagating speeds greater than the spreading speed in that direction. In this paper, first assuming the existence of principal eigenvalue, we extend the results to the case that the kernel is asymmetric and supported on a non-compact region. In addition, without the assumption of the existence of principal eigenvalue, we explore the existence of semicontinuous traveling wave solutions.</p>

Nonlinear nonlocal reaction-diffusion problem with local reaction

<p style='text-indent:20px;'>In this paper we analyse the asymptotic behaviour of some nonlocal diffusion problems with local reaction term in general metric measure spaces. We find certain classes of nonlinear terms, including logistic type terms, for which solutions are globally defined with initial data in Lebesgue spaces. We prove solutions satisfy maximum and comparison principles and give sign conditions to ensure global asymptotic bounds for large times. We also prove that these problems possess extremal ordered equilibria and solutions, asymptotically, enter in between these equilibria. Finally we give conditions for a unique positive stationary solution that is globally asymptotically stable for nonnegative initial data. A detailed analysis is performed for logistic type nonlinearities. As the model we consider here lack of smoothing effect, important focus is payed along the whole paper on differences in the results with respect to problems with local diffusion, like the Laplacian operator.</p>

Global well-posedness of the three-dimensional viscous primitive equations with bounded delays

<p style='text-indent:20px;'>The study of delay is one of the important problems in fluid mechanics. When we attempt to control the fluid in some sense, this delay may occur by applying a force that takes into account not only the current state of the system, but also the known history. In this paper, the three-dimensional viscous primitive equations with bounded delays are considered. We prove the existence of weak and strong solutions, and obtain the uniqueness of the strong solution. We also obtain the exponential decay behavior of the weak solutions and get some higher order estimates for strong solution. Under appropriate assumptions, we prove that the time-dependent weak solutions converge exponentially to the unique stationary solution.</p>

Gradient flow structure of a multidimensional nonlinear sixth-order quantum-diffusion equation

A nonlinear parabolic equation of sixth order is analyzed. The equation arises as a reduction of a model from quantum statistical mechanics, and also as the gradient flow of a second-order information functional with respect to the $L^2$-Wasserstein metric. First, we prove global existence of weak solutions for initial conditions of finite entropy by means of the time-discrete minimizing movement scheme. Second, we calculate the linearization of the dynamics around the unique stationary solution, for which we can explicitly compute the entire spectrum. A key element in our approach is a particular relation between the entropy, the Fisher information and the second order functional that generates the gradient flow under consideration.

The fragmentation equation with size diffusion: Well posedness and long-term behaviour

The dynamics of the fragmentation equation with size diffusion is investigated when the size ranges in $(0,\infty)$ . The associated linear operator involves three terms and can be seen as a nonlocal perturbation of a Schrödinger operator. A Miyadera perturbation argument is used to prove that it is the generator of a positive, analytic semigroup on a weighted $L_1$ -space. Moreover, if the overall fragmentation rate does not vanish at infinity, then there is a unique stationary solution with given mass. Assuming further that the overall fragmentation rate diverges to infinity for large sizes implies the immediate compactness of the semigroup and that it eventually stabilizes at an exponential rate to a one-dimensional projection carrying the information of the mass of the initial value.

Asymptotics of Reinforcement Learning with Neural Networks

We prove that a single-layer neural network trained with the Q-learning algorithm converges in distribution to a random ordinary differential equation as the size of the model and the number of training steps become large. Analysis of the limit differential equation shows that it has a unique stationary solution that is the solution of the Bellman equation, thus giving the optimal control for the problem. In addition, we study the convergence of the limit differential equation to the stationary solution. As a by-product of our analysis, we obtain the limiting behavior of single-layer neural networks when trained on independent and identically distributed data with stochastic gradient descent under the widely used Xavier initialization.

Parabolic Kirchhoff equations with non-homogeneous flux boundary conditions: well-posedness, regularity and asymptotic behavior

We investigate well-posedness, regularity and asymptotic behavior of parabolic Kirchhoff equations on bounded domains of , N ⩾ 2, with non-homogeneous flux boundary conditions of Neumann or Robin type. The data in the problem satisfy (f, g, u(0)) ∈ L 2(Ω) × L 2(∂Ω) × H 1(Ω). Approximated solutions are constructed using time rescaling and a complete set in H 1(Ω) relating the equation and the boundary condition. Uniform global estimates are derived and used to prove existence, uniqueness, continuous dependence on data, a priori estimates and higher regularity for the parabolic problem. Existence and uniqueness of stationary solutions are shown, as well as a description about their role on the asymptotic behavior regarding to the evolutionary equation. Furthermore, a sufficient condition for the existence of isolated local energy minimizers is provided. They are shown to be asymptotically stable stationary solutions for the parabolic equation.

Stochastic mSQG equations with multiplicative transport noises: White noise solutions and scaling limit

We consider the modified Surface Quasi-Geostrophic (mSQG) equation on the 2D torus T2, perturbed by multiplicative transport noise. The equation admits the white noise measure on T2 as the invariant measure. We first prove the existence of white noise solutions to the stochastic equation via the method of point vortex approximation, then, under a suitable scaling limit of the noise, we show that the solutions converge weakly to the unique stationary solution of the dissipative mSQG equation driven by space–time white noise. The weak uniqueness of the latter equation is also proved by following Gubinelli and Perkowski’s approach in Gubinelli and Perkowski (2020).

Asymptotic stability of exogenous chemotaxis systems with physical boundary conditions

In this paper, we consider the exogenous chemotaxis system with physical mixed zero-flux and Dirichlet boundary conditions in one dimension. Since the Dirichlet boundary condition can not contribute necessary estimates for the cross-diffusion structure in the system, the global-in-time existence and asymptotic behavior of solutions remain open up to date. In this paper, we overcome this difficulty by employing the technique of taking anti-derivative so that the Dirichlet boundary condition can be fully used, and show that the system admits global strong solutions which exponentially stabilize to the unique stationary solution as time tends to infinity against some suitable small perturbations. To the best of our knowledge, this is the first result obtained on the global well-posedness and asymptotic behavior of solutions to the exogenous chemotaxis system with physical boundary conditions.

Effect of Asynchronous Communications on Stationary Solutions for Discrete-Time Multiagent Systems

This paper investigates the effect of asynchronous communications on stationary solutions for discrete-time multiagent systems with a fixed topology. The asynchrony in this context implies that there is no global clock to synchronize the update time in each distributed agent and also each agent updates its own state asynchronously without waiting for values received from neighboring agents. It has been noted without much attention that asynchronous stationary solutions may not be unique and disagree with a synchronous stationary value. In some applications, this discrepancy may cause serious consequences, and thus it is critical to analyze the effect of asynchrony on stationary solutions. In this paper, we develop necessary and sufficient conditions for the uniqueness of stationary solutions for discrete-time multiagent systems. In the case of nonunique stationary solutions, the normalized expected error between synchronous and asynchronous stationary solutions is also analyzed. To validate the proposed methods, we provide three example problems: 1) containment control; 2) fixed-point iteration; and 3) distributed averaging formulated in a discrete-time setting.

Impulsive stabilization and stability analysis for Gilpin–Ayala competition model involved in harmful species via LMI approach and variational methods

Firstly, a dynamic analysis for reaction–diffusion Gilpin–Ayala competition model involved in harmful species is considered under Dirichlet boundary value condition. Existence of multiple stationary solutions is verified by way of Mountain Pass lemma, and the local stability result of the null solution is obtained by employing linear approximation principle. Secondly, the authors utilize variational methods and linear matrix inequality (LMI) technique to deduce the LMI-based global exponential stability criterion on the null solution which becomes the unique stationary solution of a Markovian jumping ecosystem with delayed feedback under a reasonable boundedness assumption on population densities. Particularly, LMI criterion is involved in free weight coefficient matrix, which reduces the conservatism of the algorithm. In addition, a new impulse control stabilization criterion is also derived, in which no differentiable assumptions on time-delayed functions are proposed. Finally, three numerical examples show the effectiveness of the proposed methods. It is worth mentioning that the obtained stability criteria of null solution presented some useful hints on how to eliminate pests and bacteria.

Large Deviations and Entropy Production in Viscous Fluid Flows

We study the motion of a particle in a random time-dependent vector field defined by the 2D Navier–Stokes system with a noise. Under suitable non-degeneracy hypotheses we prove that the empirical measures of the trajectories of the pair (velocity field, particle) satisfy the LDP with a good rate function. Moreover, we show that the law of a unique stationary solution restricted to the particle component possesses a positive smooth density with respect to the Lebesgue measure in any finite time. This allows one to define a natural concept of the entropy production, and to show that its time average is a bounded function of the trajectory. The proofs are based on a new criterion for the validity of the level-3 LDP for Markov processes and an application of a general result on the image of probability measures under smooth maps to the laws associated with the motion of the particle.

The exponential behavior and stabilizability of a stochastic 3D magnetohydrodynamic – Alpha model with cylindrical multiplicative noise

We study in this article, a stochastic evolution equation for the motion of magneto-fluid filling a periodic domain of : The existence and uniqueness of stationary solution is established and the long time behavior of solution is studied. The proof relies on a Galerkin approximation. Mainly, we prove that under some conditions on the forcing terms, the variational solution converges exponentially in the mean square and almost surely exponentially to the stationary solution. We also prove a result related to the stabilization of the model.

Ergodic theory for energetically open compressible fluid flows

The ergodic hypothesis is examined for energetically open fluid systems represented by the barotropic Navier–Stokes equations with general inflow/outflow boundary conditions. We show that any globally bounded trajectory generates a stationary statistical solution, which is interpreted as a stochastic process with continuous trajectories supported by the family of weak solutions of the problem. The abstract Birkhoff–Khinchin theorem is applied to obtain convergence (in expectation and a.s.) of ergodic averages for any bounded Borel measurable function of state variables associated to any stationary solution. Finally, we show that validity of the ergodic hypothesis is determined by the behavior of entire solutions (i.e. a solution defined for any t∈R). In particular, the ergodic averages converge for any trajectory provided its ω-limit set in the trajectory space supports a unique (in law) stationary solution.

A novel approach for fractional Kawahara and modified Kawahara equations using Atangana-Baleanu derivative operator

In this manuscript, iterative laplace transform method is applied to obtain approximate solutions of the nonlinear time fractional Kawahara and modified Kawahara equations based on Atangana-Baleanu derivative operator. The noticeable features of the manuscript is to providing the existence and uniqueness conditions of solution for proposed technique and the graphical presentations of numerical solution of the concerned equations for various specific cases. The obtained approximate solutions are compared with the exact solutions to verify the applicability, efficiency and accuracy of the method.

The exponential behavior of 3D stochastic primitive equations driven by fractional noise

The long‐time behavior of flows is a very interesting and important problem in the theory of stochastic hydrodynamic equations. Inspired by the theory of dynamic system, we develop a new method to the study of the pathwise exponential behavior of 3D stochastic primitive equations driven by fractional Brownian motion. That is, we show that the weak solutions to the 3D stochastic primitive equations converge exponentially to the unique stationary solution. The result and method presented in this article can be widely applied to the study of long‐time behavior of other hydrodynamic equations with noises including Brownian motion and Lévy noise.

Analysis of a nonlinear free-boundary tumor model with angiogenesis and a connection between the nonnecrotic and necrotic phases

This paper is concerned with a nonlinear free boundary problem modeling the growth of spherically symmetric tumors with angiogenesis, set with a Robin boundary condition, in which, both nonnecrotic tumors and necrotic tumors are taken into consideration. The well-posedness and asymptotic behavior of solutions are studied. It is shown that there exist two thresholds, denoted by σ̃ and σ∗, on the surrounding nutrient concentration σ̄. If σ̄≤σ̃, then the considered problem admits no stationary solution and all evolutionary tumors will finally vanish, while if σ̄>σ̃, then it admits a unique stationary solution and all evolutionary tumors will converge to this dormant tumor; moreover, the dormant tumor is nonnecrotic if σ̃<σ̄≤σ∗ and necrotic if σ̄>σ∗. The connection and mutual transition between the nonnecrotic and necrotic phases are also given.

Distinguishing Fuzzballs from Black Holes through Their Multipolar Structure.

Within general relativity, the unique stationary solution of an isolated black hole is the Kerr spacetime, which has a peculiar multipolar structure depending only on its mass and spin. We develop a general method to extract the multipole moments of arbitrary stationary spacetimes and apply it to a large family of horizonless microstate geometries. The latter can break the axial and equatorial symmetry of the Kerr metric and have a much richer multipolar structure, which provides a portal to constrain fuzzball models phenomenologically. We find numerical evidence that all multipole moments are typically larger (in absolute value) than those of a Kerr black hole with the same mass and spin. Current measurements of the quadrupole moment of black-hole candidates could place only mild constraints on fuzzballs, while future gravitational-wave detections of extreme mass-ratio inspirals with the space mission LISA will improve these bounds by orders of magnitude.

Existence and Large Time Behavior of Entropy Solutions to One-Dimensional Unipolar Hydrodynamic Model for Semiconductor Devices with Variable Coefficient Damping

In this paper, we investigate the global existence and large time behavior of entropy solutions to one-dimensional unipolar hydrodynamic model for semiconductors in the form of Euler-Possion equations with time and spacedependent damping in a bounded interval. Firstly, we prove the existence of entropy solutions through vanishing viscosity method and compensated compactness framework. Based on the uniform estimates of density, we then prove the entropy solutions converge to the corresponding unique stationary solution exponentially with time. We generalize the existing results to the variable coefficient damping case.