Modeling of the unsteady flow through a channel with an artificial outflow condition by the Navier–Stokes variational inequality

We prove the global in time existence of a weak solution to the variational inequality of the Navier–Stokes type, simulating the unsteady flow of a viscous fluid through the channel, with the so‐called “do nothing” boundary condition on the outflow. The condition that the solution lies in a certain given, however arbitrarily large, convex set and the use of the variational inequality enables us to derive an energy‐type estimate of the solution. We also discuss the use of a series of other possible outflow “do nothing” boundary conditions.     

On the Palais–Smale condition

We present a new sufficient assumption weaker than the classical Ambrosetti–Rabinowitz condition which guarantees the boundedness of (PS) sequences. Moreover, we relax the standard subcritical polynomial growth condition ensuring the compactness of a bounded (PS) sequences. We also revise the Costa–Magalhaes condition [8] to obtain Cerami condition. As a consequence, some existence results derived by minimax methods were proved. Finally, we establish the existence of positive solution under the subcritical polynomial growth condition, while the strong superlinear condition is only required along an unbounded sequence. In other words, a certain degraded oscillation is allowed.     

Finite-time singularity formation at a magnetic neutral line in Hall magnetohydrodynamics

The formation of a current sheet in a weakly collisional plasma can be modelled as a finite-time singularity solution of magnetohydrodynamic equations. We use an exact self-similar solution to confirm and generalise a previous finding that, in sharp contrast to two-dimensional solutions in standard MHD, a finite-time collapse to a current sheet can occur in Hall MHD. We derive a criterion for the finite-time singularity in terms of initial conditions, and we use an intermediate asymptotic solution for the evolution of an axial magnetic field to obtain a general expression for the singularity formation time. We illustrate the analytical results by numerical solutions.     

Optimal control in nonlinear infinite-dimensional systems with nondifferentiability of two types

We consider the optimal control problem for systems described by nonlinear equations of elliptic type. If the nonlinear term in the equation is smooth and the nonlinearity increases at a comparatively low rate of growth, then necessary conditions for optimality can be obtained by well-known methods. For small values of the nonlinearity exponent in the smooth case, we propose to approximate the state operator by a certain differentiable operator. We show that the solution of the approximate problem obtained by standard methods ensures that the optimality criterion for the initial problem is close to its minimal value. For sufficiently large values of the nonlinearity exponent, the dependence of the state function on the control is nondifferentiable even under smoothness conditions for the operator. But this dependence becomes differentiable in a certain extended sense, which is sufficient for obtaining necessary conditions for optimality. Finally, if there is no smoothness and no restrictions are imposed on the nonlinearity exponent of the equation, then a smooth approximation of the state operator is possible. Next, we obtain necessary conditions for optimality of the approximate problem using the notion of extended differentiability of the solution of the equation approximated with respect to the control, and then we show that the optimal control of the approximated extremum problem minimizes the original functional with arbitrary accuracy.     

Optimization of discrete broadcast under uncertainty using conditional value-at-risk

The paper considers the problem of scheduling packets in wireless broadcast systems under uncertainty with the conditional value-at-risk (CVaR) constraints. In such systems, a server periodically transmits a stream of packets over a broadcast channel. The client that needs to access the data, tunes in to the channel and waits for the next packet. This allows one to serve a large number of clients in an efficient way and also keep clients’ location secret. We formulate and solve two alternative stochastic optimization problems that minimize the transmission time subject to CVaR constraints. Our results indicate that it is possible to derive an analytical solution to the problems in certain cases of practical interest. We also propose a methodology to obtain numerical solutions for the general case.     

Limited self-control and long-run growth

This paper integrates imperfect self-control into the standard model of endogenous growth. In their long-run savings decisions individuals take into account a cost of self-control, which depends on the consumption temptations of their impatient short-run self. I obtain a closed-form solution for consumption and show that within a certain range of self-control an investment subsidy can be useful in order to reduce consumption and to increase investment, growth, and welfare of the long-run self. A consumption tax, perhaps surprisingly, is found to be counterproductive. It induces individuals with limited self-control to consume even more.     

Obstacle-type problems for minimal surfaces

We study certain obstacle-type problems involving standard and nonlocal minimal surfaces. We obtain optimal regularity of the solution and a characterization of the free boundary.     

Remarks for the axisymmetric Navier-Stokes equations

We are concerned with 3-D incompressible Navier-Stokes equations when the initial data and the domain are cylindrically symmetric. We show that there exists a solution in a weighted space and certain weighted norms of vorticity of the solution remain finite if they are finite initially. Consequently, we can estimate the growth rate of the solution both spatially and temporally.     

Uniqueness result for non-negative solutions of semi-linear inequalities on Riemannian manifolds

We consider certain semi-linear partial differential inequalities on complete connected Riemannian manifolds and provide a simple condition in terms of volume growth for the uniqueness of a non-negative solution. We also show the sharpness of this condition.     

Generalized solution of a nonlinear wave equation with nonlinear boundary conditions of the first,second, and third kinds

For the wave equation, we construct a generalized solution that has maximum possible analytic properties. We establish a pointwise estimate for the growth of the solution relative to the initial data, prove that certain linear combinations of the first partial derivatives of the solution have integrable derivatives along the characteristics, justify an integration-by-parts formula, and obtain the adjoint of the original differential operator.     

Integrability of Solutions to Mixed Stochastic Differential Equations

We prove that the standard conditions for the unique solvability of a mixed stochastic differential equation guarantee that its solution possesses finite moments. We also give conditions supplying the existence of exponential moments. For a special equation whose coefficients do not satisfy the linear growth condition, we prove the integrability of its solution.     

3‐D spatio–temporal structures of biofilms in a water channel

We develop a numerical predictive tool for multiphase fluid mixtures consisting of biofilms grown in a viscous fluid matrix by implementing a second‐order finite difference discretization of the multiphase biofilm model developed recently on a general purpose graphic processing unit. With this numerical tool, we study a 3‐D biomass–flow interaction resulting in biomass growth, structure formation, deformation, and detachment phenomena in biofilms grown in a water channel in quiescent state and subject to a shear flow condition, respectively. The numerical investigation is limited in the viscous regime of the biofilm–solvent mixture. In quiescent flows, the model predicts growth patterns consistent with experimental findings for single or multiple adjacent biofilm colonies, the so‐called mushroom shape growth pattern. The simulated biomass growth both in density and thickness matches very well with the experimentally grown biofilm in a water channel. When shear is imposed at a boundary, our numerical studies reproduce wavy patterns, pinching, and streaming phenomena observed in biofilms grown in a water channel. Copyright © 2014 John Wiley & Sons, Ltd.     

Global solutions of semilinear evolution equations satisfying an energy inequality

We prove the global existence (in time) for any solution of an abstract semilinear evolution equation in Hilbert space provided the solution satisfies an energy inequality and the nonlinearity does not exceed a certain growth rate. When applied to semilinear parabolic initial-boundary-value problems the result admits also the limiting growth rates which were given by Sobolevskii and Friedman, but which where not permitted in their theorem. The Navier-Stokes system in two dimensions is a special case of our general result. The method is based on the theories of semigroups and fractional powers of regularly accretive linear operators and on a nonlinear integral inequality which gives the crucial a-priori estimate for global existence.     

On a price formation free boundary model by Lasry and Lions: The Neumann problem

We discuss local and global existence and uniqueness for the price formation free boundary model with homogeneous Neumann boundary conditions introduced by Lasry and Lions in 2007. The results are based on a transformation of the problem to the heat equation with nonstandard boundary conditions. The free boundary becomes the zero level set of the solution of the heat equation. The transformation allows us to construct an explicit solution and discuss the behavior of the free boundary. Global existence can be verified under certain conditions on the free boundary and examples of non-existence are given.     

Blowup in Korteweg-de Vries-type systems

We consider a pair Korteweg-de Vries system of the Boussinesq type and its symmetric analogue. Such systems, which describe the behavior of a liquid in a channel, are shown to have no solutions defined globally in time under certain conditions. Using the method of nonlinear capacity, we obtain sufficient conditions for the solution blowup and estimate the blowup time for both these systems and for a generalized multicomponent Korteweg-de Vries-type system.     

Sparse solutions to random standard quadratic optimization problems

The standard quadratic optimization problem (StQP) refers to the problem of minimizing a quadratic form over the standard simplex. Such a problem arises from numerous applications and is known to be NP-hard. In this paper we focus on a special scenario of the StQP where all the elements of the data matrix Q are independently identically distributed and follow a certain distribution such as uniform or exponential distribution. We show that the probability that such a random StQP has a global optimal solution with k nonzero elements decays exponentially in k. Numerical evaluation of our theoretical finding is discussed as well.     

Cauchy problem on non-globally hyperbolic space-times

We consider solutions of the Cauchy problem for hyperbolic equations on non-globally hyperbolic space-times containing closed timelike curves (time machines). We prove that for the wave equation on such space-times, there exists a solution of the Cauchy problem that is discontinuous and in some sense unique for arbitrary initial conditions given on a hypersurface at a time preceding the formation of closed timelike curves. If the hypersurface of initial conditions intersects the region containing closed timelike curves, then the solution of the Cauchy problem exists only for initial conditions satisfying a certain self-consistency requirement. To Vasilii Sergeevich Vladimirov with best wishes on his 85th birthday __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 334–344, December, 2008.     

Contractivity of domain decomposition splitting methods for nonlinear parabolic problems

This work deals with the efficient numerical solution of nonlinear parabolic problems posed on a two-dimensional domain Ω. We consider a suitable decomposition of domain Ω and we construct a subordinate smooth partition of unity that we use to rewrite the original equation. Then, the combination of standard spatial discretizations with certain splitting time integrators gives rise to unconditionally contractive schemes. The efficiency of the resulting algorithms stems from the fact that the calculations required at each internal stage can be performed in parallel.     

Multiple-scale Homogenization for Weakly Nonlinear Conservation Laws with Rapid Spatial Fluctuations

We consider hyperbolic conservation laws with rapid periodic spatial fluctuations and study initial value problems that correspond to small perturbations about a steady state. Weakly nonlinear solutions are computed asymptotically using multiple spatial and temporal scales to capture the homogenized solution as well as its long-term behavior. We show that the linear problem may be destabilized through interactions between two solution modes and the periodic structure. We also show that a discontinuity, either in the initial data or due to shock formation, introduces rapid spatial and temporal fluctuations to leading order in its zone of influence. The evolution equations we derive for the homogenized leading-order solution are more general than their counterparts for conservation laws having no rapid spatial variations. In particular, these equations may be diffusive for certain general flux vectors. Selected examples are solved numerically to substantiate the asymptotic results.     

Low rank methods for a class of generalized Lyapunov equations and related issues

In this paper, we study possible low rank solution methods for generalized Lyapunov equations arising in bilinear and stochastic control. We show that under certain assumptions one can expect a strong singular value decay in the solution matrix allowing for low rank approximations. Since the theoretical tools strongly make use of a connection to the standard linear Lyapunov equation, we can even extend the result to the $d$ -dimensional case described by a tensorized linear system of equations. We further provide some reasonable extensions of some of the most frequently used linear low rank solution techniques such as the alternating directions implicit (ADI) iteration and the Krylov-Plus-Inverted-Krylov (K-PIK) method. By means of some standard numerical examples used in the area of bilinear model order reduction, we will show the efficiency of the new methods.