On minimax optimization problems

We give a short proof that in a convex minimax optimization problem ink dimensions there exist a subset ofk + 1 functions such that a solution to the minimax problem with thosek + 1 functions is a solution to the minimax problem with all functions. We show that convexity is necessary, and prove a similar theorem for stationary points when the functions are not necessarily convex but the gradient exists for each function.     

Symmetry Properties of Positive Solutions of Parabolic Equations on ℝ N : II. Entire Solutions

We consider nonautonomous quasilinear parabolic equations satisfying certain symmetry conditions. We prove that each positive bounded solution u on ? ^N  × (?∞, T) decaying to zero at spatial infinity uniformly with respect to time is radially symmetric around some origin in ? ^N . The origin depends on the solution but is independent of time. We also consider the linearized equation along u and prove that each bounded (positive or not) solution is a linear combination of a radially symmetric solution and (nonsymmetric) spatial derivatives of u. Theorems on reflectional symmetry are also given.     

Asymptotic expansions for hyperbolic–parabolic systems with a small parameter

In this paper we consider initial-boundary value problems for systems with a small parameter ?. The problems are mixed hyperbolic–parabolic when ? > 0 and hyperbolic when ? = 0. Often the solution can be expanded asymptotically in ? and to first approximation it consists of the solution of the corresponding hyperbolic problem and a boundary layer part. We prove sufficient conditions for the expansion to exist and give estimates of the remainder. We also examine how the boundary conditions should be choosen to avoid O(1) boundary layers.     

A new stability number of the Bresse‐Cattaneo system

In this paper, we consider the Bresse‐Cattaneo system with a frictional damping term and prove some optimal decay results for the L²‐norm of the solution and its higher order derivatives. In fact, we show that there is a completely new stability number δ that controls the decay rate of the solution. To prove our results, we use the energy method in the Fourier space to build some very delicate Lyapunov functionals that give the desired results. We also prove the optimality of the results by using the eigenvalues expansion method. In addition, we show that for the absence of the frictional damping term, the solution of our problem does not decay at all. This result improves some early results     

Transport Equation with Nonlocal Velocity in Wasserstein Spaces: Convergence of Numerical Schemes

Motivated by pedestrian modelling, we study evolution of measures in the Wasserstein space. In particular, we consider the Cauchy problem for a transport equation, where the velocity field depends on the measure itself. We deal with numerical schemes for this problem and prove convergence of a Lagrangian scheme to the solution, when the discretization parameters approach zero. We also prove convergence of an Eulerian scheme, under more strict hypotheses. Both schemes are discretizations of the push-forward formula defined by the transport equation. As a by-product, we obtain existence and uniqueness of the solution. All the results of convergence are proved with respect to the Wasserstein distance. We also show that L ¹ spaces are not natural for such equations, since we lose uniqueness of the solution.     

Existence and uniqueness for P-area minimizers in the Heisenberg group

In [3] we studied p-mean curvature and the associated p-minimal surfaces in the Heisenberg group from the viewpoint of PDE and differential geometry. In this paper, we look into the problem through the variational formulation. We study a generalized p-area and associated ( p-) minimizers in general dimensions. We prove the existence and investigate the uniqueness of minimizers. Since this is reduced to solving a degenerate elliptic equation, we need to consider the effect of the singular set and this requires a careful study. We define the notion of weak solution and prove that in a certain Sobolev space, a weak solution is a minimizer and vice versa. We also give many interesting examples in dimension 2. An intriguing point is that, in dimension 2, a C ²-smooth solution from the PDE viewpoint may not be a minimizer. However, this statement is true for higher dimensions due to the relative smallness of the size of the singular set.     

Solvability of the Stokes Immersed Boundary Problem in Two Dimensions

We study coupled motion of a 1-D closed elastic string immersed in a 2-D Stokes flow, known as the Stokes immersed boundary problem in two dimensions. Using the fundamental solution of the Stokes equation and the Lagrangian coordinate of the string, we write the problem into a contour dynamic formulation, which is a nonlinear nonlocal equation solely keeping track of evolution of the string configuration. We prove existence and uniqueness of local-in-time solution starting from an arbitrary initial configuration that is an H^5/2-function in the Lagrangian coordinate satisfying the so-called well-stretched assumption. We also prove that when the initial string configuration is sufficiently close to an equilibrium, which is an evenly parametrized circular configuration, then a global-in-time solution uniquely exists and it will converge to an equilibrium configuration exponentially as t → + ∞. The technique in this paper may also apply to the Stokes immersed boundary problem in three dimensions. © 2018 Wiley Periodicals, Inc.     

Symmetry Properties of Positive Solutions of Parabolic Equations on ℝ N : I. Asymptotic Symmetry for the Cauchy Problem

We consider quasilinear parabolic equations on ? ^N satisfying certain symmetry conditions. We prove that bounded positive solutions decaying to zero at spatial infinity are asymptotically radially symmetric about a center. The asymptotic center of symmetry is not fixed a priori (and depends on the solution) but it is independent of time. We also prove a similar theorem on reflectional symmetry.     

The Dirichlet Problem for the Total Variation Flow

We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we use the nonlinear semigroup theory and we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions.     

Exponential decay of non‐linear wave equation with a viscoelastic boundary condition

We study in this paper the global existence and exponential decay of solutions of the non‐linear unidimensional wave equation with a viscoelastic boundary condition. We prove that the dissipation induced by the memory effect is strong enough to secure global estimates, which allow us to show existence of global smooth solution for small initial data. We also prove that the solution decays exponentially provided the resolvent kernel of the relaxation function, k decays exponentially. When k decays polynomially, the solution decays polynomially and with the same rate. Copyright © 2000 John Wiley & Sons, Ltd.     

Global strong solution to the three dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity

In this paper, we consider an initial boundary value problem for the 3‐dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity coefficients over a bounded smooth domain. Global in time unique strong solution is proved to exist when the L² norms of initial vorticity and current density are both suitably small with arbitrary large initial density, and the vacuum of initial density is also allowed. Finally, we revisit the Navier‐Stokes model without electromagnetic effect. We find that this initial boundary problem also admits a unique global strong solution under other conditions. In particular, we prove small kinetic‐energy strong solution exists globally in time, which extends the recent result of Huang and Wang.     

On the splitting problem for selections

We investigate when does the Repovš-Semenov splitting problem for selections have an affirmative solution for continuous set-valued mappings in finite-dimensional Banach spaces. We prove that this happens when images of set-valued mappings or even their graphs are P-sets (in the sense of Balashov) or strictly convex sets. We also consider an example which shows that there is no affirmative solution of this problem even in the simplest case in R³. We also obtain affirmative solution of the approximate splitting problem for Lipschitz continuous selections in the Hilbert space.     

Mathematical investigation of the Gerber-Shiu function in the case of dependent inter-claim time and claim size

In this paper we investigate the well-known Gerber-Shiu expected discounted penalty function in the case of dependence between the inter-claim times and the claim amounts. We set up an integral equation for it and we prove the existence and uniqueness of its solution in the set of bounded functions. We show that if δ>0, the limit property of the solution is not a regularity condition, but the characteristic of the solution even in the case when the net profit condition is not fulfilled. It is the consequence of the choice of the penalty function for a given density function. We present an example when the Gerber-Shiu function is not bounded, consequently, it does not tend to zero. Using an operator technique we also prove exponential boundedness.     

On a model selection problem from high-dimensional sample covariance matrices

Modern random matrix theory indicates that when the population size p is not negligible with respect to the sample size n, the sample covariance matrices demonstrate significant deviations from the population covariance matrices. In order to recover the characteristics of the population covariance matrices from the observed sample covariance matrices, several recent solutions are proposed when the order of the underlying population spectral distribution is known. In this paper, we deal with the underlying order selection problem and propose a solution based on the cross-validation principle. We prove the consistency of the proposed procedure.     

Global well posedness for the Maxwell–Navier–Stokes system in 2D

We prove global existence of regular solutions to the full MHD system (or more precisely the Maxwell–Navier–Stokes system) in 2D. We also provide an exponential growth estimate for the Hs norm of the solution when the time goes to infinity.     

Nonlinear Dynamics Equation in p-Adic String Theory

We investigate nonlinear pseudodifferential equations with infinitely many derivatives. These are equations of a new class, and they originally appeared in p-adic string theory. Their investigation is of interest in mathematical physics and its applications, in particular, in string theory and cosmology. We undertake a systematic mathematical investigation of the properties of these equations and prove the main uniqueness theorem for the solution in an algebra of generalized functions. We discuss boundary problems for bounded solutions and prove the existence theorem for spatially homogeneous solutions for odd p. For even p, we prove the absence of a continuous nonnegative solution interpolating between two vacuums and indicate the possible existence of discontinuous solutions. We also consider the multidimensional equation and discuss soliton and q-brane solutions.     

On minimal nonscattering solution for focusing mass-subcritical nonlinear Schrödinger equation

We consider time global behavior of solutions to the focusing mass-subcritical NLS equation in a weighted L² space. We prove that there exists a threshold solution such that (i) it does not scatter; (ii) with respect to a certain scale-invariant quantity, this solution attains minimum value in all nonscattering solutions. In the mass-critical case, it is known that ground states are this kind of threshold solution. However, in our case, it turns out that the above threshold solution is not a standing wave solution.     

Curve Shortening Flow in Arbitrary Dimensional Euclidian Space

In this paper, curve shortening flow in Euclidian space R^n（n≥3） is studied, and S. Altschuler＇s results about flow for space curves are generalized. We prove that the curve shortening flow converges to a straight line in infinite time if the initial curve is a ramp. We also prove the planar phenomenon when the curve shortening flow blows up.     

Strong Convergence Theorem for Multiple Sets Split Feasibility Problems in Banach Spaces

The purpose of this article is to study the iterative approximation of solution to multiple sets split feasibility problems in p-uniformly convex real Banach spaces that are also uniformly smooth. We propose an iterative algorithm for solving multiple sets split feasibility problems and prove a strong convergence theorem of the sequence generated by our algorithm under some appropriate conditions in p-uniformly convex real Banach spaces that are also uniformly smooth.     

Existence and uniqueness results for a nonlinear stationary system

We prove a few existence results of a solution for a static system with a coupling of thermoviscoelastic type. As this system involves L⊃ coupling terms we use the techniques of renormalized solutions for elliptic equations with L⊃ data. We also prove partial uniqueness results.AMS Subject Classification: 32D05, 74D99.