Viscosity solution theory of a class of nonlinear degenerate parabolic equations I. Uniqueness and existence of viscosity solutions

In this part we construct a unique bounded Hölder continuous viscosity solution for the nonlinear PDEs with the evolutionp-Laplacian equation and its anisotropic version as typical examples. The existence and properties of free boundaries will be discussed in part II.This research is supported by the National Natural Sciences Foundation of China.     

The inviscid limit of the incompressible anisotropic Navier–Stokes equations with the non-slip boundary condition

In this paper, we study the asymptotic behavior for the incompressible anisotropic Navier–Stokes equations with the non-slip boundary condition in a half space of ${\mathbb{R}^3}$ when the vertical viscosity goes to zero. Firstly, by multi-scale analysis, we formally deduce an asymptotic expansion of the solution to the problem with respect to the vertical viscosity, which shows that the boundary layer appears in the tangential velocity field and satisfies a nonlinear parabolic–elliptic coupled system. Also from the expansion, it is observed that away from the boundary the solution of the anisotropic Navier–Stokes equations formally converges to a solution of a degenerate incompressible Navier–Stokes equation. Secondly, we study the well-posedness of the problems for the boundary layer equations and then rigorously justify the asymptotic expansion by using the energy method. We obtain the convergence results of the vanishing vertical viscosity limit, that is, the solution to the incompressible anisotropic Navier–Stokes equations tends to the solution to degenerate incompressible Navier–Stokes equations away from the boundary, while near the boundary, it tends to the boundary layer profile, in both the energy space and the L ^∞ space.     

Nonconforming rotated Q 1 element on non-tensor product anisotropic meshes

In this paper, we consider the nonconforming rotated Q ₁ element for the second order elliptic problem on the non-tensor product anisotropic meshes, i.e. the anisotropic affine quadrilateral meshes. Though the interpolation error is divergent on the anisotropic meshes, we overcome this difficulty by constructing another proper operator. Then we give the optimal approximation error and the consistency error estimates under the anisotropic affine quadrilateral meshes. The results of this paper provide some hints to derive the anisotropic error of some finite elements whose interpolations do not satisfy the anisotropic interpolation properties. Lastly, a numerical test is carried out, which coincides with our theoretical analysis.     

Hölder continuous periodic solution of Boussinesq equation with partial viscosity

We show the existence of Hölder continuous periodic solution with compact support in time of the Boussinesq equations with partial viscosity. The Hölder regularity of the solution we constructed is anisotropic which is compatible with partial viscosity of the equations.     

Global Regularity of the 3D Axi-Symmetric Navier–Stokes Equations with Anisotropic Data

In this paper, we study the 3D axisymmetric Navier–Stokes equations with swirl. We prove the global regularity of the 3D Navier–Stokes equations for a family of large anisotropic initial data. Moreover, we obtain a global bound of the solution in terms of its initial data in some L ^p norm. Our results also reveal some interesting dynamic growth behavior of the solution due to the interaction between the angular velocity and the angular vorticity fields.     

An extension of the edge covering problem

In this paper we define a weightedr-covering problem, and show that there exists an optimum solution of ther-covering problem which can be decomposed into the sum of a rounded down solution of its linear relaxation and an optimal solution of a weighted edge covering problem on a reduced graph. Vertexr-packing problem can also be reduced to ther-covering problem.     

On the sensitivity matrix of the Nash bargaining solution

In this note we provide a characterization of a subclass of bargaining problems for which the Nash solution has the property of disagreement point monotonicity. While the original d-monotonicity axiom and its stronger notion, strong d-monotonicity, were introduced and discussed by Thomson (J Econ Theory, 42: 50–58, 1987), this paper introduces local strong d-monotonicity and derives a necessary and sufficient condition for the Nash solution to be locally strongly d-monotonic. This characterization is given by using the sensitivity matrix of the Nash bargaining solution w.r.t. the disagreement point d. Moverover, we present a sufficient condition for the Nash solution to be strong d-monotonic.     

Local limit of nonlocal traffic models: Convergence results and total variation blow-up

Consider a nonlocal conservation law where the flux function depends on the convolution of the solution with a given kernel. In the singular local limit obtained by letting the convolution kernel converge to the Dirac delta one formally recovers a conservation law. However, recent counter-examples show that in general the solutions of the nonlocal equations do not converge to a solution of the conservation law. In this work we focus on nonlocal conservation laws modeling vehicular traffic: in this case, the convolution kernel is anisotropic. We show that, under fairly general assumptions on the (anisotropic) convolution kernel, the nonlocal-to-local limit can be rigorously justified provided the initial datum satisfies a one-sided Lipschitz condition and is bounded away from 0. We also exhibit a counter-example showing that, if the initial datum attains the value 0, then there are severe obstructions to a convergence proof.     

A <Emphasis Type="Italic">d</Emphasis>-person Differential Game with State Space Constraints

We consider a network of d companies (insurance companies, for example) operating under a treaty to diversify risk. Internal and external borrowing are allowed to avert ruin of any member of the network. The amount borrowed to prevent ruin is viewed upon as control. Repayment of these loans entails a control cost in addition to the usual costs. Each company tries to minimize its repayment liability. This leads to a d -person differential game with state space constraints. If the companies are also in possible competition a Nash equilibrium is sought. Otherwise a utopian equilibrium is more appropriate. The corresponding systems of HJB equations and boundary conditions are derived. In the case of Nash equilibrium, the Hamiltonian can be discontinuous; there are d interlinked control problems with state constraints; each value function is a constrained viscosity solution to the appropriate discontinuous HJB equation. Uniqueness does not hold in general in this case. In the case of utopian equilibrium, each value function turns out to be the unique constrained viscosity solution to the appropriate HJB equation. Connection with Skorokhod problem is briefly discussed.     

Global strong solutions to 3-D Navier-Stokes system with strong dissipation in one direction Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday

We consider three-dimensional incompressible Navier-Stokes equations(NS) with different viscous coefficients in the vertical and horizontal variables. In particular, when one of these viscous coefficients is large enough compared with the initial data, we prove the global well-posedness of this system. In fact, we obtain the existence of a global strong solution to(NS) when the initial data verifies an anisotropic smallness condition which takes into account the different roles of the horizontal and vertical viscosity.     

Stochastic differential games with reflection and related obstacle problems for Isaacs equations

In this paper we first investigate zero-sum two-player stochastic differential games with reflection, with the help of theory of Reflected Backward Stochastic Differential Equations (RBSDEs). We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straightforward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations with obstacles, respectively. The method differs significantly from those used for control problems with reflection, with new techniques developed of interest on its own. Further, we also prove a new estimate for RBSDEs being sharper than that in the paper of El Karoui, Kapoudjian, Pardoux, Peng and Quenez (1997), which turns out to be very useful because it allows us to estimate the L ^p -distance of the solutions of two different RBSDEs by the p-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution to the approximating Isaacs equation constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.     

Global well-posedness of the three dimensional incompressible anisotropic Navier–Stokes system

In this paper, we study the global well-posed problem for the three dimensional incompressible anisotropic Navier–Stokes system (ANS) with initial data in the scaling invariant Besov–Sobolev type spaces. We prove that (ANS) has a unique global solution provided that the initial vertical velocity is large while initial horizontal data are sufficiently small compared with the horizontal viscosity. In particular, our result implies the global well-posedness of (ANS) with highly oscillating initial data.     

Continuity in law with respect to the Hurst index of some additive functionals of sub-fractional Brownian motion

In this article, first, we prove some properties of the sub-fractional Brownian motion introduced by Bojdecki et al. [Statist. Probab. Lett. 69(2004):405–419]. Second, we prove the continuity in law, with respect to small perturbations of the Hurst index, in some anisotropic Besov spaces, of some continuous additive functionals of the sub-fractional Brownian motion. We prove that our result can be obtained easily, by using the decomposition in law of the sub-fractional Brownian motion given by Bardina and Bascompte [Collect. Math. 61(2010):191–204] and Ruiz de Chavez and Tudor [Math. Rep. 11(2009):67–74], without using the result of Wu and Xiao [Stoch. Proc. Appl. 119(2009):1823–1844] by connecting the sub-fractional Brownian motion to its stationary Gaussian process through Lamperti’s transform. This decomposition in law leads to a better understanding and simple proof of our result.     

A decomposition algorithm for linear relaxation of the weightedr-covering problem

In this paper the linear relaxation of the weightedr-covering problem (r-LCP) is considered. The dual problem (c-LMP) is the linear relaxation of the well-knownc-matching problem and hence can be solved in polynomial time. However, we describe a simple, but nonpolynomial algorithm in which ther-LCP is decomposed into a sequence of 1-LCP’s and its optimal solution is obtained by adding the optimal solutions of these 1-LCP’s. An 1-LCP can be solved in polynomial time by solving its dual as a max-flow problem on a bipartite graph. An accelerated algorithm based on this decomposition scheme to solve ar-LCP is also developed and its average case behaviour is studied.     

On the properties of sequences of fuzzy-valued Choquet integrable functions

In this paper, we focus on investigating the properties of sequences of fuzzy-valued Choquet (for short, (C)-) integrable functions. Firstly, the concept of uniform (C)-integrabiliy and other new concepts like uniform absolute continuity and uniform boundedness for sequences of fuzzy-valued (C)-integrable functions are introduced and then the relations among them are discussed. As the applications of these concepts, we also present several convergence theorems for sequences of fuzzy-valued (C)-integrable functions by using uniform (C)-integrability.     

On the Convergence of Newton-Like Methods Based on M-Fréchet Differentiable Operators and Applications in Radiative Transfer

In this study we approximate a locally unique solution of a nonlinear operator equation in Banach space using Newton-like methods. A complete error analysis of our method is also given. Our new theorem uses Lipschitz or Hölder continuity assumptions on m-Fréchet-differentiable operators where m 2 is a positive integer. A numerical example is given to show that our results provide a better information on the location of the solution as well as finer error bounds on the distances involved than earlier results. A second numerical example shows how to solve a nonlinear integral equation appearing in radiative transfer.     

Asymptotic analysis of the steady stokes equation with randomly perturbed viscosity in a thin tube structure

The Stokes equation with the nonconstant viscosity is considered in a thin tube structure, i.e., in a connected union of thin rectangles with heights of order ε ≪ 1 and bases of order 1 with smoothened boundary. An asymptotic expansion of the solution is constructed. In the case of random perturbations of the constant viscosity, we prove that the leading term for the velocity is deterministic, while for the pressure it is random, but the expectations of the pressure satisfies the deterministic Darcy equation. Estimates for the difference between the exact solution and its asymptotic approximation are proved. Bibliography: 11 titles. Illustrations: 3 figures.     

A Note on Fuzzy Relation Programming Problems with Max-Strict-t-Norm Composition

The fuzzy relation programming problem is a minimization problem with a linear objective function subject to fuzzy relation equations using certain algebraic compositions. Previously, Guu and Wu considered a fuzzy relation programming problem with max-product composition and provided a necessary condition for an optimal solution in terms of the maximum solution derived from the fuzzy relation equations. To be more precise, for an optimal solution, each of its components is either 0 or the corresponding component's value of the maximum solution. In this paper, we extend this useful property for fuzzy relation programming problem with max-strict-t-norm composition and present it as a supplemental note of our previous work.     

Nonlinear scalarizing functions in set optimization problems

In this paper, we obtain Hölder continuity of the nonlinear scalarizing function for l-type less order relation, which is introduced by Hernández and Rodríguez-Marín (J. Math. Anal. Appl. 2007;325:1–18). Moreover, we introduce the nonlinear scalarizing function for u-type less order relation and establish continuity, convexity and Hölder continuity of the nonlinear scalarizing function for u-type less order relation. As applications, we firstly obtain Lipschitz continuity of solution mapping to the parametric equilibrium problems and then establish Lipschitz continuity of strongly approximate solution mappings for l-type less order relation, u-type less order relation and set less order relation to the parametric set optimization problems by using convexity and Hölder continuity of the nonlinear scalarizing functions.     

Global well‐posedness of 2D nonlinear Boussinesq equations with mixed partial viscosity and thermal diffusivity

In this paper, we discuss with the global well‐posedness of 2D anisotropic nonlinear Boussinesq equations with any two positive viscosities and one positive thermal diffusivity. More precisely, for three kinds of viscous combinations, we obtain the global well‐posedness without any assumption on the solution. For other three difficult cases, under the minimal regularity assumption, we also derive the unique global solution. To the authors' knowledge, our result is new even for the simplified model, that is, F(θ) = θe₂. Copyright © 2017 John Wiley & Sons, Ltd.