Steady solutions of the Navier–Stokes equations with threshold slip boundary conditions

We establish the wellposedness of the time‐independent Navier–Stokes equations with threshold slip boundary conditions in bounded domains. The boundary condition is a generalization of Navier's slip condition and a restricted Coulomb‐type friction condition: for wall slip to occur the magnitude of the tangential traction must exceed a prescribed threshold, independent of the normal stress, and where slip occurs the tangential traction is equal to a prescribed, possibly nonlinear, function of the slip velocity. In addition, a Dirichlet condition is imposed on a component of the boundary if the domain is rotationally symmetric. We formulate the boundary‐value problem as a variational inequality and then use the Galerkin method and fixed point arguments to prove the existence of a weak solution under suitable regularity assumptions and restrictions on the size of the data. We also prove the uniqueness of the solution and its continuous dependence on the data. Copyright © 2006 John Wiley & Sons, Ltd.     

Rotationally symmetric 1-harmonic maps from D 2 to S 2

We consider rotationally symmetric 1-harmonic maps from D ² to S ² subject to Dirichlet boundary conditions. We prove that the corresponding energy—a degenerate non-convex functional with linear growth—admits a unique minimizer, and that the minimizer is smooth in the bulk and continuously differentiable up to the boundary. We also show that, in contrast with 2-harmonic maps, a range of boundary data exists such that the energy admits more than one smooth critical point: more precisely, we prove that the corresponding Euler–Lagrange equation admits a unique (up to scaling and symmetries) global solution, which turns out to be oscillating, and we characterize the minimizer and the smooth critical points of the energy as the monotone, respectively non-monotone, branches of such solution. R. Dal Passo passed away on 8th August 2007. Endowed with great strength, creativity and humanity, Roberta has been an outstanding mathematician, an extraordinary teacher and a wonderful friend. Farewell, Roberta.     

Rotationally symmetric viscous gas flows

The Dirichlet boundary value problem for the Navier–Stokes equations of a barotropic viscous compressible fluid is considered. The flow region and the data of the problem are assumed to be invariant under rotations about a fixed axis. The existence of rotationally symmetric weak solutions for all adiabatic exponents from the interval (γ*,∞) with a critical exponent γ* < 4/3 is proved.     

Graphs with prescribed Levi form trace

Abstract We prove existence and uniqueness of a viscosity solution of the Dirichlet problem related to the prescribed Levi mean curvature equation, under suitable assumptions on the boundary data and on the Levi curvature of the domain. We also show that such a solution is Lipschitz continuous by proving that it is the uniform limit of a sequence of classical solutions of elliptic problems and by building Lipschitz continuous barriers. Keywords: Levi mean curvature, Quasilinear degenerate elliptic PDE’s, Viscosity solutions, Comparison principle, Global Lipschitz estimates     

Nonuniformly nonlinear elliptic equations of p-biharmonic type

This paper deals with the existence and multiplicity of weak solutions to nonlinear differential equations involving a general p-biharmonic operator (in particular, p-biharmonic operator) under Dirichlet boundary conditions or Navier boundary conditions. Our method is mainly based on variational arguments.     

EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A CLASS OF p（x）-BIHARMONIC EQUATIONS

In this paper,we study a class of p(x)-biharmonic equations with Navier boundary condition.Using the mountain pass theorem,fountain theorem,local linking theorem and symmetric mountain pass theorem,we establish the existence of at least one solution and infinitely many solutions of this problem,respectively.     

Existence results for a class of degenerate quasilinear elliptic systems

We establish existence results for weak solutions of degenerate quasilinear elliptic systems. By using the variational method we obtain the existence of a solution for an elliptic system with Dirichlet boundary condition under some restriction on λ.     

Coexistence of Two Species in a Strongly Coupled Schoener’s Competitive Model

This paper deals with the existence of positive solution to a strongly coupled system with homogeneous Dirichlet boundary conditions describing a Schoener’s competitive interaction of two species. Making use of the Schauder fixed point theorem, a sufficient condition is given for the system to have a coexistence. And true solutions are constructed based on monotone iterative method. Our results show that this model possesses at least one coexistence state if cross-diffusions and intra-specific competitions are weak.     

Boundary stabilization for a coupled system

We investigate in this work the global existence of weak solutions for a nonlinear coupled system with mixed type boundary conditions. More precisely, Dirichlet and feedback boundary conditions. Further, we also prove the exponential decay of the energy associated with these solutions.     

On Caffarelli–Kohn–Nirenberg-type Inequalities for the Weighted Biharmonic Operator in Cones

We investigate Caffarelli–Kohn–Nirenberg-type inequalities for the weighted biharmonic operator in cones, both under Navier and Dirichlet boundary conditions. Moreover, we study existence and qualitative properties of extremal functions. In particular, we show that in some cases extremal functions do change sign; when the domain is the whole space, we prove some breaking symmetry phenomena.     

2m阶Dirichlet边值问题的多解(英文)

本文讨论了非线性2m阶Dirichlet边值问题多解的存在性.在非线性项满足一定条件时,通过有效地利用锥中的不动点指数理论和解的反对称延拓法,得到关于变号解的一些新的存在结果.确切地说,得到该2m阶边值问题至少存在两个正解,两个负解和多个变号解.     

A note on spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients

In this note, by constructing suitable approximate solutions, we prove the existence of global weak solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients in the whole space or exterior domain, when the initial data are spherically symmetric. In particular, we prove the existence of spherically symmetric solutions to the Saint-Venant model for shallow water in the whole space (or exterior domain).     

Reproductivity for a nematic liquid crystal model

In this paper we prove existence of weak solution with the reproductivity in time property, for a penalized PDE’s system related to a nematic liquid crystal model. This problem is relatively explict when time-independent Dirichlet boundary conditions are imposed for the orientation of crystal molecules. Nevertheless, for the time-dependent case, the treatment of the problem is completely different. The verification of a maximum principle for weak reproductive solutions is fundamental in the argument.     

Partial regularity for the Navier-Stokes-Fourier system

This paper addresses a nonstationary flow of heat-conductive incompressible Newtonian fluid with temperature-dependent viscosity coupled with linear heat transfer with advection and a viscous heat source term, under Navier/Dirichlet boundary conditions. The partial regularity for the velocity of the fluid is proved for each proper weak solution, that is, for such weak solutions which satisfy some local energy estimates in a similar way to the suitable weak solutions of the Navier-Stokes system. Finally, we study the nature of the set of points in space and time upon which proper weak solutions could be singular.     

On Stability of Solutions to Equations Describing Incompressible Heat-Conducting Motions Under Navier’s Boundary Conditions

In this paper we prove existence of global strong-weak two-dimensional solutions to the Navier-Stokes and heat equations coupled by the external force dependent on temperature and the heat dissipation, respectively. The existence is proved in a bounded domain with the Navier boundary conditions for velocity and the Dirichlet boundary condition for temperature. Next, we prove existence of 3d global strong solutions via stability.     

Positivity preserving property for a class of biharmonic elliptic problems

The lack of a general maximum principle for biharmonic equations suggests to study under which boundary conditions the positivity preserving property holds. We show that this property holds in general domains for suitable linear combinations of Dirichlet and Navier boundary conditions. The spectrum of this operator exhibits some unexpected features: radial data may generate nonradial solutions. These boundary conditions are also of some interest in semilinear equations, since they enable us to give explicit radial singular solutions to fourth order Gelfand-type problems.     

一类非局部非线性扩散方程解的全局爆破

主要研究在Dirichlet边界条件或Neumann边界条件下的一类非局部非线性的扩散方程问题.在适当的假设下,证明解的存在性、唯一性、比较原则、以及解对初边值条件的连续依赖性,并就给定的初边值条件,证明解在有限时刻全局爆破.     

Sobolev spaces of symmetric functions and applications

We prove sharp pointwise estimates for functions in the Sobolev spaces of radial functions defined in a ball. As a consequence, we obtain some imbeddings of such Sobolev spaces in weighted Lq-spaces. We also prove similar imbeddings for Sobolev spaces of functions with partial symmetry. Our techniques lead to new Hardy type inequalities. It is important to observe that we do not require any vanishing condition on the boundary to obtain all our estimates. We apply these imbeddings to obtain radial solutions and partially symmetric solutions for a biharmonic equation of the Hénon type under both Dirichlet and Navier boundary conditions. The delicate question of the regularity of these solutions is also established.     

On non‐stationary viscous incompressible flow through a cascade of profiles

The paper deals with theoretical analysis of non‐stationary incompressible flow through a cascade of profiles. The initial‐boundary value problem for the Navier–Stokes system is formulated in a domain representing the exterior to an infinite row of profiles, periodically spaced in one direction. Then the problem is reformulated in a bounded domain of the form of one space period and completed by the Dirichlet boundary condition on the inlet and the profile, a suitable natural boundary condition on the outlet and periodic boundary conditions on artificial cuts. We present a weak formulation and prove the existence of a weak solution. Copyright © 2006 John Wiley & Sons, Ltd.     

Stochastic 3D tamed Navier–Stokes equations: Existence,uniqueness and small time large deviation principles

In this paper, we first give a direct approach to the existence and uniqueness of strong solutions to stochastic 3D tamed Navier–Stokes equations in case of Dirichlet boundary conditions for the Stokes–Laplacian. Then we prove a small time large deviation principle for the solutions.