Positive solutions to the Laplace equation with nonlinear boundary conditions on the half space

We consider positive solutions of $\varDelta u=0$ in $\mathbf{R}_+^n$ , $\partial _{\nu }u=u^q$ on $\partial \mathbf{R}_+^n$ , where $n\ge 3$ and $q>n/(n-2)$ . We investigate the qualitative property of positive $x_n$ -axial symmetric solutions. In particular, we are concerned with the asymptotic expansion and the intersection property of positive $x_n$ -axial symmetric solutions.     

Nano boundary layer equation with nonlinear Navier boundary condition

At the micro and nano scale the standard no slip boundary condition of classical fluid mechanics does not apply and must be replaced by a boundary condition that allows some degree of tangential slip. In this study the classical laminar boundary layer equations are studied using Lie symmetries with the no-slip boundary condition replaced by a nonlinear Navier boundary condition. This boundary condition contains an arbitrary index parameter, denoted by n>0, which appears in the coefficients of the ordinary differential equation to be solved. The case of a boundary layer formed in a convergent channel with a sink, which corresponds to n=1/2, is solved analytically. Another analytical but non-unique solution is found corresponding to the value n=1/3, while other values of n for n>1/2 correspond to the boundary layer formed in the flow past a wedge and are solved numerically. It is found that for fixed slip length the velocity components are reduced in magnitude as n increases, while for fixed n the velocity components are increased in magnitude as the slip length is increased.     

Numerical investigation on nano boundary layer equation with Navier boundary condition

In this paper, by applying rational Legendre collocation technique and relaxation method, the classical laminar boundary layer equations with the nonlinear Navier boundary conditions are investigated. The features of the flow characteristics for different values of n are discussed. Numerical approaches are used to find solutions for the cases n > 1 / 2 corresponding to the flow past a wedge and n = 1 / 2 corresponding to the flow in a convergent channel. During the comparison, the effectivity and stability of the applied methods are demonstrated. The effects of the varying slip length, index parameter, components of velocity, and tangential stress are analyzed. Copyright © 2012 John Wiley & Sons, Ltd.     

On regularity of a weak solution to the Navier–Stokes equation with generalized impermeability boundary conditions

We consider the 3D Navier–Stokes equation with generalized impermeability boundary conditions. As auxiliary results, we prove the local in time existence of a strong solution (‘strong’ in a limited sense) and a theorem on structure. Then, taking advantage of the boundary conditions, we formulate sufficient conditions for regularity up to the boundary of a weak solution by means of requirements on one of the eigenvalues of the rate of deformation tensor. Finally, we apply these general results to the case of an axially symmetric flow with zero angular velocity.     

Kreiss symmetrizer and boundary conditions for the Euler-Korteweg system in a half space

The Euler-Korteweg system is a third order, dispersive system of PDEs, obtained from the standard Euler equations for compressible fluids by adding the so-called Korteweg stress tensor - encoding capillarity effects. Various results of well-posedness have been obtained recently for the Cauchy problem associated with the Euler-Korteweg system in the whole space. As to mixed problems, with initial and boundary value data, they are still mostly open. Here the linearized Euler-Korteweg system is studied in a half space by the use of normal mode analysis, which yields a generalized Kreiss-Lopatinski? condition that must be satisfied by the boundary conditions for the boundary value problem to be well-posed.Conversely, under the uniform Kreiss-Lopatinski? condition, generalized Kreiss symmetrizers are constructed in one space dimension for an extended system originally introduced for the Cauchy problem, which displays crucial quasi-homogeneity properties. A priori estimates without loss of derivatives are thus derived, and finally the well-posedness of the mixed problem is obtained by combining the estimates for the pure boundary value problem and trace results for solutions of the pure Cauchy problem.     

On positive solutions for a fourth order asymptotically linear elliptic equation under Navier boundary conditions

A result on existence of positive solution for a fourth order nonlinear elliptic equation under Navier boundary conditions is established. The nonlinear term involved is asymptotically linear both at the origin and at infinity. We exploit topological degree theory and global bifurcation.     

Blowup behaviors for diffusion system coupled through nonlinear boundary conditions in a half space

This paper deals with the blowup estimates near the blowup time for the system of heat equations in a half space coupled through nonlinear boundary conditions. The upper and lower bounds of blowup rate are established. The uniqueness and nonunique-ness results for the system with vanishing initial value are given.     

A uniqueness condition for the polyharmonic equation in free space

Consider the polyharmonic wave equation ?u + (? Δ)^mu = f in ?ⁿ × (0, ∞) with time-independent right-hand side. We study the asymptotic behaviour of u ( x , t) as t → ∞ and show that u( x , t) either converges or increases with order t^α or In t as t → ∞. In the first case we study the limit $ u_0 \left({\bf x} \right) \colone \mathop {\lim }\limits_{t \to \infty } \,u\left({{\bf x},t} \right) $ and give a uniqueness condition that characterizes u₀ among the solutions of the polyharmonic equation ( ? Δ)^mu = f in ?ⁿ. Furthermore we prove in the case 2m ? n that the polyharmonic equation has a solution satisfying the uniqueness condition if and only if f is orthogonal to certain solutions of the homogeneous polyharmonic equation.     

Solvability of some boundary-value problems for polyharmonic equation with Hadamard-Marchaud boundary operator

We investigate solvability conditions for certain non-classical boundary-value problems for the polyharmonic equation. As the boundary operators we consider fractional differential operators in the sense of Hadamard-Marchaud. The considered problems generalize well-known Dirichlet and von Neumann boundary-value problems for boundary operators of fractional type.     

Blowing up boundary conditions for a nonlocal nonlinear diffusion equation in several space dimensions

We analyze boundary value problems prescribing Dirichlet or Neumann boundary conditions for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation in a bounded smooth domain Ω∈R^N with N≥1. First, we prove existence and uniqueness of solutions and the validity of a comparison principle for these problems. Next, we impose boundary data that blow up in finite time and study the behavior of the solutions.     

Burgers equation with random boundary conditions

We prove an existence and uniqueness theorem for stationary solutions of the inviscid Burgers equation on a segment with random boundary conditions. We also prove exponential convergence to the stationary distribution.

     

Navier–Stokes equations with periodic boundary conditions and pressure loss

We present in this note the existence and uniqueness results for the Stokes and Navier–Stokes equations which model the laminar flow of an incompressible fluid inside a two-dimensional channel of periodic sections. The data of the pressure loss coefficient enables us to establish a relation on the pressure and to thus formulate an equivalent problem.     

On the global null controllability of a Navier-Stokes system with Navier slip boundary conditions

In this paper, we deal with a two-dimensional Navier-Stokes system in a rectangle with Navier slip boundary conditions on the horizontal sides. We establish the global null controllability of the system by controlling the normal component and the vorticity of the velocity on the vertical sides. The linearized control system around zero is controllable but one does not know how to deduce global controllability results for the nonlinear system. Our proof uses the return method together with a local exact controllability result by Fursikov and Imanuvilov.     

Problems with nonlinear boundary conditions for a hyperbolic equation

Two problems with nonlinear boundary conditions are studied. Existence and uniqueness theorems are proved for generalized solutions to each problem.     

On a stochastic wave equation with unilateral boundary conditions

We prove the existence and uniqueness of solutions to the initial boundary value problem for a one-dimensional wave equation with unilateral boundary conditions and random noise. We also establish the existence of an invariant measure.

     

Boundary conditions for the volume potential for the polyharmonic equation

In a bounded connected domain, we obtain boundary conditions for the volume potential for the polyharmonic equation.