Boundary observability for the space-discretizations of the 1 — d wave equation

We consider the finite-difference and finite-element space discretization of the 1 — d wave equation with homogeneous Dirichlet boundary conditions in a bounded interval. We analyze the problem of estimating the total energy of solutions in terms of the energy concentrated on the boundary, uniformly as the net-spacing h → 0. We prove that there is no such a uniform bound due to spurious high frequencies. We prove however an uniform bound in suitable subspaces of solutions that eventually cover the whole energy space.     

Exact solutions of a class of second-order nonlocal boundary value problems and applications

In this paper, solutions of a class of second-order differential equations with some multi-point boundary conditions are studied. We give exact expressions of the solutions for the linear m-point boundary problems by the Green’s functions. As applications, we study uniqueness and iteration of the positive solutions for a nonlinear singular second-order m-point boundary value problem.     

Boundary asymptotic and uniqueness of solutions to the p-Laplacian with infinite boundary values

We prove boundary asymptotics to solutions of weighted p-Laplacian equations that take infinite value on the boundary of a bounded domain. Uniqueness of such solutions would then follow as a consequence. Our results extend previously known results by allowing weights that are unbounded in the domain.     

An elliptic problem with an indefinite nonlinearity and a parameter in the boundary condition

We establish the existence of solutions of nonlinear elliptic boundary value problems involving a positive parameter on the boundary. We also examine a profile of solutions of problem (1.2) when a parameter λ tends to 0.     

Strong full bounded solutions of nonlinear parabolic equations with nonlinear boundary conditions

We study the existence of (generalized) bounded solutions existing for all times for nonlinear parabolic equations with nonlinear boundary conditions on a domain that is bounded in space and unbounded in time (the entire real line). We give a counterexample which shows that a (weak) maximum principle does not hold in general for linear problems defined on the entire real line in time. We consider a boundedness condition at minus infinity to establish (one-sided) L^∞-a priori estimates for solutions to linear boundary value problems and derive a weak maximum principle which is valid on the entire real line in time. We then take up the case of nonlinear problems with (possibly) nonlinear boundary conditions. By using comparison techniques, some (delicate) a priori estimates obtained herein, and nonlinear approximation methods, we prove the existence and, in some instances, positivity and uniqueness of strong full bounded solutions existing for all times.     

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.     

Delay nonlinear boundary conditions as limit of reactions concentrating in the boundary

In this work, we are interested in the dynamic behavior of a parabolic problem with nonlinear boundary conditions and delay in the boundary. We construct a reaction–diffusion problem with delay in the interior, where the reaction term is concentrated in a neighborhood of the boundary and this neighborhood shrinks to boundary, as a parameter ? goes to zero. We analyze the limit of the solutions of this concentrated problem and prove that these solutions converge in certain continuous function spaces to the unique solution of the parabolic problem with delay in the boundary. This convergence result allows us to approximate the solution of equations with delay acting on the boundary by solutions of equations with delay acting in the interior and it may contribute to analyze the dynamic behavior of delay equations when the delay is at the boundary.     

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.     

Nodal solutions of multi-point boundary value problems

We study the nonlinear boundary value problem consisting of the equation y^″+w(t)f(y)=0 on [a,b] and a multi-point boundary condition. By relating it to the eigenvalues of a linear Sturm-Liouville problem with a two-point separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. We also discuss the changes in the existence question for different types of nodal solutions as the problem changes.     

Lipschitz continuity of solutions of a free boundary problem involving the p-Laplacian

We prove local interior and boundary Lipschitz continuity of solutions of a free boundary problem involving the p-Laplacian.     

Exponential decay for the solutions of nonlinear elliptic systems posed in unbounded cylinders

We study the asymptotic behavior at infinity of the solutions of a nonlinear elliptic system posed in a cylinder of infinite length. The problem is written in a variational formulation, where we ask the derivative of the solutions to be in Lp. We show that an exponential decay at infinity for the second member implies exponential decay for the derivative of the solutions. We also give an application of this result to the study of boundary layers problems.     

Characteristic boundary layers for parabolic perturbations of quasilinear hyperbolic problems

In this paper, we study the existence and nonlinear stability of the totally characteristic boundary layer for the quasilinear equations with positive definite viscosity matrix under the assumption that the boundary matrix vanishes identically on the boundary x=0. We carry out a series of weighted estimates to the boundary layer equations—Prandtl type equations to get the regularity and the far field behavior of the solutions. This allows us to perform a weighted energy estimate for the error equation to prove the stability of the boundary layers. The stability result finally implies the asymptotic limit of the viscous solutions.     

Vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system

In this paper, we study the vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system in a bounded domain. We first show the local existence of smooth solutions of the Euler/Allen-Cahn equations by modified Galerkin method. Then using the boundary layer function to deal with the mismatch of the boundary conditions between Navier-Stokes and Euler equations, and assuming that the energy dissipation for Navier-Stokes equation in the boundary layer goes to zero as the viscosity tends to zero, we prove that the solutions of the Navier-Stokes/Allen-Cahn system converge to that of the Euler/Allen-Cahn system in a proper small time interval. In addition, for strong solutions of the Navier-Stokes/Allen-Cahn system in 2D, the convergence rate is cν1/2.     

The effects of indefinite nonlinear boundary conditions on the structure of the positive solutions set of a logistic equation

We investigate a semilinear elliptic equation with a logistic nonlinearity and an indefinite nonlinear boundary condition, both depending on a parameter λ. Overall, we analyze the effect of the indefinite nonlinear boundary condition on the structure of the positive solutions set. Based on variational and bifurcation techniques, our main results establish the existence of three nontrivial non-negative solutions for some values of λ, as well as their asymptotic behavior. These results suggest that the positive solutions set contains an S-shaped component in some case, as well as a combination of a C-shaped and an S-shaped components in another case.     

Nano boundary layers over stretching surfaces

In this paper, we present similarity solutions for the nano boundary layer flows with Navier boundary condition. We consider viscous flows over a two-dimensional stretching surface and an axisymmetric stretching surface. The resulting nonlinear ordinary differential equations are solved analytically by the Homotopy Analysis Method. Numerical solutions are obtained by using a boundary value problem solver, and are shown to agree well with the analytical solutions. The effects of the slip parameter K and the suction parameter s on the fluid velocity and on the tangential stress are investigated and discussed. As expected, we find that for such fluid flows at nano scales, the shear stress at the wall decreases (in an absolute sense) with an increase in the slip parameter K.     

Uniqueness and existence for bounded boundary value problems

The existence and uniqueness of solutions for the boundary value problems with general linear point evaluation boundary conditions is established. We assume that f is bounded and that there is uniqueness on a homogeneous problem and on the linear variational problems.     

A note on parabolic equation with nonlinear dynamical boundary condition

We consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition that is related to the so-called Wentzell boundary condition. First we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we derive a suitable ?ojasiewicz-Simon type inequality to show the convergence of global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms f,g are real analytic. Moreover, we provide an estimate for the convergence rate.     

Approximate solutions and micro-regularity in the Denjoy-Carleman classes

We begin with a sequence M of positive real numbers and we consider the Denjoy-Carleman class CM. We show how to construct M-approximate solutions for complex vector fields with CM coefficients. We then use our construction to study micro-local properties of boundary values of approximate solutions in general M-involutive structures of codimension one, where the approximate solution is defined in a wedge whose edge (where the boundary value exists) is a maximally real submanifold. We also obtain a CM version of the Edge-of-the-Wedge Theorem.     

Estimates for solutions of reduced hyperbolic equations of the second order with a large parameter

We consider solutions of inhomogeneous, reduced hyperbolic equations of the second order, with a large parameter multiplying the unknown function. These solutions are defined on the m-dimensional region outside a star-shaped body. They satisfy an “outgoing” radiation condition at infinity and a Dirichlet boundary condition.We obtain a priori estimates for these solutions, at every point outside or on the surface of a two- or three-dimensional star-shaped body, that hold for sufficiently large values of the parameter. We prove that each solution is bounded by a linear combination of (i) the maximum norm of its prescribed boundary values, (ii) the L₂ norm of the prescribed values of its tangential derivative, (iii) an L₂ norm of the source term. This result is based on similar inequalities that we first obtain for a certain L₂ norm of the gradient, and of the normal derivative on the boundary, of each solution defined outside an m-dimensional star-shaped body.For the special case of the reduced wave equation, Morawetz and Ludwig [1] have obtained similar estimates. Just as their results have been used in [3] to confirm the geometrical theory of diffraction, the estimates obtained in this paper can be used to establish the validity of certain formal asymptotic solutions of reduced hyperbolic equations.     

The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations

In this paper, we address some fundamental issues concerning “time marching” numerical schemes for computing steady state solutions of boundary value problems for nonlinear partial differential equations. Simple examples are used to illustrate that even theoretically convergent schemes can produce numerical steady state solutions that do not correspond to steady state solutions of the boundary value problem. This phenomenon must be considered in any computational study of nonunique solutions to partial differential equations that govern physical systems such as fluid flows. In particular, numerical calculations have been used to “suggest” that certain Euler equations do not have a unique solution. For Burgers' equation on a finite spatial interval with Neumann boundary conditions the only steady state solutions are constant (in space) functions. Moreover, according to recent theoretical results, for any initial condition the corresponding solution to Burgers' equation must converge to a constant as t → ∞. However, we present a convergent finite difference scheme that produces false nonconstant numerical steady state “solutions.” These erroneous solutions arise out of the necessary finite floating point arithmetic inherent in every digital computer. We suggest the resulting numerical steady state solution may be viewed as a solution to a “nearby” boundary value problem with high sensitivity to changes in the boundary conditions. Finally, we close with some comments on the relevance of this paper to some recent “numerical based proofs” of the existence of nonunique solutions to Euler equations and to aerodynamic design.