Elliptic boundary value problems with λ-dependent boundary conditions

In this paper second order elliptic boundary value problems on bounded domains Ω⊂Rn with boundary conditions on ∂Ω depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework. For a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure. In the special case of rational Nevanlinna or Riesz-Herglotz functions on the boundary the solution operator is obtained in an explicit form in the product Hilbert space L²(Ω)⊕(L²m(∂Ω)), which is a natural generalization of known results on λ-linear elliptic boundary value problems and λ-rational boundary value problems for ordinary second order differential equations.     

Differential recursion relations for Laguerre functions on symmetric cones

Let Ω be a symmetric cone and V the corresponding simple Euclidean Jordan algebra. In our previous papers (some with G. Zhang) we considered the family of generalized Laguerre functions on Ω that generalize the classical Laguerre functions on R⁺. This family forms an orthogonal basis for the subspace of L-invariant functions in L²(Ω,dμν), where dμν is a certain measure on the cone and where L is the group of linear transformations on V that leave the cone Ω invariant and fix the identity in Ω. The space L²(Ω,dμν) supports a highest weight representation of the group G of holomorphic diffeomorphisms that act on the tube domain T(Ω)=Ω+iV. In this article we give an explicit formula for the action of the Lie algebra of G and via this action determine second order differential operators which give differential recursion relations for the generalized Laguerre functions generalizing the classical creation, preservation, and annihilation relations for the Laguerre functions on R⁺.     

Long-time behavior of reaction-diffusion equations with dynamical boundary condition

In this paper, we study the long-time behavior of the reaction-diffusion equation with dynamical boundary condition, where the nonlinear terms f and g satisfy the polynomial growth condition of arbitrary order. Some asymptotic regularity of the solution has been proved. As an application of the asymptotic regularity results, we can not only obtain the existence of a global attractor A in (H¹(Ω)∩L^p(Ω))×L^q(Γ) immediately, but also can show further that A attracts every L²(Ω)×L²(Γ)-bounded subset with (H¹(Ω)∩L^p+δ(Ω))×L^q+κ(Γ)-norm for any δ,κ∈[0,∞).     

Grand Sobolev spaces and their applications in geometric function theory and PDEs

Function spaces that are slightly larger than the Lebesgue L ^p (Ω) spaces (even larger than the Marcinkiewicz L ^p,∞ (Ω) spaces) have been introduced by Iwaniec and Sbordone [Arch. Ration. Mech. Anal. 119 (1992), 129–143] in connection with integrability properties of the Jacobian. These are the grand Lebesgue spaces L ^p)(Ω). In this survey we collect a number of results which prove that these spaces are useful in various classical settings of geometric function theory and partial differential equations (PDEs).     

The attractors for the nonhomogeneous nonautonomous Navier-Stokes equations

In this paper, we consider the attractors for the two-dimensional nonautonomous Navier-Stokes equations in nonsmooth bounded domain Ω with nonhomogeneous boundary condition u=φ on ∂Ω. Assuming , which is translation compact and φ∈L^∞(∂Ω), we establish the existence of the uniform attractor in L²(Ω) and .     

Convergence analysis of an upwind mixed element method for advection diffusion problems

We consider a upwinding mixed element method for a system of first order partial differential equations resulting from the mixed formulation of a general advection diffusion problem. The system can be used to model the transport of a contaminant carried by a flow. We use the lowest order Raviart-Thomas mixed finite element space. We show the first order convergence both for concentration and concentration flux in L²(Ω).     

The convergence of non-Newtonian fluids to Navier-Stokes equations

In this paper, the convergence of solutions for incompressible dipolar viscous non-Newtonian fluids is investigated. We obtain the conclusion that the solutions of non-Newtonian fluids converge to the solutions of Navier-Stokes equations in the sense of L²-norm (resp. H¹-norm), as the viscosities tend to zero and the initial data belong to H¹(Ω) (resp. H²(Ω)). Moreover, we obtain L^∞-norm convergence of solutions if the initial data belong to H²(Ω).     

Global existence and decay of energy to systems of wave equations with damping and supercritical sources

This paper is concerned with a system of nonlinear wave equations with supercritical interior and boundary sources and subject to interior and boundary damping terms. It is well-known that the presence of a nonlinear boundary source causes significant difficulties since the linear Neumann problem for the single wave equation is not, in general, well-posed in the finite-energy space H ¹(Ω) × L ²(?Ω) with boundary data from L ²(?Ω) (due to the failure of the uniform Lopatinskii condition). Additional challenges stem from the fact that the sources considered in this article are non-dissipative and are not locally Lipschitz from H ¹(Ω) into L ²(Ω) or L ²(?Ω). With some restrictions on the parameters in the system and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution and establish (depending on the behavior of the dissipation in the system) exponential and algebraic uniform decay rates of energy. Moreover, we prove a blow-up result for weak solutions with nonnegative initial energy.     

Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source

This paper deals with a mathematical model of cancer invasion of tissue. The model consists of a system of reaction-diffusion-taxis partial differential equations describing interactions between cancer cells, matrix degrading enzymes, and the host tissue. In two space dimensions, we prove global existence and uniqueness of classical solutions to this model for any μ>0 (where μ is the logistic growth rate of cancer cells). The crucial point of proof is to raise the regularity estimate of a solution from L¹(Ω) to L³(Ω×(0,T)) (where Ω⊂R² is some bounded domain and T>0 is some constant). This paper develops new estimate techniques and improves greatly our previous results [Y. Tao, M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion, Nonlinearity 21 (2008) 2221-2238] in 2 dimensions.     

Asymptotic behaviour of nonlocal reaction-diffusion equations

The existence of a global attractor in L²(Ω) is established for a reaction-diffusion equation on a bounded domain Ω in R^d with Dirichlet boundary conditions, where the reaction term contains an operator F:L²(Ω)→L²(Ω) which is nonlocal and possibly nonlinear. Existence of weak solutions is established, but uniqueness is not required. Compactness of the multivalued flow is obtained via estimates obtained from limits of Galerkin approximations. In contrast with the usual situation, these limits apply for all and not just for almost all time instants.     

Regularity of hyperbolic equations underL 2(0,T; L 2 (Γ))-Dirichlet boundary terms

With Ω an open bounded domain inR ⁿ with boundary Γ, letf(t; f ₀,f ₁;u) be the solution to a second order linear hyperbolic equation defined on Ω, under the action of the forcing termu(t) applied in the Dirichlet B.C., and with initial dataf ₀ ∈L ₂ (Ω) andf ₁ ∈H ^?1 (Ω). In a previous paper [6], we proved (among other things) that the mapu → f ? f _t , from the Dirichlet input into the solution is continuous fromL ₂(0,T; L ₂ (Γ)) intoL ₂(0,T; L ₂(Ω))?L₂ (0, T; H ^?1 (Ω)). Here, we make crucial use of this result to present the following marked improvement: the mapu → f ?f _t is continuous fromL ₂ (0, T; L ₂ (Γ)) intoC([0, T]; L ₂ (Ω))?C([0, T]; H ^?1 (Ω)). Our approach uses the cosine operator model introduced in [6], along with crucial relevant fact from cosine operator theory. A new trace theory result, on which we base our proof here, plays also a decisive role in the corresponding quadratic optimal control problem [7]. Whenu, instead, acts in the Neumann B. C. and Ω is either a sphere or a parallelepiped, the same approach leads to the same improvement over results obtained in [6] to the regularity int of the solution (i.e., fromL ₂ (0, T) toC[0, T]).     

Boundary trace embedding theorems for variable exponent Sobolev spaces

We study boundary trace embedding theorems for variable exponent Sobolev space W1,p(⋅)(Ω). Let Ω be an open (bounded or unbounded) domain in RN satisfying strong local Lipschitz condition. Under the hypotheses that p∈L^∞(Ω), 1?infp(x)?supp(x)<N, |∇p|∈Lγ(⋅)(Ω), where γ∈L^∞(Ω) and infγ(x)>N, we prove that there is a continuous boundary trace embedding W1,p(⋅)(Ω)→Lq(⋅)(∂Ω) provided q(⋅), a measurable function on ∂Ω, satisfies condition for x∈∂Ω.     

Crank-Nicolson split least-squares procedure for nonlocal reactive flow in porous media

In this paper, we introduce a Crank-Nicolson split least-squares Galerkin finite element procedure for parabolic integro-differential equations, arising in the modeling of nonlocal reactive flows in porous media. By selecting the least-squares functional properly, the procedure can be split into two independent sub-procedures, one of which is for the primitive unknown and the other is for the flux. By carefully choosing projections, we get optimal order H ¹(Ω) and L ²(Ω) norm error estimates for u and sub-optimal (L ²(Ω)) ^d norm error estimate for σ with second-order accuracy in time increment. The numerical examples are given to testify the efficiency of the introduced scheme.     

A basic inequality for the Stokes operator related to the Navier boundary condition

We show that ‖Au+Δu_‖L²(Ωε)?C(ε‖∇u_‖L²(Ωε)+‖u_‖L²(Ωε)), where Ωε is a thin domain in R³ of depth ε, the vector field u belongs to the domain of A, which is the Stokes operator for divergence-free vector fields on Ωε satisfying the Navier boundary condition.     

On the Navier problem for the stationary Navier-Stokes equations

The Navier problem is to find a solution of the steady-state Navier-Stokes equations such that the normal component of the velocity and a linear combination of the tangential components of the velocity and the traction assume prescribed value a and s at the boundary. If Ω is exterior it is required that the velocity converges to an assigned constant vector u₀ at infinity. We prove that a solution exists in a bounded domain provided ‖a_‖L²(∂Ω) is less than a computable positive constant and is unique if ‖a_‖W1/2,2(∂Ω)+‖s_‖L²(∂Ω) is suitably small. As far as exterior domains are concerned, we show that a solution exists if ‖a_‖L²(∂Ω)+‖a−u₀⋅n_‖L²(∂Ω) is small.     

A global Lipschitz continuity result for a domain-dependent Neumann eigenvalue problem for the Laplace operator

Let Ω be an open connected subset of Rⁿ of finite measure for which the Poincaré-Wirtinger inequality holds. We consider the Neumann eigenvalue problem for the Laplace operator in the open subset φ(Ω) of Rⁿ, where φ is a locally Lipschitz continuous homeomorphism of Ω onto φ(Ω). Then, we show Lipschitz-type inequalities for the reciprocals of the eigenvalues delivered by the Rayleigh quotient. Then, we further assume that the imbedding of the Sobolev space W^1,2(Ω) into the space L²(Ω) is compact, and we prove the same type of inequalities for the projections onto the eigenspaces upon variation of φ.     

Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes

A general class of nonconforming meshes has been recently studied for stationary anisotropic heterogeneous diffusion problems, see Eymard et al. (IMA J. Numer. Anal. 30 (2010), 1009–1043). Thanks to the basic ideas developed in the stated reference for stationary problems, we derive a new discretization scheme in order to approximate the nonstationary heat problem. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. We derive error estimates in discrete norms L ^∞(0, T;H ₀ ¹ (Ω)) and W ^1,∞(0, T;L ²(Ω)), and an error estimate for an approximation of the gradient, in a general framework in which the discrete bilinear form involved in the finite volume scheme satisfies some ellipticity condition.     

Hypercyclicity and mixing for cosine operator functions generated by second order partial differential operators

For Ω⊂Rd open, we characterize when cosine operator functions generated by second order partial differential operators on Lp(Ω,μ) and C0,ρ(Ω), respectively, are hypercyclic and prove that this happens if and only if they are weakly mixing. In the case of d=1 we give an easy to check characterization of when this happens. Moreover, mixing of these cosine operator functions is also characterized.     

An indefinite Laplacian on a rectangle

In this note, we investigate the nonelliptic differential expression

$$A = - div\operatorname{sgn} \nabla $$

on a rectangular domain Ω in the plane. The seemingly simple problem of associating a self-adjoint operator with the differential expression A in L²(Ω) is solved here. Such indefinite Laplacians arise in mathematical models of metamaterials characterized by negative electric permittivity and/or negative magnetic permeability.