Periodic solutions of the Navier–Stokes equations in a perturbed half‐space and an aperture domain

We shall construct a periodic strong solution of the Navier–Stokes equations for some periodic external force in a perturbed half‐space and an aperture domain of the dimension n?3. Our proof is based on L^p–L^q estimates of the Stokes semigroup. We apply L^p–L^q estimates to the integral equation which is transformed from the original equation. As a result, we obtain the existence and uniqueness of periodic strong solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Evolution equations for shapes and geometries

The first object of this paper is to introduce a new evolution equation for the characteristic function of the boundary Γ of a Lipschitzian domain Ω in the N-dimensional Euclidean space under the influence of a smooth time-dependent velocity field. The originality of this equation is that the evolution takes place in an L^p-space with respect to the (N − 1)-Hausdorff measure. A second more speculative objective is to discuss how that equation can be relaxed to rougher velocity fields via some weak formulation. A candidate is presented and some of the technical difficulties and open issues are discussed. Continuity results in several metric topologies are also presented. The paper also specializes the results on the evolution of the oriented distance function to initial sets with zero N-dimensional Lebesgue measure.     

Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right‐hand side in Lp (p⩾1)

Accurate modelling of heat transfer in high‐temperature situations requires accounting for the effect of heat radiation. In complex industrial applications involving dissipative heating, we hardly can expect from the mathematical theory that the heat sources will be in a better space than L¹. In this paper, we focus on a stationary heat equation with nonlocal boundary conditions and L^p right‐hand side, with p?1 being arbitrary. Thanks to new coercivity results, we are able to produce energy estimates that involve only the L^p norm of the heat sources and to prove the existence of weak solutions. Copyright © 2008 John Wiley & Sons, Ltd.     

A compatible‐incompatible decomposition of symmetric tensors in Lp with application to elasticity

In this paper, we prove the Saint‐Venant compatibility conditions in L^p for p∈(1,+∞), in a simply connected domain of any space dimension. As a consequence, alternative, simple, and direct proofs of some classical Korn inequalities in L^p are provided. We also use the Helmholtz decomposition in L^p to show that every symmetric tensor in a smooth domain can be decomposed in a compatible part, which is the symmetric part of a displacement gradient, and in an incompatible part, which is the incompatibility of a certain divergence‐free tensor. Moreover, under a suitable Dirichlet boundary condition, this Beltrami‐type decomposition is proved to be unique. This decomposition result has several applications, one of which being in dislocation models, where the incompatibility part is related to the dislocation density and where 1 < p < 2. This justifies the need to generalize and prove these rather classical results in the Hilbertian case (p = 2), to the full range p∈(1,+∞). Copyright © 2015 John Wiley & Sons, Ltd.     

A posteriori error estimates for finite element approximations to the wave equation with discontinuous coefficients

We derive residual‐based a posteriori error estimates of finite element method for linear wave equation with discontinuous coefficients in a two‐dimensional convex polygonal domain. A posteriori error estimates for both the space‐discrete case and for implicit fully discrete scheme are discussed in L^∞(L²) norm. The main ingredients used in deriving a posteriori estimates are new Clément type interpolation estimates in conjunction with appropriate adaption of the elliptic reconstruction technique of continuous and discrete solutions. We use only an energy argument to establish a posteriori error estimates with optimal order convergence in the L^∞(L²) norm.     

Global weak solution to the flow of liquid crystals system

In this paper, we study a simplified system for the flow of nematic liquid crystals in a bounded domain in the three‐dimensional space. We derive the basic energy law which enables us to prove the global existence of the weak solutions under the condition that the initial density belongs to L^γ(Ω) for any $\gamma >\frac{3}{2}$. Especially, we also obtain that the weak solutions satisfy the energy inequality in integral or differential form. Copyright © 2009 John Wiley & Sons, Ltd.     

Estimates of difference norms for functions in anisotropic Sobolev spaces

We investigate the spaces of functions on ?ⁿ for which the generalized partial derivatives Dequation/tex2gif-sup-2.gif_kf exist and belong to different Lorentz spaces Lequation/tex2gif-sup-3.gif . For the functions in these spaces, the sharp estimates of the Besov type norms are found. The methods used in the paper are based on estimates of non‐increasing rearrangements. These methods enable us to cover also the case when some of the p_k's are equal to 1. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weak solutions to a time-dependent heat equation with nonlocal radiation boundary condition and arbitrary <Emphasis Type="Italic">p</Emphasis>-summable right-hand side

We consider a model for transient conductive-radiative heat transfer in grey materials. Since the domain contains an enclosed cavity, nonlocal radiation boundary conditions for the conductive heat-flux are taken into account. We generalize known existence and uniqueness results to the practically relevant case of lower integrable heat-sources, and of nonsmooth interfaces. We obtain energy estimates that involve only the L ^p norm of the heat sources for exponents p close to one. Such estimates are important for the investigation of models in which the heat equation is coupled to Maxwell’s equations or to the Navier-Stokes equations (dissipative heating), with many applications such as crystal growth.     

Weak solutions to a nonlinear variational wave equation and some related problems

In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and some related problems. We first introduce the main tools, the L ^p Young measure theory and related compactness results, in the first section. Then we use the L ^p Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed. In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic equation, which is also the so-called vortex density equation arising from sup-conductivity.     

Approximate identities in variable L p spaces

We give conditions for the convergence of approximate identities, both pointwise and in norm, in variable L ^p spaces. We unify and extend results due to Diening [8], Samko [18] and Sharapudinov [19]. As applications, we give criteria for smooth functions to be dense in the variable Sobolev spaces, and we give solutions of the Laplace equation and the heat equation with boundary values in the variable L ^p spaces. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spectral collocation methods for nonlinear weakly singular Volterra integro‐differential equations

In this paper, a spectral collocation approximation is proposed for neutral and nonlinear weakly singular Volterra integro‐differential equations (VIDEs) with non‐smooth solutions. We use some suitable variable transformations to change the original equation into a new equation, so that the solution of the resulting equation possesses better regularity, and the the Jacobi orthogonal polynomial theory can be applied conveniently. Under reasonable assumptions on the nonlinearity, we carry out a rigorous error analysis in L^∞ norm and weighted L² norm. To perform the numerical simulations, some test examples (linear and nonlinear) are considered with nonsmooth solutions, and numerical results are presented. Further more, the comparative study of the proposed methods with some existing numerical methods is provided.     

Sobolev Inequalities on Forms and <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Superscript>-Cohomology on Complete Riemannian Manifolds

We prove some Sobolev inequalities on differential forms over a class of complete non-compact Riemannian manifolds with suitable geometric conditions. Moreover, we establish some L ^p,q -estimates and existence theorems of the Cartan-De Rham equation and the Hodge systems. As applications, we prove some vanishing theorems of the L ^p,q -cohomology and prove the L ^q -solvability of the nonlinear p-Laplace equation on forms on complete non-compact Riemannian manifolds with suitable geometric conditions.     

The resolvent problem for the stokes system in exterior domains: An elementary approach

We consider the Stokes system with resolvent parameter in an exterior domain: under Dirichlet boundary conditions. Here Ω is a bounded domain with C² boundary, and [λ??\] ? [∞, 0], ν >0. Using the method of integral equations, we are able to construct solutions ( u , π) in L^p spaces. Our approach yields an integral representation of these solutions. By evaluating the corresponding integrals, we obtain L^p estimates that imply in particular that the Stokes operator on exterior domains generates an analytic semigroup in L^p.     

On the convergence of Fourier series of computable Lebesgue integrable functions

This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of L^p‐computable functions (computable Lebesgue integrable functions) with a size notion, by introducing L^p‐computable Baire categories. We show that L^p‐computable Baire categories satisfy the following three basic properties. Singleton sets {f } (where f is L^p‐computable) are meager, suitable infinite unions of meager sets are meager, and the whole space of L^p‐computable functions is not meager. We give an alternative characterization of meager sets via Banach‐Mazur games. We study the convergence of Fourier series for L^p‐computable functions and show that whereas for every p > 1, the Fourier series of every L^p‐computable function f converges to f in the L^p norm, the set of L¹‐computable functions whose Fourier series does not diverge almost everywhere is meager (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global solution on Cauchy problem in nonlinear non-simple thermoelastic materials

We prove a theorem about global existence (in time) of the solution to the initial-value problem for a nonliear system of coupled partial differential equations of fourth order describing the thermoelasticity of non-simple materias. We consider such the case of thim system in which some nonlinear coeffcients can depend not only on the temperature and the gradient of displacement and also on the second derivative of displacement. The corresponding global existence theorem has been proved using the L ^p – L ^q time decay estmates for the solution of the associated linearized problem. Next, we proved the energy estimate in the Sobolev space with constant independent of time. Such an energy estimate allows us to apply the standard continuation argument and to continue the local solution to one desired for all t ∈ (0, ∞)     

Adaptive FE–BE coupling for strongly nonlinear transmission problems with Coulomb friction

We analyze an adaptive finite element/boundary element procedure for scalar elastoplastic interface problems involving friction, where a nonlinear uniformly monotone operator such as the p-Laplacian is coupled to the linear Laplace equation on the exterior domain. The problem is reduced to a boundary/domain variational inequality, a discretized saddle point formulation of which is then solved using the Uzawa algorithm and adaptive mesh refinements based on a gradient recovery scheme. The Galerkin approximations are shown to converge to the unique solution of the variational problem in a suitable product of L ^p - and L ²-Sobolev spaces.     

Composite finite element approximation for nonlinear parabolic problems in nonconvex polygonal domains

We study a new class of finite elements so‐called composite finite elements (CFEs), introduced earlier by Hackbusch and Sauter, Numer. Math., 1997; 75:447‐472, for the approximation of nonlinear parabolic equation in a nonconvex polygonal domain. A two‐scale CFE discretization is used for the space discretizations, where the coarse‐scale grid discretized the domain at an appropriate distance from the boundary and the fine‐scale grid is used to resolve the boundary. A continuous, piecewise linear CFE space is employed for the spatially semidiscrete finite element approximation and the temporal discretizations is based on modified linearized backward Euler scheme. We derive almost optimal‐order convergence in space and optimal order in time for the CFE method in the L^∞(L²) norm. Numerical experiment is carried out for an L‐shaped domain to illustrate our theoretical findings.     

A probabilistic interpretation and stochastic particle approximations of the 3-dimensional Navier-Stokes equations

We develop a probabilistic interpretation of local mild solutions of the three dimensional Navier-Stokes equation in the L^p spaces, when the initial vorticity field is integrable. This is done by associating a generalized nonlinear diffusion of the McKean-Vlasov type with the solution of the corresponding vortex equation. We then construct trajectorial (chaotic) stochastic particle approximations of this nonlinear process. These results provide the first complete proof of convergence of a stochastic vortex method for the Navier-Stokes equation in three dimensions, and rectify the algorithm conjectured by Esposito and Pulvirenti in 1989. Our techniques rely on a fine regularity study of the vortex equation in the supercritical L^p spaces, and on an extension of the classic McKean-Vlasov model, which incorporates the derivative of the stochastic flow of the nonlinear process to explain the vortex stretching phenomenon proper to dimension three. Supported by Fondecyt Project 1040689 and Nucleus Millennium Information and Randomness ICM P01-005.     

The scalar Oseen operator − Δ + ∂/2202<Emphasis Type="Italic">x</Emphasis><Subscript>1</Subscript> in ℝ<Superscript>2</Superscript>

This paper solves the scalar Oseen equation, a linearized form of the Navier-Stokes equation. Because the fundamental solution has anisotropic properties, the problem is set in a Sobolev space with isotropic and anisotropic weights. We establish some existence results and regularities in L ^p theory.     

A Concentration‐Collapse Decomposition for L2 Flow Singularities

We exhibit a concentration‐collapse decomposition of singularities of fourth‐order curvature flows, including the L² curvature flow and Calabi flow, in dimensions n ≤ 4. The proof requires the development of several new a priori estimates. First, we develop a smoothing result for initial metrics with small energy and a volume growth lower bound, in the vein of Perelman's pseudo locality result. Next, we generalize our technique from prior work to exhibit local smoothing estimates for the L² flow in the presence of a curvature‐related bound. A final key ingredient is a new local ?‐regularity result for L² critical metrics with possibly nonconstant scalar curvature. Applications of these results include new compactness and diffeomorphism‐finiteness theorems for smooth compact 4‐manifolds satisfying the necessary and effectively minimal hypotheses of L² curvature pinching and a volume‐noncollapsing condition. © 2015 Wiley Periodicals, Inc.