Estimates In Sobolev Norms Research Articles

Local error estimates and post processing for the Galerkin BEM on polygons

In this paper we give local error estimates in Sobolev norms for the Galerkin method applied to strongly elliptic pseudodifferential equations on a polygon. By using the K-operator, an operator which averages the values of the Galerkin solution, we construct improved approximations.

Well‐posedness and decay for 3D non‐isothermal Navier–Stokes–Allen–Cahn system in ℝ3

In this paper, we investigate the local existence, global existence and time decay estimates for the nonisentropic Navier–Stokes–Allen–Cahn system in . This system describes the flow of a two‐phase immiscible heat‐conducting viscous incompressible mixture. In order to overcome the difficulties caused by the derivatives of double‐well potential and the nonlinear terms, we rewrite the Cahn–Hilliard part of the system as a new equivalent equation with “good” linear principle parts. Then, on the basis of the higher order norm estimates of solutions and the mollifier technique, we obtain the local well‐posedness of strong solutions. Moreover, by using pure energy method and standard continuity argument together with negative Sobolev norm estimates, one proves the global well‐posedness and time decay estimates provided that the initial data are sufficiently small.

A class of three dimensional Cahn-Hilliard equation with nonlinear diffusion

In this paper, we study the well-posedness and asymptotic behavior for a class of Cahn-Hilliard equation with nonlinear diffusion in R3. In order to overcome the difficulties caused by the derivatives of multi-well potential and the nonlinear terms, we “borrow” a linear principle part from the derivatives of multi-well potential, rewrite the equation as an equivalent equation with linear principle part. Then, on the basis of the higher order norm estimates of solutions and the mollifier technique, we obtain the local well-posedness of strong solutions. Moreover, by using pure energy method, standard continuity argument together with negative Sobolev norm estimates, one proves the global well-posedness and time decay estimates provided that the H4-norm of initial data is sufficiently small. In the end, based on parabolic interpolation inequality, bootstrap argument and some weighted estimates, we also establish the space-time decay estimates of strong solutions.

Global well-posedness and large time behavior of epitaxy thin film growth model

We consider the global well-posedness and large time behavior of solutions for epitaxy thin film growth model in mathbb{R}^{d} with the dimensional dgeq 3. First, using the pure energy method and a standard continuity argument, we prove that there exists a unique global strong solution under the condition that the initial data is sufficiently small. Moreover, we also establish the suitable negative Sobolev norm estimates and obtain the optimal decay rates of the higher-order spatial derivatives of the strong solution.

Computability of magnetic Schrödinger and Hartree equations on unbounded domains

AbstractWe study the computability of global solutions to linear Schrödinger equations with magnetic fields and the Hartree equation on . We show that the solution can always be globally computed with error control on the entire space if there exist a priori decay estimates in generalized Sobolev norms on the initial state. Using weighted Sobolev norm estimates, we show that the solution can be computed with uniform computational runtime with respect to initial states and potentials. We finally study applications in optimal control theory and provide numerical examples.

A priori estimates for the free boundary problem of incompressible inviscid Boussinesq and MHD-Boussinesq equations without heat diffusion

<abstract><p>For all physical spatial dimensions $ n = 2 $ and $ 3 $, we establish a priori estimates of Sobolev norms for free boundary problem of inviscid Boussinesq and MHD-Boussinesq equations without heat diffusion under the Taylor-type sign condition on the initial free boundary. It is different from MHD equations because the energy of the system is not conserved.</p></abstract>

A Class of Sixth Order Viscous Cahn-Hilliard Equation with Willmore Regularization in ℝ3

The main purpose of this paper is to study the Cauchy problem of sixth order viscous Cahn–Hilliard equation with Willmore regularization. Because of the existence of the nonlinear Willmore regularization and complex structures, it is difficult to obtain the suitable a priori estimates in order to prove the well-posedness results, and the large time behavior of solutions cannot be shown using the usual Fourier splitting method. In order to overcome the above two difficulties, we borrow a fourth-order linear term and a second-order linear term from the related term, rewrite the equation in a new form, and introduce the negative Sobolev norm estimates. Subsequently, we investigate the local well-posedness, global well-posedness, and decay rate of strong solutions for the Cauchy problem of such an equation in R3, respectively.

On well-posedness and large time behavior for smectic-A liquid crystals equations in $$\mathbb {R}^3$$

The main purpose of this manuscript is to study the well-posedness and decay estimates for strong solutions to the Cauchy problem of 3D smectic-A liquid crystals equations. First, applying Banach fixed point theorem, we prove the local existence and uniqueness of strong solutions. Then, by establishing some nontrivial estimates with energy method and a standard continuity argument, we prove that there exists a unique global strong solution provided that the initial data are sufficiently small. Moreover, we also establish the suitable negative Sobolev norm estimates and obtain the optimal decay rates of the higher-order spatial derivatives of the strong solutions.

Sobolev Mapping of Some Holomorphic Projections

Sobolev irregularity of the Bergman projection on a family of domains containing the Hartogs triangle is shown. On the Hartogs triangle itself, a sub-Bergman projection is shown to satisfy better Sobolev norm estimates than its Bergman projection.

On the Cauchy problem of a sixth-order Cahn–Hilliard equation arising in oil-water-surfactant mixtures

We study the well-posedness and asymptotic behavior of solutions to the Cauchy problem of a three-dimensional sixth-order Cahn–Hilliard equation arising in oil-water-surfactant mixtures. First, by using the pure energy method and a standard continuity argument, we prove that there exists a unique global strong solution provided that the H 2 -norm of the initial data is sufficiently small. Moreover, we establish suitable negative Sobolev norm estimates and obtain the optimal decay rates of the higher-order spatial derivatives of the strong solution.

Justification of the NLS Approximation for the Euler–Poisson Equation

The nonlinear Schrodinger (NLS) equation can be derived as a formal approximation equation describing the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the ion Euler–Poisson equation. In this paper, we rigorously justify such approximation by giving error estimates in Sobolev norms between exact solutions of the ion Euler–Poisson system and the formal approximation obtained via the NLS equation. The justification consists of several difficulties such as the resonances and loss of regularity, due to the quasilinearity of the problem. These difficulties are overcome by introducing normal form transformation and cutoff functions and carefully constructed energy functional of the equation.

Existence of long time solutions and validity of the nonlinear Schrödinger approximation for a quasilinear dispersive equation

We consider a nonlinear dispersive equation with a quasilinear quadratic term. We establish two results. First, we show that solutions to this equation with initial data of order O(ε) in Sobolev norms exist for a time span of order O(ε−2) for sufficiently small ε. Secondly, we derive the Nonlinear Schrödinger (NLS) equation as a formal approximation equation describing slow spatial and temporal modulations of the envelope of an underlying carrier wave, and justify this approximation with the help of error estimates in Sobolev norms between exact solutions of the quasilinear equation and the formal approximation obtained via the NLS equation.The proofs of both results rely on estimates of appropriate energies whose constructions are inspired by the method of normal-form transforms. To justify the NLS approximation, we have to overcome additional difficulties caused by the occurrence of resonances. We expect that the method developed in the present paper will also allow to prove the validity of the NLS approximation for a larger class of quasilinear dispersive systems with resonances.

Justification of the Nonlinear Schrödinger Approximation for a Quasilinear Klein–Gordon Equation

We consider a nonlinear Klein-Gordon equation with a quasilinear quadratic term. The Nonlinear Schr\"odinger (NLS) equation can be derived as a formal approximation equation describing the evolution of the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the quasilinear Klein-Gordon equation. It is the purpose of this paper to present a method which allows one to prove error estimates in Sobolev norms between exact solutions of the quasilinear Klein-Gordon equation and the formal approximation obtained via the NLS equation. The paper contains the first validity proof of the NLS approximation of a nonlinear hyperbolic equation with a quasilinear quadratic term by error estimates in Sobolev spaces. We expect that the method developed in the present paper will allow an answer to the relevant question of the validity of the NLS approximation for other quasilinear hyperbolic systems.

A priori estimates for the free boundary problem of incompressible neo-Hookean elastodynamics

A free boundary problem for the incompressible neo-Hookean elastodynamics is studied in two and three spatial dimensions. The a priori estimates in Sobolev norms of solutions with the physical vacuum condition are established through a geometrical point of view of Christodoulou and Lindblad (2000) [3]. Some estimates on the second fundamental form and velocity of the free surface are also obtained.

A Priori Estimates for Free Boundary Problem of Incompressible Inviscid Magnetohydrodynamic Flows

In the present paper, we prove the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid MHD equations in all physical spatial dimensions $n=2$ and 3 by adopting a geometrical point of view used in Christodoulou-Lindblad CPAM 2000, and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition that the outer normal derivative of the total pressure including the fluid and magnetic pressures is negative on the free boundary, which is similar to the physical condition (Taylor sign condition) for the incompressible Euler equations of fluids.

Sobolev norm estimates of solutions for the sublinear Emden-Fowler equation

We study the sublinear Emden-Fowler equation in small domains. As the domain becomes smaller, so does any solution. We investigate the convergence rate of the Sobolev norm of solutions as the volume of the domain converges to zero. The result is obtained by estimating the first eigenvalue of the Laplacian with the help of the variational method.

Mapping techniques for isogeometric analysis of elliptic boundary value problems containing singularities

The Method of Auxiliary Mapping (MAM), introduced by Babuška and Oh [2], is an effective method for dealing with singularities in elasticity [16]. MAM was extended to boundary element method (BEM) in the framework of mesh free particle methods [17], reproducing polynomial particle methods [20], and also to infinite domain problems [18].Similarly, we consider NURBS geometrical mappings that are able to generate crack singularities for isogeometric analysis of elliptic boundary value problems. However, the mapping techniques proposed in this paper are different from MAM. In order to generate singular shape functions, MAM uses conformal mappings that locally change the physical domain, whereas the NURBS mappings used for design of engineering system are not allowed to alter the physical domain for isogeometric analysis. Moreover, unlike MAM, the proposed method makes it possible to independently control the radial and angular direction of the function to be approximated as far as the point singularities are concerned.We prove error estimates in Sobolev norms and demonstrate that the proposed mapping technique is highly effective for isogeometric analysis of elliptic boundary value problems with singularities. Mesh refinements to deal with singularities are compared with the mapping technique in the isogeometic analysis framework.

Plane wave approximation of homogeneous Helmholtz solutions

In this paper, we study the approximation of solutions of the homogeneous Helmholtz equation Δu + ω2u = 0 by linear combinations of plane waves with different directions. We combine approximation estimates for homogeneous Helmholtz solutions by generalized harmonic polynomials, obtained from Vekua’s theory, with estimates for the approximation of generalized harmonic polynomials by plane waves. The latter is the focus of this paper. We establish best approximation error estimates in Sobolev norms, which are explicit in terms of the degree of the generalized polynomial to be approximated, the domain size, and the number of plane waves used in the approximations.

On the linear problem related to the stability of uniformly rotating self-gravitating liquid

The paper is concerned with “maximal regularity” estimates of Sobolev norms of solutions of a linearized evolution problem for the perturbations of the velocity and pressure of a uniformly rotating liquid. Bibliography: 12 titles.

On the Cauchy problem of degenerate hyperbolic equations

In this paper, we study a class of degenerate hyperbolic equations and prove the existence of smooth solutions for Cauchy problems. The existence result is based on a priori estimates of Sobolev norms of solutions. Such estimates illustrate a loss of derivatives because of the degeneracy.