Asymptotic regularity of solutions to Hartree–Fock equations with Coulomb potential

We study the asymptotic regularity of solutions to Hartree–Fock (HF) equations for Coulomb systems. To deal with singular Coulomb potentials, Fock operators are discussed within the calculus of pseudo‐differential operators on conical manifolds. First, the non‐self‐consistent‐field case is considered, which means that the functions that enter into the nonlinear terms are not the eigenfunctions of the Fock operator itself. We introduce asymptotic regularity conditions on the functions that build up the Fock operator, which guarantee ellipticity for the local part of the Fock operator on the open stretched cone ?₊ × S². This proves the existence of a parametrix with a corresponding smoothing remainder from which it follows, via a bootstrap argument, that the eigenfunctions of the Fock operator again satisfy asymptotic regularity conditions. Using a fixed‐point approach based on Cancès and Le Bris analysis of the level‐shifting algorithm, we show via another bootstrap argument that the corresponding self‐consistent‐field solutions to the HF equation have the same type of asymptotic regularity. Copyright © 2008 John Wiley & Sons, Ltd.     

Global regularity of the elastic fields of a power‐law model on Lipschitz domains

In this paper, we study the global regularity of the displacement and stress fields of a nonlinear elastic model of power‐law type. It is assumed that the underlying domains are Lipschitz domains which satisfy an additional geometric condition near those points, where the type of the boundary conditions changes. The proof of the global regularity result relies on a difference quotient technique. Finally, a global regularity result for the stress fields of the elastic, perfect plastic Hencky model is derived. This model appears as a limit model of the power‐law model. Copyright © 2006 John Wiley & Sons, Ltd.     

Well‐posedness of time‐space fractional stochastic evolution equations driven by α‐stable noise

This paper is devoted to the well‐posedness for time‐space fractional Ginzburg‐Landau equation and time‐space fractional Navier‐Stokes equations by α‐stable noise. The spatial regularity and the temporal regularity of the nonlocal stochastic convolution are firstly established, and then the existence and uniqueness of the global mild solution are obtained by the Banach fixed point theorem and Mittag‐Leffler functions, respectively. Numerical simulations for time‐space fractional Ginzburg‐Landau equation are provided to verify the analysis results.     

Non‐linear initial boundary value problems of hyperbolic–parabolic type. A general investigation of admissible couplings between systems of higher order. Part 2: abstract quasilinear theory

This is the second part of an article that is devoted to the theory of non‐linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2 at hand, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (3.4) using the so‐called energy method. In the above sense the regularity assumptions (A6) and (A7) about the coefficients and right‐hand sides define the admissible couplings. In part 3, we extend the results of part 2 to non‐linear initial boundary value problems. In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.     

A regularity criterion for the density‐dependent magnetohydrodynamic equations

In this paper, regularity criterion for the 3‐D density‐dependent magnetohydrodynamic equation is considered. It is proved that the solution keeps smoothness only under an integrable condition on the velocity field in multiplier spaces. Hence, it turns out that the velocity field plays a dominant role in the regularity criteria of the weak solutions to this nonlinear coupling problem even with density. Copyright © 2009 John Wiley & Sons, Ltd.     

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation,II

This paper is a continuation of our recent paper 8 . We will consider the semi‐linear Cauchy problem for wave models with scale‐invariant time‐dependent mass and dissipation and power non‐linearity. The goal is to study the interplay between the coefficients of the mass and the dissipation term to prove global existence (in time) of small data energy solutions assuming suitable regularity on the L² scale with additional L¹ regularity for the data. In order to deal with this L² regularity in the non‐linear part, we will develop and employ some tools from Harmonic Analysis.     

New asymptotic estimates of solutions for generalized Rosenau equations

This paper presents new methods widely applicable to expand solutions for wave equations with damping terms such as Rosenau‐type equations. Some of them have the diffusion structure that appears strongly in the low‐frequency region, and some detailed analysis on diffusion waves is seen in this report. In the high‐frequency region, difficulties arising from the regularity‐loss type are overcome by a new discovery of suitable asymptotic profiles and expanding techniques of solutions even if regularity assumptions on the initial data are not imposed. It is also shown that stronger regularity assumptions on the initial data give better asymptotic estimates.     

Regularity of weak solutions of Maxwell's equations with mixed boundary‐conditions

In this paper global H^s‐ and L^p‐regularity results for the stationary and transient Maxwell equations with mixed boundary conditions in a bounded spatial domain are proved. First it is shown that certain elements belonging to the fractional‐order domain of the Maxwell operator belong to H^s(Ω) for sufficiently small s > 0. It follows from this regularity result that H^s(Ω) is an invariant subspace of the unitary group corresponding to the homogeneous Maxwell equations with mixed boundary conditions. In the case that a possibly non‐linear conductivity is present a L^p‐regularity theorem for the transient equations is proved. Copyright © 1999 John Wiley & Sons, Ltd.     

Minimal regularity of the solutions of some transmission problems

We consider some transmission problems for the Laplace operator in two‐dimensional domains. Our goal is to give minimal regularity of the solutions, better than H¹, with or without conditions on the (positive) material constants. Under a monotonicity or quasi‐monotonicity condition on the constants (or on the inverses according to the boundary conditions), we study the behaviour of the solution near vertex and near interior nodes and show in each case that the given regularity is sharp. Without condition we prove that the regularity near a corner is of the form H^1+ρ, where ρ is a given bound depending on the material constants. Numerical examples are presented which confirm the sharpness of our lower bounds. Copyright © 2003 John Wiley & Sons, Ltd.     

Regularity inheritance in pseudorandom graphs

Advancing the sparse regularity method, we prove one‐sided and two‐sided regularity inheritance lemmas for subgraphs of bijumbled graphs, improving on results of Conlon, Fox, and Zhao. These inheritance lemmas also imply improved H‐counting lemmas for subgraphs of bijumbled graphs, for some H.     

Uniform convergence of Galerkin‐multigrid methods for nonconforming finite elements for second‐order problems with less than full elliptic regularity

In this article we prove uniform convergence estimates for the recently developed Galerkin‐multigrid methods for nonconforming finite elements for second‐order problems with less than full elliptic regularity. These multigrid methods are defined in terms of the “Galerkin approach,” where quadratic forms over coarse grids are constructed using the quadratic form on the finest grid and iterated coarse‐to‐fine intergrid transfer operators. Previously, uniform estimates were obtained for problems with full elliptic regularity, whereas these estimates are derived with less than full elliptic regularity here. Applications to the nonconforming P₁, rotated Q₁, and Wilson finite elements are analyzed. The result applies to the mixed method based on finite elements that are equivalent to these nonconforming elements. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 203–217, 2002; DOI 10.1002/num.10004     

Non‐linear initial boundary value problems of hyperbolic–parabolic type. A general investigation of admissible couplings between systems of higher order. Part 3: applications

This is the third part of an article that is devoted to the theory of non‐linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to quasilinear initial boundary value problems using the so‐called energy method. In the above sense the regularity assumptions about the coefficients and right‐hand sides define the admissible couplings. In part 3 at hand, we extend the results of part 2 to the nonlinear initial boundary value problem (4.2). In particular, assumptions (B8) and (B9) about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit assumptions (B8) and (B9) for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Well‐posedness of a parabolic free boundary problem driven by diffusion and surface tension

Well‐posedness and regularity results are shown for a class of free boundary problems consisting of diffusion on a free domain where the boundary movement depends on its mean curvature of the boundary and the diffusion on the boundary, and initial conditions are radially symmetric. Short‐time existence and uniqueness of solutions in a suitable Sobolev space are shown using a fixed‐point argument. Higher regularity is a posteriori. Finally, it is shown that solutions exist globally in time and converge to equilibrium if the boundary movement depends on the mean curvature of the boundary and diffusion in a specific way. A mathematical model describing the swelling of a cell due to osmosis is treated as an example. Copyright © 2014 John Wiley & Sons, Ltd.     

Regularity criterion on energy equality for compressible Cahn‐Hilliard‐Navier‐Stokes equations

This paper is concerned with the compressible Cahn‐Hilliard‐Navier‐Stokes system. We establish a sufficient regularity condition such that every weak solution conserves its energy equality for every t > 0. Our approach is based on the commutator and mollification approximation.     

Non‐linear initial boundary value problemsof hyperbolic–parabolic type. A general investigationof admissible couplings between systems of higher order. Part 1: linear theory

In this article (which is divided in three parts) we investigate the non‐linear initial boundary value problems (1.2) and (1.3). In both cases we consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1 at hand, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (1.2) using the so‐called energy method. In the above sense, the regularity assumptions about the coefficients and right‐hand sides define the admissible couplings. In part 3, we extend the results of part 2 to the non‐linear initial boundary value problem (1.3). In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.     

On Monge‐Ampère equations with homogenous right‐hand sides

We study the regularity and behavior at the origin of solutions to the two‐dimensional degenerate Monge‐Ampère equation det D²_u = |x|^α with α > ?2. We show that when α > 0, solutions admit only two possible behaviors near the origin, radial and nonradial, which in turn implies C^{2, δ}‐regularity. We also show that the radial behavior is unstable. For α < 0 we prove that solutions admit only the radial behavior near the origin. © 2008 Wiley Periodicals, Inc.     

Loss of regularity for the solutions to hyperbolic equations with non‐regular coefficients—an application to Kirchhoff equation

We consider the Cauchy problem for second‐order strictly hyperbolic equations with time‐depending non‐regular coefficients. There is a possibility that singular coefficients make a regularity loss for the solution. The main purpose of this paper is to derive an optimal singularity for the coefficient that the Cauchy problem is C^∞ well‐posed. Moreover, we will apply such a result to the estimate of the existence time of the solution for Kirchhoff equation. Copyright © 2003 John Wiley & Sons, Ltd.     

New global a priori estimates for the third‐grade fluid equations

This note bridges the gap between the existence and regularity classes for the third‐grade Rivlin–Ericksen fluid equations. We obtain a new global a priori estimate, which conveys the precise regularity conditions that lead to the existence of a global in time regular solution. Copyright © 2006 John Wiley & Sons, Ltd.     

Regularity theory for fully nonlinear integro‐differential equations

We consider nonlinear integro‐differential equations like the ones that arise from stochastic control problems with purely jump Lévy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C^{1, α} regularity for general fully nonlinear integro‐differential equations. Our estimates remain uniform as the degree of the equation approaches 2, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations. © 2008 Wiley Periodicals, Inc.     

Partial regularity of suitable weak solutions to the four‐dimensional incompressible magneto‐hydrodynamic equations

In this paper, we study the partial regularity of suitable weak solutions to the incompressible magneto‐hydrodynamic equations in dimension four by borrowing and improving the arguments given by Caffarelli, Kohn, and Nirenberg for incompressible Navier–Stokes equations. The so‐called ε‐regularity criteria are established for suitable weak solutions. As an application, an estimate on Hausdorff dimension of the possible singular points set for a suitable weak solution is given. Finally, we present further information on distribution of the possible singular points if the given initial data decay sufficiently rapidly or are not too singular at the origin, in some sense. Copyright © 2012 John Wiley & Sons, Ltd.