Oscillatory asymptotic expansion for semilinear wave equation

We study oscillatory properties of the solution to semilinear wave equation, assuming oscillatory terms in initial data have sufficiently small amplitude. The main result gives an a priori estimate of the remainder in the approximation by means of the method of geometric optics. The method of establishing this estimate is based on a combination between energy type estimates for transport equation and Sobolev embedding. Copyright © 2001 John Wiley & Sons, Ltd.     

Pointwise supercloseness of pentahedral finite elements

This article derives the weak estimate of the first type for pentahedral finite elements over uniform partitions of the domain for the Poisson equation. The estimate for the W^1,1‐seminorm of the discrete derivative Green's function is also given. Using these two estimates, we obtain the pointwise supercloseness of derivatives of the pentahedral finite element approximation and the interpolant to the true solution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

New Bounds for the Inhomogenous Burgers and the Kuramoto-Sivashinsky Equations

We give a substantially simplified proof of the near-optimal estimate on the Kuramoto-Sivashinsky equation from a previous paper of the third author, at the same time slightly improving the result. That result relied on two ingredients: a regularity estimate for capillary Burgers and an a novel priori estimate for the inhomogeneous inviscid Burgers equation, which works out that in many ways the conservative transport nonlinearity acts as a coercive term. It is the proof of the second ingredient that we substantially simplify by proving a modified Kármán-Howarth-Monin identity for solutions of the inhomogeneous inviscid Burgers equation. We show that this provides a new interpretation of recent results obtained by Golse and Perthame.     

Low regularity solutions of two fifth-order KDV type equations

The Kawahara and modified Kawahara equations are fifth-order KdV type equations that have been derived to model many physical phenomena such as gravity-capillary waves and magneto-sound propagation in plasmas. This paper establishes the local well-posedness of the initial-value problem for the Kawahara equation in H ^s (R) with s ≥ − 7/4 and the local well-posedness for the modified Kawahara equation in H ^s (R) with s ≥ − 1/4. To prove these results, we derive a fundamental estimate on dyadic blocks for the Kawahara equation through the [k; Z]_multiplier norm method of Tao [14] and use this to obtain new bilinear and trilinear estimates in suitable Bourgain spaces.     

Phragmén–Lindelöf type theorem for <Emphasis Type="Italic">m</Emphasis>-hessian equations

We derive an interior estimate for the gradient of a solution u to the m-Hessian equation with a certain right-hand side. The estimate depends on the oscillation of u and properties of the right-hand side of the equation. The proof is based on a modification of some ideas of Trudinger (1997). As a consequence of the main result, a theorem of Fragmén–Lindel?f type is obtained for solutions to the m-Hessian equations in the entire space . Bibliography: 4 titles. Translated from Problems in Mathematical Analysis 39 February, 2009, pp. 147–155.     

Fractional Crank–Nicolson–Galerkin finite element scheme for the time‐fractional nonlinear diffusion equation

This article presents a finite element scheme with Newton's method for solving the time‐fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank–Nicolson scheme based on backward Euler convolution quadrature. We discuss the existence‐uniqueness results for the fully discrete problem. A new discrete fractional Gronwall type inequality for the backward Euler convolution quadrature is established. A priori error estimate for the fully discrete problem in L²(Ω) norm is derived. Numerical results based on finite element scheme are provided to validate theoretical estimates on time‐fractional nonlinear Fisher equation and Huxley equation.     

Multilinear estimates and well‐posedness for the vortex filament fourth‐order Schrödinger equations

This paper is concerned with the initial value problem for the fourth‐order nonlinear Schrödinger type equation related to the theory of vortex filament. By deriving a fundamental estimate on dyadic blocks for the fourth‐order Schrödinger through the [k,Z]‐multiplier norm method. we establish multilinear estimates for this nonlinear fourth‐order Schrödinger type equation. The local well‐posedness for initial data in with s > 1 ∕ 2 is implied by the multilinear estimates. Copyright © 2012 John Wiley & Sons, Ltd.     

Stability and convergence of the spectral Galerkin method for the Cahn‐Hilliard equation

A spectral Galerkin method in the spatial discretization is analyzed to solve the Cahn‐Hilliard equation. Existence, uniqueness, and stabilities for both the exact solution and the approximate solution are given. Using the theory and technique of a priori estimate for the partial differential equation, we obtained the convergence of the spectral Galerkin method and the error estimate between the approximate solution u_N(t) and the exact solution u(t). © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations

The estimate of the probability of the large deviation or the statistical random field is the key to ensure the convergence of moments of the associated estimator, and it also plays an essential role to prove mathematical validity of the asymptotic expansion of the estimator. For non-linear stochastic processes, it involves technical difficulties to show a standard exponential type estimate; besides, it is not necessary for these purposes. In this paper, we propose a polynomial-type large deviation inequality which is easily verified by the L ^p -boundedness of certain functionals; usually they are simple additive functionals. We treat a statistical random field with multi-grades and discuss M and Bayesian type estimators. As an application, we show the behavior of those estimators, including convergence of moments, for the statistical random field in the quasi-likelihood analysis of the stochastic differential equation that is possibly multi-dimensional and non-linear. The results are new even for stochastic differential equations, while they obviously apply to other various statistical models.     

A local regularity of the complex Monge–Ampère equation

We prove a local regularity (and a corresponding a priori estimate) for plurisubharmonic solutions of the nondegenerate complex Monge–Ampère equation assuming that their W ^{2, p} -norm is under control for some p > n(n − 1). This condition is optimal. We use in particular some methods developed by Trudinger and an estimate for the complex Monge–Ampère equation due to Kołodziej.     

Optimal Error Estimate of the Penalty Finite Element Method for the Micropolar Fluid Equations

We present an optimal error estimate of the numerical velocity, pressure, and angular velocity for the fully discrete penalty finite element method of the micropolar equations when the parameters ?, Δ t, and h are sufficiently small. In order to obtain this estimate, we present the time discretization of the penalty micropolar equation that is based on the backward Euler scheme; the spatial discretization of the time discretized penalty micropolar equation is based on a finite elements space pair (X _h , M _h ) that satisfies some approximations properties.     

Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I

We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solution of an ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of Wiener processes in probability in the metric of space C [0, T]. As an example, we consider an ordinary differential equation perturbed by a rapidly oscillating centered process that is a 1-periodic function of a solution of a time-homogeneous stochastic differential equation with 1-periodic coefficients. We obtain an estimate for the rate of approach of a solution of this equation to a solution of the corresponding It? stochastic equation.     

Stability of Cauchy-type Singular Integral Equations over an Interval

About the Cauchy type of singular integral equations over an interval, the paper presents three main results: the unified normalization form, the stability property on the smooth function space C^m,, and the pointwise error estimate of the solution for their perturbed equations. As the applications of the main results, the approximate methods for the equation are mentioned. Mathematics Subject Classification (2000) 45E05.     

Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients

In this paper we extend the results of Caffarelli, Jerison, and Kenig [Ann. of Math. (2) 155 (2002)] and Caffarelli and Kenig [Amer. J. Math. 120 (1998)] by establishing an almost monotonicity estimate for pairs of continuous functions satisfying in an infinite strip (global version) or a finite parabolic cylinder (localized version), where ${\cal L}$ is a uniformly parabolic operator with double Dini continuous ${\cal A}$ and uniformly bounded b and c. We also prove the elliptic counterpart of this estimate. This closes the gap between the known conditions in the literature (both in the elliptic and parabolic case) imposed on u_± in order to obtain an almost monotonicity estimate. At the end of the paper, we demonstrate how to use this new almost monotonicity formula to prove the optimal C^1,1‐regularity in a fairly general class of quasi‐linear obstacle‐type free boundary problems. © 2010 Wiley Periodicals, Inc.     

An estimate of the deformation of a closed convex surface with a change in its curvature

We consider closed convex surfaces ℱ of the space R³ containing a fixed point 0 in the interior. A central projection from 0 enables us to transfer the curvature ω(u) of the surface ℱ, regarded as a function of a set uɛℱ, onto a sphere with center 0. A. D. Aleksandrov established the fact that the surface ℱ is determined (moreover, uniquely) to·within a homothetic transformation with center 0 by prescribing the curvature transferred in this way onto the sphere. In this paper we give an estimate of the variation of the distances τ _F (B) of points of the surface from 0 as a function of the variation of the curvature transferred onto the sphere. The derivation of this estimate relies substantially on nondegeneracy of the surface ℱ; as a measure of nondegeneracy we take the ratio R/ζ, of the radii ℱ of balls with center ℱ, circumscribed and inscribed, respectively, about 0. Also, in this paper, we introduce and study those characteristics ℒ _F and τ _F of the curvature of the surface ℱ, which make it possible to estimate R/ζ from above and, by the same token, to obtain an estimate of how τ _F (B) varies in terms only of the curvature of the surface and its variation. An analytical treatment shows that basically our result yields an estimate of the maximum of the modulus of the change in the solution of a Monge—Ampere type equation on a sphere in terms of the change in its right-hand side in some integral norm, while the estimate of R/ζ, yields an a priori estimate of the modulus of the solution of this equation. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 45, pp. 83–110, 1974.     

On a new generalization of dispersionless Kadomtsev–Petviashvili hierarchy and its reductions

The dispersionless Kadomtsev–Petviashvili hierarchy is generalized by introducing two new time series γ_n and σ_k with two parameters η_n and λ_k. By this hierarchy, we obtain the first type, the second type as well as mixed type of dispersionless Kadomtsev–Petviashvili equation with self‐consistent sources and their related conservation equations. In addition, the reduction and constrained flow of this new hierarchy are studied. The first type, the second type and the mixed type of dispersionless Korteweg–de Vries equation with self‐consistent sources and of dispersionless Boussinesq equation with self‐consistent sources are obtained. Copyright © 2013 John Wiley & Sons, Ltd.     

Modulation equations and parabolic limits of reaction random-walk systems

Reaction random-walk systems are hyperbolic models to describe spatial motion (in one dimension) with finite speed and reactions of particles. Here we present two approaches which relate reaction random-walk equations with reaction diffusion equations. First, we consider the case of high particle speeds (parabolic limit). This leads to a singular perturbation analysis of a semilinear damped wave equation. A initial layer estimate is given. Secondly, we consider the case of a transcritical bifurcation. We use techniques similar to that of the Ginzburg–Landau method to find a modulation equation for the amplitude of the first unstable mode. It turns out that the modulation equation is Fisher's equation, hence near the bifurcation point travelling wave solutions are obtained. The approximation result and the corresponding estimate is given in terms of the bifurcation parameter. Both results are based on an a priori estimate for classical solutions which follows from explicit representations of the solution of the linear telegraph equation. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

The conjugate heat equation and Ancient solutions of the Ricci flow

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of Type I κ-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman?s previous result on backward limits of κ-solutions in dimension 3, in which case the curvature operator is nonnegative (it follows from Hamilton–Ivey curvature pinching estimate). As an application, this also addresses an issue left in Naber (2010) [23], where Naber proves the interesting result that there exists a Type I dilation limit that converges to a gradient shrinking Ricci soliton, but that soliton might be flat. The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.     

Quaternionic Monge-Ampère equation and Calabi problem for HKT-manifolds

A quaternionic version of the Calabi problem on the Monge-Ampère equation is introduced, namely a quaternionic Monge-Ampère equation on a compact hypercomplex manifold with an HKT-metric. The equation is non-linear elliptic of second order. For a hypercomplex manifold with holonomy in SL(n,ℍ), uniqueness (up to a constant) of a solution is proven, aas well as the zero order a priori estimate. The existence of a solution is conjectured, similar to the Calabi-Yau theorem. We reformulate this quaternionic equation as a special case of the complex Hessian equation, making sense on any complex manifold.     

Extremal solutions of boundary value problems

To prove the existence of a solution of a two-point boundary value problem for an nth-order operator equation by the a priori estimate method, we study extremal solutions of auxiliary boundary value problems for an nth-order differential equation with simplest right-hand side, which have a unique solution under certain restrictions on the boundary conditions.