An Explicit Solution with Correctors for the Green–Naghdi Equations

In this paper, the water waves problem for uneven bottoms in a highly nonlinear regime is studied. It is well known that, for such regimes, a generalization of the Boussinesq equations called the Green–Naghdi equations can be derived and justified when the bottom is variable (Lannes and Bonneton in Phys Fluids 21, 2009). Moreover, the Green–Naghdi and Boussinesq equations are fully nonlinear and dispersive systems. We derive here new linear asymptotic models of the Green–Naghdi and Boussinesq equations so that they have the same accuracy as the standard equations. We solve explicitly the new linear models and numerically validate the results.     

Low‐curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes

We consider a class of fourth‐order nonlinear diffusion equations motivated by Tumblin and Turk's “low‐curvature image simplifiers” for image denoising and segmentation. The PDE for the image intensity u is of the form where g(s) = k²/(k² + s²) is a “curvature” threshold and λ denotes a fidelity‐matching parameter. We derive a priori bounds for Δu that allow us to prove global regularity of smooth solutions in one space dimension, and a geometric constraint for finite‐time singularities from smooth initial data in two space dimensions. This is in sharp contrast to the second‐order Perona‐Malik equation (an ill‐posed problem), on which the original LCIS method is modeled. The estimates also allow us to design a finite difference scheme that satisfies discrete versions of the estimates, in particular, a priori bounds on the smoothness estimator in both one and two space dimensions. We present computational results that show the effectiveness of such algorithms. Our results are connected to recent results for fourth‐order lubrication‐type equations and the design of positivity‐preserving schemes for such equations. This connection also has relevance for other related fourth‐order imaging equations. © 2004 Wiley Periodicals, Inc.     

Uniqueness of Radial Solutions for the Fractional Laplacian

We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian (?Δ)^s with s ? (0,1) for any space dimensions N ≥ 1. By extending a monotonicity formula found by Cabré and Sire , we show that the linear equation has at most one radial and bounded solution vanishing at infinity, provided that the potential V is radial and nondecreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schrödinger operator H = (?Δ)^s + V are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half‐space , we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation for arbitrary space dimensions N ≥ 1 and all admissible exponents α > 0. This generalizes the nondegeneracy and uniqueness result for dimension N = 1 recently obtained by the first two authors and, in particular, the uniqueness result for solitary waves of the Benjamin‐Ono equation found by Amick and Toland .© 2016 Wiley Periodicals, Inc.     

Blowup in a three‐dimensional vector model for the Euler equations

We present a three‐dimensional vector model given in terms of an infinite system of nonlinearly coupled ordinary differential equations. This model has structural similarities with the Euler equations for incompressible, inviscid fluid flows. It mimics certain important properties of the Euler equations, namely, conservation of energy and divergence‐free velocity. It is proven for certain families of initial data that the model system permits local existence in time for initial conditions in Sobolev spaces H^s, s > ; and blowup occurs in the sense that the H^{3/2 + ?} norm becomes unbounded in finite time. © 2004 Wiley Periodicals, Inc.     

Cauchy problem for quasilinear hyperbolic systems of shallow water equations

This article investigates the Cauchy problem for two different models (modified and classical), governed by quasilinear hyperbolic systems that arise in shallow water theory. Under certain reasonable hypotheses on the initial data, we obtain the global smooth solutions for both the systems. The bounds on simple wave solutions of the modified system are shown to depend on the parameter H characterizing the advective transport of impulse. Similarly the bounds on simple wave solutions of the classical system describing the flow over a sloping bottom with profile b(x) are shown to depend on the bottom topography. On the other hand, if the initial data are specified differently, then it is shown that solutions for both the systems exhibit finite time blow-up from specific smooth initial data. Moreover, we show that an increase in H and convexity of b would reduce the time taken for the solutions to blow up.     

Long‐time dynamics of KdV solitary waves over a variable bottom

We study the variable‐bottom, generalized Korteweg—de Vries (bKdV) equation ?_tu = ??_x(?u + f(u) ? b(t,x)u), where f is a nonlinearity and b is a small, bounded, and slowly varying function related to the varying depth of a channel of water. Many variable‐coefficient KdV‐type equations, including the variable‐coefficient, variable‐bottom KdV equation, can be rescaled into the bKdV. We study the long‐time behavior of solutions with initial conditions close to a stable, b = 0 solitary wave. We prove that for long time intervals, such solutions have the form of the solitary wave whose center and scale evolve according to a certain dynamical law involving the function b(t,x) plus an H¹(?)‐small fluctuation. © 2005 Wiley Periodicals, Inc.     

Decay property of Timoshenko system in thermoelasticity

We investigate the decay property of a Timoshenko system of thermoelasticity in the whole space for both Fourier and Cattaneo laws of heat conduction. We point out that although the paradox of infinite propagation speed inherent in the Fourier law is removed by changing to the Cattaneo law, the latter always leads to a solution with the decay property of the regularity‐loss type. The main tool used to prove our results is the energy method in the Fourier space together with some integral estimates. We derive L² decay estimates of solutions and observe that for the Fourier law the decay structure of solutions is of the regularity‐loss type if the wave speeds of the first and the second equations in the system are different. For the Cattaneo law, decay property of the regularity‐loss type occurs no matter what the wave speeds are. In addition, by restricting the initial data to with a suitably large s and γ ∈ [0,1], we can derive faster decay estimates with the decay rate improvement by a factor of t^?γ/2. Copyright © 2011 John Wiley & Sons, Ltd.     

On the Cauchy problem for the generalized shallow water wave equation

In this paper we mainly study the Cauchy problem for the generalized shallow water wave equation in the Sobolev space Hs of lower order s. Using the crucial bilinear estimates in the Fourier transform restriction spaces related to the shallow water wave equation, we establish local well-posedness in Hs with any .     

Global existence of solutions for 2‐D semilinear wave equations with dissipation localized near infinity in an exterior domain

We shall derive some global existence results to semilinear wave equations with a damping coefficient localized near infinity for very special initial data in H×L². This problem is dealt with in the two‐dimensional exterior domain with a star‐shaped complement. In our result, a power p on the non‐linear term |u|^p is strictly larger than the two‐dimensional Fujita‐exponent. Copyright © 2005 John Wiley & Sons, Ltd.     

A two‐grid method for mixed finite‐element solution of reaction‐diffusion equations

We present a scheme for solving two‐dimensional, nonlinear reaction‐diffusion equations, using a mixed finite‐element method. To linearize the mixed‐method equations, we use a two grid scheme that relegates all the Newton‐like iterations to a grid Δ_H much coarser than the original one Δ_h, with no loss in order of accuracy so long as the mesh sizes obey . The use of a multigrid‐based solver for the indefinite linear systems that arise at each coarse‐grid iteration, as well as for the similar system that arises on the fine grid, allows for even greater efficiency. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 317–332, 1999     

Linear stability of solitary waves of the Green‐Naghdi equations

We investigate the eigenvalue problem obtained from linearizing the Green‐Naghdi equations about solitary wave solutions. Unlike weakly nonlinear water wave models, the physical system considered here has nonlinearity in its highest derivative term. This results in more detailed asymptotic analysis of the eigenvalue problem in the presence of a large parameter. Combining the technique of singular perturbation with the Evans function, we show that for solitary waves of small amplitude, the problem has no eigenvalues of positive real part and the Evans function is nonvanishing everywhere except the origin. This fact then leads to the linear stability of these solitary waves. © 2001 John Wiley & Sons, Inc.     

The Cauchy problem for Kawahara equation in Sobolev spaces with low regularity

This paper is devoted to studying the initial‐value problem of the Kawahara equation. By establishing some crucial bilinear estimates related to the Bourgain spaces X_{s, b}(R²) introduced by Bourgain and homogeneous Bourgain spaces, which is defined in this paper and using I‐method as well as L² conservation law, we show that this fifth‐order shallow water wave equation is globally well‐posed for the initial data in the Sobolev spaces H^s(R) with $s{>}-\frac{63}{58}$. Copyright © 2010 John Wiley & Sons, Ltd.     

Unconditional superconvergence analysis of an H1‐galerkin mixed finite element method for nonlinear Sobolev equations

An efficient H¹‐Galerkin mixed finite element method (MFEM) is presented with and zero order Raviart‐Thomas elements for the nonlinear Sobolev equations. On one hand, the existence and uniqueness of the solutions of the semidiscrete approximation scheme are proved and the super close results of order for the original variable u in a broken H¹ norm and the auxiliary variable in norm are deduced without the boundedness of the numerical solution in ‐norm. Conversely, a linearized Crank‐Nicolson fully discrete scheme with the unconditional super close property is also developed through a new approach, while previous literature always require certain time step conditions (see the references below). Finally, a numerical experiment is included to illustrate the feasibility of the proposed method. Here h is the subdivision parameter and τ is the time step.     

OnH 2 minimization of theH ∞ suboptimal solutions

In this paper we show that theH ² minimization of theH suboptimal solutions for a class of suboptimalH distance problems can be reduced to a finite dimensional nonlinear optimization problem. This extends a result of [7] where the same problem is considered in the Caratheodory-Schur interpolation case.     

Superconvergence of a two‐grid method for fourth‐order dispersive‐dissipative wave equations

In this paper, the superconvergence analysis of a two‐grid method (TGM) with low‐order finite elements is presented for the fourth‐order dispersive‐dissipative wave equations for a second order fully discrete scheme. The superclose estimates in the H¹‐norm on the two grids are obtained by the combination technique of the interpolation and Ritz projection. Then, with the help of the interpolated postprocessing technique, the global superconvergence properties are deduced. Finally, numerical results are provided to show the performance of the proposed TGM for conforming bilinear element and nonconforming element, respectively. It shows that the TGM is an effective method to the problem considered of our paper compared with the traditional Galerkin finite element method (FEM).     

Traveling waves in a diffusive epidemic model with criss‐cross mechanism

In this paper, we propose a reaction‐diffusion system to describe the spread of infectious diseases within two population groups by self and criss‐cross infection mechanism. Firstly, based on the eigenvalues, we give two methods for the calculation of the critical wave speed c^?. Secondly, by constructing a pair of upper‐lower solutions and using the Schauder fixed‐point theorem, we prove that the system admits positive traveling wave solutions, which connect the initial disease‐free equilibrium at t = ?∞, but the traveling waves need not connect the final disease‐free equilibrium at t = +∞. Hence, we study the asymptotic behaviors of the traveling wave solutions to show that the traveling wave solutions converge to at t = +∞. Finally, by the two‐sided Laplace transform, we establish the nonexistence of traveling waves for the model. The approach in this paper provides an effective method to deal with the existence of traveling wave solutions for the nonmonotone reaction‐diffusion systems consisting of four equations.     

Existence of positive solutions for nonlinear singular boundary value problem with p‐Laplacian

In this paper, we study the existence of positive solutions for the following Sturm–Liouville‐like four‐point singular boundary value problem (BVP) with p‐Laplacian where ?_p(s)=|s|^p?2 s, p>1, f is a lower semi‐continuous function. Using the fixed‐point theorem of cone expansion and compression of norm type, the existence of positive solution and infinitely many positive solutions for Sturm–Liouville‐like singular BVP with p‐Laplacian are obtained. Copyright © 2009 John Wiley & Sons, Ltd.     

New error estimates of nonconforming mixed finite element methods for the Stokes problem

In this paper, we revisit the classical error estimates of nonconforming Crouzeix–Raviart type finite elements for the Stokes equations. By introducing some quasi‐interpolation operators and using the special properties of these nonconforming elements, it is proved that their consistency errors can be bounded by their approximation errors together with a high‐order term, especially which can be of arbitrary order provided that f in the right‐hand side is piecewise smooth enough. Furthermore, we show an interesting result that both in the energy norm and L² norm the consistency errors are dominated by the approximation errors of their finite element spaces. As byproducts, we derive the error estimates in both energy and L² norms under the regularity assumption ( u ,p) ∈ H ^{1 + s}(Ω) × H^s(Ω) with any s ∈ (0,1], which fills the gap in the a priori error estimate of these nonconforming elements with low regularity . Furthermore, a robust convergence is proved with minimal regularity assumption s = 0. These results seem to be missing in the literature. Numerical tests are provided, confirming the analysis, especially the new results on the L² convergence. Copyright © 2013 John Wiley & Sons, Ltd.     

Analysis of positive solution and Hyers‐Ulam stability for a class of singular fractional differential equations with p‐Laplacian in Banach space

This paper deals with 2 core aspects of fractional calculus including existence of positive solution and Hyers‐Ulam stability for a class of singular fractional differential equations with nonlinear p‐Laplacian operator in Caputo sense. For these aims, the suggested problem is converted into an integral equation via Green function , for ε∈(n−1,n], where n≥4. Then, the Green function is examined whether it is increasing or decreasing and positive or negative function. After these properties, some classical fixed‐point theorems are used for the existence of positive solution. Hyers‐Ulam stability of the proposed problem is also considered. For the application of the results, an expressive example is included.     

Multiple sign‐changing solutions for Kirchhoff‐type equations in

In this paper, we study the following Kirchhoff‐type equations where a>0,b⩾0,4<p<2^∗=6, and . Under some suitable conditions, we prove that the equation has three solutions of mountain pass type: one positive, one negative, and sign‐changing. Furthermore, this problem has infinitely many sign‐changing solutions. Copyright © 2017 John Wiley & Sons, Ltd.