Cauchy problem of Schrödinger-Improved Boussinesq Systems on the torus

In this paper, we will study the local well-posedness of Schrödinger-Improved Boussinesq System with additive noise in T^d${\mathbb{T}}^d$, d?1$d?1$, and we will also study the global well-posedness of dimension 1 case with the initial data (_u0,_v1,_v2)∈L²×L²×L²$(u_0,v_1,v_2)\in L^2\times L^2\times L^2$ almost surely, gaining some exponential growth of L²$L^2$ norm of v.     

Analysis of one-dimensional Helmholtz equation with PML boundary

In this paper, the linear conforming finite element method for the one-dimensional Bérenger's PML boundary is investigated and well-posedness of the given equation is discussed. Furthermore, optimal error estimates and stability in the L²$L^2$ or H¹$H^1$-norm are derived under the assumption that h$h$, h²ω²$h^2{\omega }^2$ and h²ω³$h^2{\omega }^3$ are sufficiently small, where h$h$ is the mesh size and ω$\omega$ denotes a fixed frequency. Numerical examples are presented to validate the theoretical error bounds.     

On the well-posedness of higher order viscous Burgers' equations

We consider higher order viscous Burgers' equations with generalized nonlinearity and study the associated initial value problems for given data in the L²$L^2$-based Sobolev spaces. We introduce appropriate time weighted spaces to derive multilinear estimates and use them in the contraction mapping principle   argument to prove local well-posedness for data with Sobolev regularity below L²$L^2$. We also prove ill-posedness for this type of models and show that the local well-posedness results are sharp in some particular cases viz., when the orders of dissipation p  , and nonlinearity k+1$k+1$, satisfy a relation p=2k+1$p=2k+1$.     

Approximations of Euler–Maruyama type for stochastic differential equations with Markovian switching,under non-Lipschitz conditions

We develop the Euler–Maruyama scheme for a class of stochastic differential equations with Markovian switching (SDEwMSs) under non-Lipschitz conditions  . Both L¹$L^1$ and L²$L^2$-convergence are discussed under different non-Lipschitz conditions. To overcome the mathematical difficulties arisen from the Markovian switching as well as the non-Lipschitz coefficients, several new analytical techniques have been developed in this paper which should prove to be very useful in the numerical analysis of stochastic systems.     

L -decay rates of planar viscous rarefaction wave for scalar conservation law with degenerate viscosity in n -dimensions

We investigate the Cauchy problem and the initial-boundary value problem for multi-dimensional conservation laws with degenerate viscosity in the whole space and in the half-space respectively. We give the optimal decay estimates in the W^1,p(1≤p≤∞)$W^{1,p}(1\le p\le \infty )$ norm for the perturbation from the planar viscous rarefaction wave. The analysis based on the new L^p$L^p$-energy method and L¹$L^1$-estimates.     

Korn type inequalities in Orlicz spaces

We exhibit balance conditions between a Young function A and a Young function B   for a Korn type inequality to hold between the L^B$L^B$ norm of the gradient of vector-valued functions and the L^A$L^A$ norm of its symmetric part. In particular, we extend a standard form of the Korn inequality in L^p$L^p$, with 1<p<∞$1<p<\infty$, and an Orlicz version involving a Young function A   satisfying both the _Δ2${\mathrm{\Delta }}_2$ and the _∇2${\mathrm{\nabla }}_2$ condition.     

On the inverse limit stability of endomorphisms

We present several results suggesting that the concept of C¹$C^1$-inverse (limit structural) stability is free of singularity theory. An example of a robustly transitive, C¹$C^1$-inverse stable endomorphism with a persistent critical set is given. We show that every C¹$C^1$-inverse stable, axiom A endomorphism satisfies a certain strong transversality condition (T). We prove that every attractor–repeller endomorphism satisfying axiom A and condition (T  ) is C¹$C^1$-inverse stable. The latter is applied to Hénon maps, rational functions and others. This leads us to conjecture that C¹$C^1$-inverse stable endomorphisms are exactly those which satisfy axiom A and condition (T).     

Well-posedness of a 3D parabolic–hyperbolic Keller–Segel system in the Sobolev space framework

We investigate global strong solution to a 3-dimensional parabolic–hyperbolic system arising from the Keller–Segel model. We establish the global well-posedness and asymptotic behavior in the energy functional setting. Precisely speaking, if the initial difference between cell density and its mean is small in L²$L^2$, and the ratio of the initial gradient of the chemical concentration and the initial chemical concentration is also small in H¹$H^1$, then they remain to be small in L²×H¹$L^2\times H^1$ for all time. Moreover, if the mean value of the initial cell density is smaller than some constant, then the cell density approaches its initial mean and the chemical concentration decays exponentially to zero as t goes to infinity. The proof relies on an application of Fourier analysis to a linearized parabolic–hyperbolic system and the smoothing effect of the cell density and the damping effect of the chemical concentration.     

Uniform bound of Sobolev norms of solutions to 3D nonlinear wave equations with null condition

This article concerns the time growth of Sobolev norms of classical solutions to the 3D quasi-linear wave equations with the null condition. Given initial data in H^s×H^s−1$H^s\times H^{s-1}$ with compact supports, the global well-posedness theory has been established independently by Klainerman [13] and Christodoulou [3], respectively, for a relatively large integer s  . However, the highest order Sobolev energy, namely, the H^s$H^s$ energy of solutions may have a logarithmic growth in time. In this paper, we show that the H^s$H^s$ energy of solutions is also uniformly bounded for s?5$s?5$. The proof employs the generalized energy method of Klainerman, enhanced by weighted L²$L^2$ estimates and the ghost weight introduced by Alinhac.     

The wave equation for the Bessel Laplacian

We study radial solutions of the Cauchy problem for the wave equation in the multidimensional unit ball B^d$B^d$, d≥1$d\ge 1$. In this case, the operator that appears is the Bessel Laplacian and the solution u(t,x)$u(t,x)$ is given in terms of a Fourier–Bessel expansion. We prove that, for initial L^p$L^p$ data, the series converges in the L²$L^2$ norm. The analysis of a particular operator, the adjoint of the Riesz transform for Fourier–Bessel series, is needed for our purposes, and may be of independent interest. As applications, certain L^p−L²$L^p-L^2$ estimates for the solution of the heat equation and the extension problem for the fractional Bessel Laplacian are obtained.     

Harness processes and harmonic crystals

In the Hammersley harness processes the R$\mathbb{R}$-valued height at each site i∈Z^d$i\in {\mathbb{Z}}^d$ is updated at rate 1 to an average of the neighboring heights plus a centered random variable (the noise). We construct the process “a la Harris” simultaneously for all times and boxes contained in Z^d${\mathbb{Z}}^d$. With this representation we compute covariances and show L²$L^2$ and almost sure time and space convergence of the process. In particular, the process started from the flat configuration and viewed from the height at the origin converges to an invariant measure. In dimension three and higher, the process itself converges to an invariant measure in L²$L^2$ at speed t^1−d/2$t^{1-d/2}$ (this extends the convergence established by Hsiao). When the noise is Gaussian the limiting measures are Gaussian fields (harmonic crystals) and are also reversible for the process.     

Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems

In this paper, we propose a new family of NCP-functions and the corresponding merit functions, which are the generalization of some popular NCP-functions and the related merit functions. We show that the new NCP-functions and the corresponding merit functions possess a system of favorite properties. Specially, we show that the new NCP-functions are strongly semismooth, Lipschitz continuous, and continuously differentiable; and that the corresponding merit functions have SC¹$S{C}^1$ property (i.e., they are continuously differentiable and their gradients are semismooth) and LC¹$L{C}^1$ property (i.e., they are continuously differentiable and their gradients are Lipschitz continuous) under suitable assumptions. Based on the new NCP-functions and the corresponding merit functions, we investigate a derivative free algorithm for the nonlinear complementarity problem and discuss its global convergence. Some preliminary numerical results are reported.     

The stationary Oseen equations in an exterior domain: An approach in weighted Sobolev spaces

In this work, we study the linearized Navier–Stokes equations in an exterior domain of R³${\mathbb{R}}^3$ at the steady state, that is, the Oseen equations. We are interested in the existence and the uniqueness of weak, strong and very weak solutions in L^p$L^p$-theory which makes our work more difficult. Our analysis is based on the principle that linear exterior problems can be solved by combining their properties in the whole space R³${\mathbb{R}}^3$ and the properties in bounded domains. Our approach rests on the use of weighted Sobolev spaces.     

High resolution of 2D natural convection in a cavity by the DQ method

In this paper, the differential quadrature method (DQ) is applied to solve the benchmark problem of 2D natural convection in a cavity by utilizing the velocity–vorticity form of the Navier–Stokes equations, which is governed by the velocity Poisson equation, continuity equation and vorticity transport equation as well as energy equation. The coupled equations are simultaneously solved by imposing the vorticity definition at boundary without any iterative procedure. The present model is properly utilized to obtain results in the range of Rayleigh number (10³${10}^3$–10⁷${10}^7$) and H/L$H/L$ aspect ratios varying from 1 to 3. Nusselt numbers computed for 10³?Ra?10⁷${10}^3?\text{<italic>Ra</italic>}?{10}^7$ in a cavity show excellent agreement with the results available in the literature. Additionally, the detailed features of flow phenomena such as velocity, temperature, vorticity, and streamline plots are also delineated in this work. Thus, it is convinced that the DQ method is capable of solving coupled differential equations efficiently and accurately.     

Remark on the stability of the large stationary solutions to the Navier–Stokes equations under the general flux condition

Consider stationary weak solutions of the Navier–Stokes equations in a bounded domain in R³${\mathbb{R}}^3$ under the nonhomogeneous boundary condition. We give a new approach for the stability of the stationary flow in the L²$L^2$-framework. Furthermore, we give some examples of stable solutions which may be large in L³(Ω)$L^3\left(\Omega \right)$ or W^1,3/2(Ω)$W^{1,3/2}\left(\Omega \right)$.