Global Existence and Convergence Rates of Smooth Solutions for the 3-D Compressible Magnetohydrodynamic Equations Without Heat Conductivity

In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations without heat conductivity, which is a hyperbolic-parabolic system. The global solutions are obtained by combining the local existence and a priori estimates if H³-norm of the initial perturbation around a constant states is small enough and its L¹-norm is bounded. A priori decay-in-time estimates on the pressure, velocity and magnetic field are used to get the uniform bound of entropy. Moreover, the optimal convergence rates are also obtained.     

Shallow water equations: viscous solutions and inviscid limit

We establish the inviscid limit of the viscous shallow water equations to the Saint-Venant system. For the viscous equations, the viscosity terms are more degenerate when the shallow water is close to the bottom, in comparison with the classical Navier-Stokes equations for barotropic gases; thus, the analysis in our earlier work for the classical Navier-Stokes equations does not apply directly, which require new estimates to deal with the additional degeneracy. We first introduce a notion of entropy solutions to the viscous shallow water equations and develop an approach to establish the global existence of such solutions and their uniform energy-type estimates with respect to the viscosity coefficient. These uniform estimates yield the existence of measure-valued solutions to the Saint-Venant system generated by the viscous solutions. Based on the uniform energy-type estimates and the features of the Saint-Venant system, we further establish that the entropy dissipation measures of the viscous solutions for weak entropy-entropy flux pairs, generated by compactly supported C ² test-functions, are confined in a compact set in H ^?1, which yields that the measure-valued solutions are confined by the Tartar-Murat commutator relation. Then, the reduction theorem established in Chen and Perepelitsa [5] for the measure-valued solutions with unbounded support leads to the convergence of the viscous solutions to a finite-energy entropy solution of the Saint-Venant system with finite-energy initial data, which is relative with respect to the different end-states of the bottom topography of the shallow water at infinity. The analysis also applies to the inviscid limit problem for the Saint-Venant system in the presence of friction.     

The convergence of non-Newtonian fluids to Navier-Stokes equations

In this paper, the convergence of solutions for incompressible dipolar viscous non-Newtonian fluids is investigated. We obtain the conclusion that the solutions of non-Newtonian fluids converge to the solutions of Navier-Stokes equations in the sense of L²-norm (resp. H¹-norm), as the viscosities tend to zero and the initial data belong to H¹(Ω) (resp. H²(Ω)). Moreover, we obtain L^∞-norm convergence of solutions if the initial data belong to H²(Ω).     

Advanced combinations of splitting–shooting–integrating methods for digital image transformations

An image consists of many discrete pixels with greyness of different levels, which can be quantified by greyness values. The greyness values at a pixel can also be represented by an integral as the mean of continuous greyness functions over a small pixel region. Based on such an idea, the discrete images can be produced by numerical integration; several efficient algorithms are developed to convert images under transformations. Among these algorithms, the combination of splitting–shooting–integrating methods (CSIM) is most promising because no solutions of nonlinear equations are required for the inverse transformation. The CSIM is proposed in [6] to facilitate images and patterns under a cycle transformations T⁻¹T, where T is a nonlinear transformation. When a pixel region in two dimensions is split into N² subpixels, convergence rates of pixel greyness by CSIM are proven in [8] to be only O(1/N). In [10], the convergence rates O_p(1/N^1.5) in probability and O_p(1/N²) in probability using a local partition are discovered. The CSIM is well suited to binary images and the images with a few greyness levels due to its simplicity. However, for images with large (e.g., 256) multi-greyness levels, the CSIM still needs more CPU time since a rather large division number is needed.In this paper, a partition technique for numerical integration is proposed to evaluate carefully any overlaps between the transformed subpixel regions and the standard square pixel regions. This technique is employed to evolve the CSIM such that the convergence rate O(1/N²) of greyness solutions can be achieved. The new combinations are simple to carry out for image transformations because no solutions of nonlinear equations are involved in, either. The computational figures for real images of 256×256 with 256 greyness levels display that N=4 is good enough for real applications. This clearly shows validity and effectiveness of the new algorithms in this paper.     

The pointwise estimates of solutions for a nonlinear convection diffusion reaction equation

This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension.First,the pointwise estimates of solutions are obtained,furthermore,we obtain the optimal L~p,1≤ p ≤ +∞,convergence rate of solutions for small initial data.Then we establish the local existence of solutions,the blow up criterion and the sufficient condition to ensure the nonnegativity of solutions for large initial data.Our approach is based on the detailed analysis of the Green function of the linearized equation and some energy estimates.     

Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations

In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations in R³. We prove the global existence of the smooth solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in H³-framework. Moreover, if additionally the initial data belong to L^p with , the optimal convergence rates of the solutions in L^q-norm with 2≤q≤6 and its spatial derivatives in L²-norm are obtained.     

Lp estimates of solutions of some partial differential equations

The author is concerned with the long time asymptotic behaviors of the global weak solutions of some nonlinear evolution equations. First of all, he derives some uniform L¹ and L^∞ upper bounds for the solutions, under some mild conditions. Then, by applying the well-known Fourier splitting method and the L¹ estimates, he asserts the L² decay estimates of the solutions. The rates of decay are sharp in the sense that the integral of the initial data over $\text{R}$ is nonzero.     

Asymptotic stability of stationary solutions to the compressible bipolar Navier–Stokes–Poisson equations

In this paper, we consider the compressible bipolar Navier–Stokes–Poisson equations with a non‐flat doping profile in three‐dimensional space. The existence and uniqueness of the non‐constant stationary solutions are established when the doping profile is a small perturbation of a positive constant state. Then under the smallness assumption of the initial perturbation, we show the global existence of smooth solutions to the Cauchy problem near the stationary state. Finally, the convergence rates are obtained by combining the energy estimates for the nonlinear system and the L²‐decay estimates for the linearized equations. Copyright © 2017 John Wiley & Sons, Ltd.     

Analyse d'un schéma décalé pour l'approximation de systèmes hyperboliques non conservatifs

We consider a model system made of two nonlinear equations which are non conservative. A conservation law can be obtained from these equations through linear operations only, which don't modify the shock waves. A numerical scheme based on a different mesh adapted to each variable is proposed. By choosing a shifted mesh, we have un explicit Riemann solver and we can derive a finite volume scheme. We prove a priori estimates in L^∞ norm and Total Variation for the system, which lead to a strong convergence in L¹ norm towards a solution satisfying the associated conservation law.     

Convergence of compressible Euler-Poisson system to incompressible Euler equations

In this paper, we study the quasi-neutral limit of compressible Euler-Poisson equations in plasma physics in the torus T^d. For well prepared initial data the convergence of solutions of compressible Euler-Poisson equations to the solutions of incompressible Euler equations is justified rigorously by an elaborate energy methods based on studies on an λ-weighted Lyapunov-type functional. One main ingredient of establishing uniformly a priori estimates with respect to λ is to use the curl-div decomposition of the gradient.     

Vanishing viscosity with short wave-long wave interactions for multi-D scalar conservation laws

We consider a system coupling a multidimensional semilinear Schrödinger equation and a multidimensional nonlinear scalar conservation law with viscosity, which is motivated by a model of short wave-long wave interaction introduced by Benney (1977). We prove the global existence and uniqueness of the solution of the Cauchy problem for this system. We also prove the convergence of the whole sequence of solutions when the viscosity ε and the interaction parameter α approach zero so that α=o(ε1/2). We also indicate how to extend these results to more general systems which couple multidimensional semilinear systems of Schrödinger equations with multidimensional nonlinear systems of scalar conservation laws mildly coupled.     

The relaxation-time limit in the quantum hydrodynamic equations for semiconductors

The relaxation-time limit from the quantum hydrodynamic model to the quantum drift-diffusion equations in R³ is shown for solutions which are small perturbations of the steady state. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density including the quantum Bohm potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. The relaxation-time limit is performed both in the stationary and the transient model. The main assumptions are that the steady-state velocity is irrotational, that the variations of the doping profile and the velocity at infinity are sufficiently small and, in the transient case, that the initial data are sufficiently close to the steady state. As a by-product, the existence of global-in-time solutions to the quantum drift-diffusion model in R³ close to the steady-state is obtained.     

Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms

In this paper we consider the Cauchy problem of semilinear parabolic equations with nonlinear gradient terms a(x)|u|^q−1u|∇u|^p. We prove the existence of global solutions and self-similar solutions for small initial data. Moreover, for a class of initial data we show that the global solutions behave asymptotically like self-similar solutions as t→∞.     

An application of Taylor series in the approximation of solutions to stochastic differential equations with time-dependent delay

The subject of this paper is the analytic approximation method for solving stochastic differential equations with time-dependent delay. Approximate equations are defined on equidistant partitions of the time interval, and their coefficients are Taylor approximations of the coefficients of the initial equation. It will be shown, without making any restrictive assumption for the delay function, that the approximate solutions converge in L^p-norm and with probability 1 to the solution of the initial equation. Also, the rate of the L^p convergence increases when the degrees in the Taylor approximations increase, analogously to what is found in real analysis. At the end, a procedure will be presented which allows the application of this method, with the assumption of continuity of the delay function.     

Blow-up solutions to the nonlinear Schrödinger equation with oscillating nonlinearities

In this article we investigate the possibility of finite time blow-up in H¹(R²) for solutions to critical and supercritical nonlinear Schrödinger equations with an oscillating nonlinearity. We prove that despite the oscillations some solutions blow up in finite time. Conversely, we observe that for a given initial data oscillations can extend the local existence time of the corresponding solution.     

Existence of solutions for a nonlinear system with a parameter

In this paper, the existence of solutions for a system of nonlinear equations is considered. n² nonzero real solutions are obtained by using the critical point theory. Additionally, the Dirichlet boundary value problems of even order difference equations and partial difference equations are investigated.     

The nonrelativistic limit in radiation hydrodynamics: I. Weak entropy solutions for a model problem

This paper is concerned with a model system for radiation hydrodynamics in multiple space dimensions. The system depends singularly on the light speed c and consists of a scalar nonlinear balance law coupled via an integral-type source term to a family of radiation transport equations. We first show existence of entropy solutions to Cauchy problems of the model system in the framework of functions of bounded variation. This is done by using difference schemes and discrete ordinates. Then we establish strong convergence of the entropy solutions, indexed with c, as c goes to infinity. The limit function satisfies a scalar integro-differential equation.     

Stabilization of statistics in wave and Klein-Gordon equations with mixing. Scattering theory for infinite energy solutions

We consider wave and Klein-Gordon equations in the whole space ?ⁿ with arbitraryn≥2. We assume initial data to be homogeneous random functions in ?ⁿ with zero expectation and finite mean density of energy. Moreover, we assume initial data fit mixing condition of Ibragimov-Linnik type. We consider the distributions of the random solution at the moment of timet. The main results mean the convergence of this distribution to some Gaussian measure ast→∞. This is a central limit theorem for wave and Klein-Gordon equations. The limit Gaussian measures are invariant measures for equations considered. Corresponding stationary random solutions are ergodic and mixing in time. The results are inspired by mathematical problems of statistical physics.     

Regularity and integrability of rotating shallow-water equations

We consider classical shallow-water equations for a rapidly rotating fluid layer. The Poincaré/Kelvin linear propagator describes fast oscillating waves for the linearized system. We show that solutions of the full nonlinear shallow-water equations can be decomposed as U(t,x₁,x₂) + Ũ(t,x₁,x₂) + W’(t,x₁,x₂) + r, where Ũ is a solution of the quasigeostrophic (QG) equation. Here r is a remainder, which is uniformly estimated from above by a majorant of order 1/f₀. The vector field W’(t,x₁,x₂) describes the rapidly oscillating ageostrophic (AG) component. This component is exactly solved in terms of Poincaré/Kelvin waves with phase shifts explicitly determined from the nonlinear quasigeostrophic equations. The mathematically rigorous control of the error r, based on estimates of small divisors, is used to prove the existence, on a long time interval T^*, of regular solutions to classical shallow-water equations with general initial data (T^* → +∞, as 1/f₀ → 0).     

An optimal Steffensen-type family for solving nonlinear equations

In this paper, a general family of Steffensen-type methods with optimal order of convergence for solving nonlinear equations is constructed by using Newton’s iteration for the direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub [H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Math. 21 (1974) 634-651] that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2^m−1. Its error equations and asymptotic convergence constants are obtained. Finally, it is compared with the related methods for solving nonlinear equations in the numerical examples.