Uniform Estimates Research Articles

Global non-relativistic quasi-neutral limit for a two-fluid Euler-Maxwell system

This paper concerns a two-fluid Euler-Maxwell system for plasmas with two small parameters. We study the non-relativistic quasi-neutral limit for smooth solutions of the system in whole space R3. When the initial data are smooth and sufficiently close to constant equilibrium states, we give a rigorous justification of the limit for all time. The limit system is governed by a compressible Euler system. The proof is based on uniform energy estimates and various dissipative estimates of smooth solutions with respect to the parameters and time. A key step in these estimates is to control the quasi-neutrality of the velocities by using a projection operator.

Uniform Regularity Estimates for Three Dimensional Nonhomogeneous Incompressible Heat Conducting Navier–Stokes Flows

Uniform Regularity Estimates for Three Dimensional Nonhomogeneous Incompressible Heat Conducting Navier–Stokes Flows

A higher order compact numerical approach for singularly perturbed parabolic problem with retarded term

In this work, a compact finite difference approach is constructed for singularly perturbed parabolic reaction diffusion problems with a retarded term. The time and space derivatives have been discretized using the θ-method and a compact fourth-order finite difference method on a Shishkin mesh, respectively. Parameter uniform error estimates have been calculated in the L ∞ norm. Some numerical examples have been considered to corroborate the theoretical results and compare the numerical results with existing methods in the literature. It is shown that the present approach provides improved results till date for the problem considered in this paper.

Uniform gradient estimates of generalized parabolic mean curvature type equations with oblique boundary value problems

Uniform gradient estimates of generalized parabolic mean curvature type equations with oblique boundary value problems

Asymptotic stability of solutions to quasilinear damped wave equations with variable sources

In this paper, we consider the following quasilinear damped hyperbolic equation involving variable exponents: $$ u_{tt}-\operatorname{div}( |\nabla u|^{r(x)-2}\nabla u)+|u_t|^{m(x)-2} u_t-\Delta u_t=|u|^{q(x)-2}u, $$ with homogenous Dirichlet initial boundary value condition. An energy estimate and Komornik's inequality are used to prove uniform estimate of decay rates of the solution. We also show that $u(x, t)=0$ is asymptotic stable in terms of natural energy associated with the solution of the above equation. As we know, such results are seldom seen for the variable exponent case. At last, we give some numerical examples to illustrate our results.

Study of fractional semipositone problems on R^{N}

Let \(s\in (0,1)\) and \(N\gt 2s\). In this paper, we consider the following class of nonlocal semipositone problems: \[(-\Delta)^s u= g(x)f_a(u)\text{ in }\mathbb{R}^N,\quad u \gt 0\text{ in }\mathbb{R}^N,\] where the weight \(g \in L^1(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)\) is positive, \(a\gt 0\) is a parameter, and \(f_a \in \mathcal{C}(\mathbb{R})\) is strictly negative on \((-\infty,0]\). For \(f_a\) having subcritical growth and weaker Ambrosetti-Rabinowitz type nonlinearity, we prove that the above problem admits a mountain pass solution \(u_a\), provided \(a\) is near zero. To obtain the positivity of \(u_a\), we establish a Brezis-Kato type uniform estimate of \((u_a)\) in \(L^r(\mathbb{R}^N)\) for every \(r \in [\frac{2N}{N-2s}, \infty]\).

Random dynamics for a stochastic nonlocal reaction-diffusion equation with an energy functional

<abstract><p>In this paper, the asymptotic behavior of solutions to a fractional stochastic nonlocal reaction-diffusion equation with polynomial drift terms of arbitrary order in an unbounded domain was analysed. First, the stochastic equation was transformed into a random one by using a stationary change of variable. Then, we proved the existence and uniqueness of solutions for the random problem based on pathwise uniform estimates as well as the energy method. Finally, the existence of a unique pullback attractor for the random dynamical system generated by the transformed equation is shown.</p></abstract>

Studies on invariant measures of fractional stochastic delay Ginzburg-Landau equations on Rn.

This paper is concerned with invariant measures of fractional stochastic delay Ginzburg-Landau equations on the entire space $ \mathbb{R}^n $. We first derive the uniform estimates and the mean-square uniform smallness of the tails of solutions in corresponding space. Then we deduce the weak compactness of a set of probability distributions of the solutions applying the Ascoli-Arzel$ \grave{a} $. We finally prove the existence of invariant measures by applying Krylov-Bogolyubov's method.

An iterative method for solving one class of non-linear integral equations with Nemytskii operator on the positive semi-axis

A class of non-linear integral equations with monotone Nemytskii operator on the positive semi-axis is considered. This class of integral equations appears in many areas of modern natural science. In particular, such equations, with various restrictions on non-linearity and the kernel, arise in the dynamic theory of $p$-adic strings for the scalar field of tachyons, in the kinetic theory of gases and plasma within the framework of the usual and modified non-linear Bhatnagar-Gross-Crook models for the Boltzmann kinetic equation. Equations of similar nature appear also in the non-linear radiative transfer theory in inhomogeneous media and in the mathematical theory of the spread of epidemic diseases within the framework of the modified Diekmann-Kaper model. A constructive theorem for the existence of a bounded positive and continuous solution is proved. As a result, we get a uniform estimate of the difference between the previous and subsequent iterations, the corresponding successive approximations converge uniformly to a bounded continuous solution to the equation. The asymptotic behaviour of the constructed solution at infinity is studied. In particular, it is proved that the limit of this solution at infinity exists and is positive, and is uniquely determined from a certain characteristic equation. It is also proved that the difference between the limit and the solution is a summable function on the positive semi-axis. Using some geometric estimates for convex and concave functions, and employing the theorem on integral asymptotics obtained here, we prove the uniqueness of the solution in a certain subclass of non-negative non-trivial continuous and bounded functions. The results obtained are also applied to the study of a special class of non-linear Urysohn type integral equations on the positive semi-axis. In particular, the existence of a positive bounded solution to this class of equations is verified, and some qualitative properties of the constructed solution are studied. We also give specific applied examples of the corresponding kernels and non-linearities to illustrate the importance of the results obtained.

Axisymmetric Incompressible Viscous Plasmas: Global Well-Posedness and Asymptotics

Abstract This paper is devoted to the global analysis of the three-dimensional axisymmetric Navier–Stokes–Maxwell equations. More precisely, we are able to prove that, for large values of the speed of light $c\in (c_0, \infty )$ , for some threshold $c_0>0$ depending only on the initial data, the system in question admits a unique global solution. The ensuing bounds on the solutions are uniform with respect to the speed of light, which allows us to study the singular regime $c\rightarrow \infty $ and rigorously derive the limiting viscous magnetohydrodynamic (MHD) system in the axisymmetric setting. The strategy of our proofs draws insight from recent results on the two-dimensional incompressible Euler–Maxwell system to exploit the dissipative–dispersive structure of Maxwell’s system in the axisymmetric setting. Furthermore, a detailed analysis of the asymptotic regime $c\to \infty $ allows us to derive a robust nonlinear energy estimate which holds uniformly in c. As a byproduct of such refined uniform estimates, we are able to describe the global strong convergence of solutions toward the MHD system. This collection of results seemingly establishes the first available global well-posedness of three-dimensional viscous plasmas, where the electric and magnetic fields are governed by the complete Maxwell equations, for large initial data as $c\to \infty $ .

Asymptotic stability of Solutions to Quasilinear Damped Wave Equations with Variable Sources

In this paper, we consider the following quasilinear damped hyperbolic equation involving variable exponents \[{u_{tt}} - div( {{{| {\nabla u} |}^{r(x) - 2}}\nabla u} ) + {| {{u_t}} |^{m(x) - 2}}{u_t} - \Delta {u_t} = {| u |^{q( x ) - 2}}u\] with homogenous Dirichlet initial boundary value condition. Some energy estimate and Komornik inequality are used to prove some uniform estimate of decay rates of the solution. And then, we prove that $u(x, t) = 0$ is asymptotic stable in terms of natural energy associated with the solution of the above equation. As we know, such results are seldom seen for the variable exponent case. At last, we give some numerical examples to illustrate our results.

Optimal decay rate and convergence of the incompressible viscous resistive MHD system

In this work, we investigate the optimal Lp(p≥2) time decay rate of global solutions to the incompressible viscous resistive MHD system with degeneration, and study its convergence to the incompressible viscous ideal MHD system as the resistivity ε goes to zero. The analysis is based on the high-low frequency decomposition and the uniform energy estimate of the system with respect to ε.

Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces

We consider perturbations of Minkowski space as well as more general spacetimes on which the wave operator \square_{g} is known to be essentially self-adjoint. We define complex powers (\square_{g}-i\varepsilon)^{-\alpha} by functional calculus, and show that the trace density exists as a meromorphic function of \alpha . We relate its poles to geometric quantities, in particular to the scalar curvature. The results allow us to formulate a spectral action principle which serves as a simple Lorentzian model for the bosonic part of the Chamseddine–Connes action. Our proof combines microlocal resolvent estimates, including radial propagation estimates, with uniform estimates for the Hadamard parametrix. The arguments work in Lorentzian signature directly and do not rely on transition from the Euclidean setting.

Exponential stability of damped Euler-Bernoulli beam controlled by boundary springs and dampers

In this paper, the vibration model of an elastic beam, governed by the damped Euler-Bernoulli equation ρ(x)utt+μ(x)ut+(r(x)uxx)xx=0, subject to the clamped boundary conditions u(0,t)=ux(0,t)=0 at x=0, and the boundary conditions (−r(x)uxx)x=ℓ=krux(ℓ,t)+kauxt(ℓ,t), (−(r(x)uxx)x)x=ℓ=−kdu(ℓ,t)−kvut(ℓ,t) at x=ℓ, is analyzed. The boundary conditions at x=ℓ correspond to linear combinations of damping moments caused by rotation and angular velocity and also, of forces caused by displacement and velocity, respectively. The system stability analysis based on well-known Lyapunov approach is developed. Under the natural assumptions guaranteeing the existence of a regular weak solution, uniform exponential decay estimate for the energy of the system is derived. The decay rate constant in this estimate depends only on the physical and geometric parameters of the beam, including the viscous external damping coefficient μ(x)≥0, and the boundary springs kr,kd≥0 and dampers ka,kv≥0. Some numerical examples are given to illustrate the role of the damping coefficient and the boundary dampers.

Improved error estimates of the time‐splitting methods for the long‐time dynamics of the Klein–Gordon–Dirac system with the small coupling constant

AbstractWe provide improved uniform error estimates for the time‐splitting Fourier pseudo‐spectral (TSFP) methods applied to the Klein–Gordon–Dirac system (KGDS) with the small parameter . We first reformulate the KGDS into a coupled Schrödinger–Dirac system (CSDS) and then apply the second‐order Strang splitting method to CSDS with the spatial discretization provided by Fourier pseudo‐spectral method. Based on rigorous analysis, we establish improved uniform error bounds for the second‐order Strang splitting method at up to the long time at . In addition to the conventional analysis methods, we mainly apply the regularity compensation oscillation technique for the analysis of long time dynamic simulation. The numerical results show that our method and conclusion are not only suitable for one‐dimensional problem, but also can be directly extended to higher dimensional problem and highly oscillatory problem. As far as we know there has not been any relevant long time analysis and any improved uniform error bounds for the TSFP method solving the KGDS. Our methods are novel and provides a reference for analyzing the improved error bounds of other coupled systems similar to the KGDS.

On the Fourier Analysis of the Einstein–Klein–Gordon System: Growth and Decay of the Fourier Constants

We consider the (1+3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1+3)$$\\end{document}-dimensional Einstein equations with negative cosmological constant coupled to a spherically symmetric, massless scalar field and study perturbations around the anti-de Sitter spacetime. We derive the resonant systems, pick out vanishing secular terms and discuss issues related to small divisors. Most importantly, we rigorously establish (sharp, in most of the cases) asymptotic behaviour for all the interaction coefficients. The latter is based on uniform estimates for the eigenfunctions associated to the linearized operator as well as on some oscillatory integrals.

On the incompressible and non-resistive limit of 3D compressible magnetohydrodynamic equations in bounded domains

In this paper, we investigate the incompressible and non-resistive limit for the initial boundary value problem of isentropic compressible resistive magnetohydrodynamic equations with ill-prepared initial data in three-dimensional bounded domains. We establish the higher-order uniform estimates with respect to both the Mach number and the resistivity coefficient in the framework of new type of weighted Sobolev spaces. Then we obtain the strong convergence of the magnetic field and the divergence-free component of the velocity field, as both the Mach number and the resistivity coefficient tend to zero.

Global convergence of a two-fluid non-isentropic Euler–Poisson system in one space dimension

In this paper, we study the global in time quasi-neutral limit for a two-fluid non-isentropic Euler–Poisson system in one space dimension. We prove that the system converges to the non-isentropic Euler equations as the Debye length tends to zero. This problem is studied for smooth solutions near the constant equilibrium state. To prove this result, we establish uniform energy estimates and various dissipation estimates with respect to the Debye length and the time. These estimates allow to pass to the limit to obtain the limit system by compactness arguments. In addition, the global convergence rate is obtained by use of stream function technique.

Uniform estimates for solutions of a class of nonlinear equations in a finite-dimensional space

Uniform estimates for solutions of a class of nonlinear equations in a finite-dimensional space

Improved uniform error estimates for the two-dimensional nonlinear space fractional Dirac equation with small potentials over long-time dynamics

We develop improved uniform error bounds on a second-order Strang splitting method for the long-time dynamics of the nonlinear space fractional Dirac equation (NSFDE) in two dimension (2D) with small electromagnetic potentials. First, a Strang splitting approach is implemented to discretize NSFDE in time. Afterwards the Fourier pseudospectral method is used to complete the discretization of NSFDE in space. With the aid of a second-order Strang splitting approach employed to the Dirac equation, the major local truncation error of the indicated numerical methods is established. Moreover, for the semi-discrete scheme and full-discretization, we rigorously demonstrate the improved, sharp uniform error estimates are O(ετ2) and O(h1m+h2m+ετ2) in virtue of the regularity compensation oscillation (RCO) technique. In the formulations, τ is the time step, hi(i=1,2) stands for spatial sizes in xi-directions, m is dependent on the regularity of solutions, and ε∈(0,1]. In order to verify our error bounds and to illustrate some fascinating long-time dynamical behaviors of the NSFDE with honeycomb lattice potentials for varied ε, numerical investigations are presented.