High-dimensional Equations Research Articles

Self-similarity and asymptotic stability for coupled nonlinear Schrödinger equations in high dimensions

This paper is concerned with systems of coupled Schrödinger equations with polynomial nonlinearities and dimension n ≥ 1 . We show the existence of global self-similar solutions and prove that they are asymptotically stable in a framework based on weak- L p spaces, whose elements have local finite L 2 -mass. The radial symmetry of the solutions is also addressed.

The Approximate Solutions of FPK Equations in High Dimensions for Some Nonlinear Stochastic Dynamic Systems

AbstractThe probabilistic solutions of the nonlinear stochastic dynamic (NSD) systems with polynomial type of nonlinearity are investigated with the subspace-EPC method. The space of the state variables of large-scale nonlinear stochastic dynamic system excited by white noises is separated into two subspaces. Both sides of the Fokker-Planck-Kolmogorov (FPK) equation corresponding to the NSD system is then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in another subspace is formulated. Therefore, the FPK equation in low dimensions is obtained from the original FPK equation in high dimensions and it makes the problem of obtaining the probabilistic solutions of large-scale NSD systems solvable with the exponential polynomial closure method. Examples about the NSD systems with polynomial type of nonlinearity are given to show the effectiveness of the subspace-EPC method in these cases.

Symmetric solutions of Einstein's equations in higher dimensions

We present a method of solving the Einstein equations under assumption of SO(n + 1) symmetry acting on n-dimensional spheres.

Analysis of some large-scale nonlinear stochastic dynamic systems with subspace-EPC method

The probabilistic solutions to some nonlinear stochastic dynamic (NSD) systems with various polynomial types of nonlinearities in displacements are analyzed with the subspace-exponential polynomial closure (subspace-EPC) method. The space of the state variables of the large-scale nonlinear stochastic dynamic system excited by Gaussian white noises is separated into two subspaces. Both sides of the Fokker-Planck-Kolmogorov (FPK) equation corresponding to the NSD system are then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in the other subspace is formulated. Therefore, the FPK equations in low dimensions are obtained from the original FPK equation in high dimensions and the FPK equations in low dimensions are solvable with the exponential polynomial closure method. Examples about multi-degree-offreedom NSD systems with various polynomial types of nonlinearities in displacements are given to show the effectiveness of the subspace-EPC method in these cases.

Erratum to: A KAM Theorem with Applications to Partial Differential Equations of Higher Dimensions

Erratum to: A KAM Theorem with Applications to Partial Differential Equations of Higher Dimensions

The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions

The final open part of Straussʼ conjecture on semilinear wave equations was the blow-up theorem for the critical case in high dimensions. This problem was solved by Yordanov and Zhang (2006) [18], or Zhou (2007) [21] independently. But the estimate for the lifespan, the maximal existence time, of solutions was not clarified in both papers. In this paper, we refine their theorems and introduce a new iteration argument to get the sharp upper bound of the lifespan. As a result, with the sharp lower bound by Li and Zhou (1995) [10], the lifespan T ( ε ) of solutions of u t t − Δ u = u 2 in R 4 × [ 0 , ∞ ) with the initial data u ( x , 0 ) = ε f ( x ) , u t ( x , 0 ) = ε g ( x ) of a small parameter ε > 0 , compactly supported smooth functions f and g, has an estimate exp ( c ε − 2 ) ⩽ T ( ε ) ⩽ exp ( C ε − 2 ) , where c and C are positive constants depending only on f and g. This upper bound has been known to be the last open optimality of the general theory for fully nonlinear wave equations.

Hydrodynamics at the smallest scales: a solvability criterion for Navier–Stokes equations in high dimensions

Strong global solvability is difficult to prove for high-dimensional hydrodynamic systems because of the complex interplay between nonlinearity and scale invariance. We define the Ladyzhenskaya-Lions exponent α(L)(n)=(2+n)/4 for Navier-Stokes equations with dissipation -(-Δ)(α) in R(n), for all n≥2. We review the proof of strong global solvability when α≥α(L)(n), given smooth initial data. If the corresponding Euler equations for n>2 were to allow uncontrolled growth of the enstrophy (1/2)∥∇u∥(L²)(2), then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier-Stokes equations for α<α(L)(n). The energy is critical under scale transformations only for α=α(L)(n).

Methodology for the solutions of some reduced Fokker‐Planck equations in high dimensions

AbstractIn this paper, a new methodology is formulated for solving the reduced Fokker‐Planck (FP) equations in high dimensions based on the idea that the state space of large‐scale nonlinear stochastic dynamic system is split into two subspaces. The FP equation relevant to the nonlinear stochastic dynamic system is then integrated over one of the subspaces. The FP equation for the joint probability density function of the state variables in another subspace is formulated with some techniques. Therefore, the FP equation in high‐dimensional state space is reduced to some FP equations in low‐dimensional state spaces, which are solvable with exponential polynomial closure method. Numerical results are presented and compared with the results from Monte Carlo simulation and those from equivalent linearization to show the effectiveness of the presented solution procedure. It attempts to provide an analytical tool for the probabilistic solutions of the nonlinear stochastic dynamics systems arising from statistical mechanics and other areas of science and engineering.

High frequency dispersive estimates for the Schrödinger equation in high dimensions

We prove optimal dispersive estimates at high frequency for the Schrodinger group for a class of real-valued potentials V (x) = O(� x� −δ ), δ>n − 1, and V ∈ C k (R n ), k>k n ,w heren 4a nd n−3 2 kn < n 2 . We also give a sufficient condition in terms of L 1 → L ∞ bounds for the formal iterations of Duhamel's formula, which might be satisfied for potentials

Radial Fourier multipliers in high dimensions

Given a fixed p≠2, we prove a simple and effective characterization of all radial multipliers of $ \mathcal{F}{L^p}\left( {{\mathbb{R}^d}} \right) $, provided that the dimension d is sufficiently large. The method also yields new Lq space-time regularity results for solutions of the wave equation in high dimensions.

Treatment of incompatible initial and boundary data for parabolic equations in higher dimension

A new method is proposed to improve the numerical simulation of time dependent problems when the initial and boundary data are not compatible. Unlike earlier methods limited to space dimension one, this method can be used for any space dimension. When both methods are applicable (in space dimension one), the improvements in precision are comparable, but the method proposed here is not restricted by dimension.

Stability results for higher dimensional equations on time scales

The stability of the equilibrium solution of x Δ = f(x) where f: D ⊂ ℝ n → ℝ n and x: T → ℝ n , T a time scale, with regards to the Hilger Circle is examined.

THE MASS-CRITICAL FOURTH-ORDER SCHRÖDINGER EQUATION IN HIGH DIMENSIONS

We prove global wellposedness and scattering for the Mass-critical homogeneous fourth-order Schrödinger equation in high dimensions n ≥ 5, for general L2initial data in the defocusing case, and for general initial data with Mass less than certain fraction of the Mass of the ground state in the focusing case.

Existence results for the Klein-Gordon-Maxwell equations in higher dimensions with critical exponents

In this paper we study the existence of radially symmetricsolitary waves in $R^N$ for the nonlinear Klein-Gordonequations coupled with the Maxwell's equations when thenonlinearity exhibits critical growth. The main feature of thiskind of problem is the lack of compactness arising in connectionwith the use of variational methods.

Dynamics for the energy critical nonlinear wave equation in high dimensions

T. Duyckaerts and F. Merle (2008) studied the variational structure near the ground state solution W W of the energy critical wave equation and classified the solutions with the threshold energy E ( W , 0 ) E(W,0) in dimensions d = 3 , 4 , 5 d=3,4,5 . In this paper, we extend the results to all dimensions d ≥ 6 d\ge 6 . The main issue in high dimensions is the non-Lipschitz continuity of the nonlinearity which we get around by making full use of the decay property of W W .

Traveling wave solutions of the Schrödinger map equation

AbstractWe first construct traveling wave solutions for the Schrödinger map in ℝ2 of the form m(x1, x2 − ϵ t), where m has exactly two vortices at approximately $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}(\pm {{1}\over{2 \epsilon}}, 0) \in \R^2$ of degree ±1. We use a perturbative approach that gives a complete characterization of the asymptotic behavior of the solutions. With a few modifications, a similar construction yields traveling wave solutions of Schrödinger map equations in higher dimensions. © 2010 Wiley Periodicals, Inc.

Future geodesic completeness of some spatially homogeneous solutions of the vacuum Einstein equations in higher dimensions

It is known that all spatially homogeneous solutions of the vacuum Einstein equations in four dimensions which exist for an infinite proper time towards the future are future geodesically complete. This paper investigates whether the analogous statement holds in higher dimensions. A positive answer to this question is obtained for a large class of models which can be studied with the help of Kaluza–Klein reduction to solutions of the Einstein-scalar field equations in four dimensions. The proof of this result makes use of a criterion for geodesic completeness which is applicable to more general spatially homogeneous models.

Q-deformed harmonic and Clifford analysis and the q-Hermite and Laguerre polynomials

We define a q-deformation of the Dirac operator, inspired by the one-dimensional q-derivative. This implies a q-deformation of the partial derivatives. By taking the square of this Dirac operator we find a q-deformation of the Laplace operator. This allows us to construct q-deformed Schrödinger equations in higher dimensions. The equivalence of these Schrödinger equations with those defined on q-Euclidean space in quantum variables is shown. We also define the m-dimensional q-Clifford–Hermite polynomials and show their connection with the q-Laguerre polynomials. These polynomials are orthogonal with respect to an m-dimensional q-integration, which is related to integration on q-Euclidean space. The q-Laguerre polynomials are the eigenvectors of an suq(1|1)-representation.

On the impact of the scheme for solving the higher dimensional equation in coupled population balance systems

AbstractPopulation balance systems are models for processes in nature and industry that lead to a coupled system of equations (Navier–Stokes equations, transport equations, etc.) where the equations are defined in domains with different dimensions. This paper will study the impact of using different schemes for solving the three‐dimensional (3D) equation of a precipitation process in a two‐dimensional flow domain. The numerical schemes for the 3D equation are assessed with respect to the median of the volume fraction of the particle size distribution and the computational costs. It turns out that in the case of a structured flow field with small variations in time all schemes give qualitatively the same results. For a highly time‐ dependent flow field, the evolution of the median of the volume fraction differs considerably between first order and higher order schemes. Copyright © 2010 John Wiley &amp; Sons, Ltd.

Boundary Controllability for the Quasilinear Wave Equation

We study the boundary exact controllability for the quasilinear wave equation in high dimensions. Our main tool is the geometric analysis. We derive the existence of long time solutions near an equilibrium, prove the locally exact controllability around the equilibrium under some checkable geometrical conditions. We then establish the globally exact controllability in such a way that the state of the quasilinear wave equation moves from an equilibrium in one location to an equilibrium in another location under some geometrical conditions. The Dirichlet action and the Neumann action are studied, respectively. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and equilibria of the quasilinear wave equation. A criterion of exact controllability is given, which based on the sectional curvature of the Riemann metric. Some examples are presented to verify the global exact controllability.