The inviscid limit for the complex Ginzburg-Landau equation

We study the inviscid limit of the complex Ginzburg-Landau equation. We observe that the solutions for the complex Ginzburg-Landau equation converge to the corresponding solutions for the nonlinear Schrödinger equation. We give its convergence rate. We estimate the integral forms of solutions for two equations.     

Exact solutions of Rayleigh's equation and sufficient conditions for inviscid instability of parallel,bounded shear flows

The primary goal of this paper is to specify sufficient conditions for the inviscid instability of a general class of plane parallel shear flows. For given complex eigenvaluec and real wave number , and for givenh(y), the real part of the adjoint eigenfunction, Rayleigh's equation is converted into a nonlinear integral equation for the basic velocity profileU(y). Sufficient conditions are deduced for the existence and uniqueness of solutions to this integral equation, subject to appropriate homogeneous boundary conditions on the eigenfunction (y); the velocity profilesU(y) so derived are guaranteed to be unstable. Also separately described in this paper is a method to obtain a general class of new, exact neutrally stable solutions of Rayleigh's equation; given any realc and , and a function (U), the velocity profileU(y) and the eigenfunction (U(y)) may be determined theoretically. A specific example of a class of neutrally stable solutions for jet-like profiles on an unbounded domain is given.     

New solutions for the one-dimensional nonconservative inviscid Burgers equation

The propagation of travelling waves is a relevant physical phenomenon. As usual the understanding of a real propagating wave depends upon a correct formulation of a idealized model. Discontinuous functions, Dirac-δ measures and their distributional derivatives are, respectively, idealizations of sharp jumps, localized high peaks and single sharp localised oscillations. In the present paper we study the propagation of distributional travelling waves for Burgers inviscid equation. This will be afforded by our theory of distributional products, and is based on a rigorous and consistent concept of solution we have introduced in [C.O.R. Sarrico, Distributional products and global solutions for nonconservative inviscid Burgers equation, J. Math. Anal. Appl. 281 (2003) 641-656]. Our approach exhibit Dirac-δ travelling solitons (they are just the “infinitesimal narrow solitons” of Maslov, Omel'yanov and Tsupin [V.P. Maslov, O.A. Omel'yanov, Asymptotic soliton-form solutions of equations with small dispersion, Russian Math. Surveys 36 (1981) 73-149; V.P. Maslov, V.A. Tsupin, Necessary conditions for the existence of infinitely narrow solitons in gas dynamics, Soviet Phys. Dokl. 24 (1979) 354-356]) and also solutions which are not measures such as for instance u(x,t)=b+δ^′(x−bt), a wave of constant speed b. Moreover, for signals with two jump discontinuities we have, in our setting, the propagation of more solitons and more values for the signal speed are allowed than those afforded within classical framework.     

Finite-difference approximation of dirichlet observation problems for weak solutions to the wave equation subject to Robin boundary conditions

For the wave equation with variable coefficients subject to Neumann and Robin boundary conditions, two mutually dual problems are considered: the Dirichlet observation problem with weak generalized solutions and the control problem with strong generalized solutions. Both problems are approximated by finite differences preserving the duality relation. The convergence of the approximate solutions is established in the norms of the corresponding dual spaces.     

Sufficient conditions for the blowup of a solution to the Boussinesq equation subject to a nonlinear Neumann boundary condition

Sufficient blowup conditions are obtained for a solution to the generalized Boussinesq equation subject to a nonlinear Neumann boundary condition.     

Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation

Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation

     

Uniform local well-posedness and inviscid limit for the Benjamin-Ono-Burgers equation

In this paper, we study the Cauchy problem for the Benjamin-Ono-Burgers equation \({\partial _t}u - \epsilon \partial _x^2u + {\cal H}\partial _x^2u + u{u_x} = 0\), where \({\cal H}\) denotes the Hilbert transform operator. We obtain that it is uniformly locally well-posed for small data in the refined Sobolev space \({\tilde H^\sigma }(\mathbb{R})\,\,(\sigma \geqslant 0)\), which is a subspace of L²(ℝ). It is worth noting that the low-frequency part of \({\tilde H^\sigma }(\mathbb{R})\) is scaling critical, and thus the small data is necessary. The high-frequency part of \({\tilde H^\sigma }(\mathbb{R})\) is equal to the Sobolev space H^σ (ℝ) (σ ⩾ 0) and reduces to L²(ℝ). Furthermore, we also obtain its inviscid limit behavior in \({\tilde H^\sigma }(\mathbb{R})\) (σ ⩾ 0).

     

On the well-posedness of the inviscid SQG equation

In this paper we consider the inviscid SQG equation on the Sobolev spaces $H^s({\mathbb{R}}^2)$, $s>2$. Using a geometric approach we show that for any $T>0$ the corresponding solution map, $\theta (0)?\theta (T)$, is nowhere locally uniformly continuous.     

Cascadic multigrid methods combined with sixth order compact scheme for poisson equation

Based on the extrapolation theory and a sixth order compact difference scheme, new extrapolation interpolation operator and extrapolation cascadic multigrid methods for two dimensional Poisson equation are presented. The new extrapolation interpolation operator is used to provide a better initial value on refined grid. The convergence of the new methods are given. Numerical experiments are shown to illustrate that the new methods have higher accuracy and efficiency.     

Statistical theory for the stochastic Burgers equation in the inviscid limit

A statistical theory is developed for the stochastic Burgers equation in the inviscid limit. Master equations for the probability density functions of velocity, velocity difference, and velocity gradient are derived. No closure assumptions are made. Instead, closure is achieved through a dimension reduction process; namely, the unclosed terms are expressed in terms of statistical quantities for the singular structures of the velocity field, here the shocks. Master equations for the environment of the shocks are further expressed in terms of the statistics of singular structures on the shocks, namely, the points of shock generation and collisions. The scaling laws of the structure functions are derived through the analysis of the master equations. Rigorous bounds on the decay of the tail probabilities for the velocity gradient are obtained using realizability constraints. We also establish that the probability density function Q(ξ) of the velocity gradient decays as |ξ|^−7/2 as ξ → − ∞. © 2000 John Wiley & Sons, Inc.     

Second-order,L 0-stable methods for the heat equation with time-dependent boundary conditions

A family of second-order,L ₀-stable methods is developed and analysed for the numerical solution of the simple heat equation with time-dependent boundary conditions. Methods of the family need only real arithmetic in their implementation. In a series of numerical experiments no oscillations, which are a feature of some results obtained usingA ₀-stable methods, are observed in the computed solutions. Splitting techniques for first- and second-order hyperbolic problems are also considered.Dedicated to Professor J. Crank on the occasion of his 80th birthday     

A method for integration of unstable systems of ordinary differential equation subject to two-point boundary conditions

Instability problems in systems of differential equations are discussed. A matrix technique is given for producing numerical solutions to a system of ordinary differential equations with boundary conditions specified at each end of the interval when the system contains dominant solutions which give rise to numerical instability in conventional integration methods. A method of bringing up the initial conditions is described, whereby the two-point nature of the problem is made use of to stabilize the system. Three numerical examples are included.     

The lifespan of classical solutions for the inviscid Surface Quasi-geostrophic equation

We consider classical solutions of the inviscid Surface Quasi-geostrophic equation that are a small perturbation ϵ from a radial stationary solution $\theta =|x|$. We use a modified energy method to prove the existence time of classical solutions from $\frac{1}{ϵ}$ to a time scale of $\frac{1}{{ϵ}^4}$. Moreover, by perturbing in a suitable direction we construct global smooth solutions, via bifurcation, that rotate uniformly in time and space.     

An inviscid regularization for the surface quasi‐geostrophic equation

Inspired by recent developments in Berdina‐like models for turbulence, we propose an inviscid regularization for the surface quasi‐geostrophic (SQG) equations. We are particularly interested in the celebrated question of blowup in finite time of the solution gradient of the SQG equations. The new regularization yields a necessary and sufficient condition, satisfied by the regularized solution, when a regularization parameter α tends to 0 for the solution of the original SQG equations to develop a singularity in finite time. As opposed to the commonly used viscous regularization, the inviscid equations derived here conserve a modified energy. Therefore, the new regularization provides an attractive numerical procedure for finite‐time blowup testing. In particular, we prove that, if the initial condition is smooth, then the regularized solution remains as smooth as the initial data for all times. © 2007 Wiley Periodicals, Inc.     

Gaussian bounds for derivatives of central Gaussian semigroups on compact groups

For symmetric central Gaussian semigroups on compact connected groups, assuming the existence of a continuous density, we show that this density admits space derivatives of all orders in certain directions. Under some additional assumptions, we prove that these derivatives satisfy certain Gaussian bounds.

     

Resolution of the Transport equation subject to constraint

We present a method for solving the Transport equation when its solution has to belong to a constrained set which is not required to be convex. An autonomous formulation of the characteristics method allows us to use the tangency condition which has been introduced for ordinary differential equations. Thus we obtain a sufficient condition for existence of solutions, which shows the interplay between the geometry of the constraints set K   and the velocity field β$\beta$. A numerical method is proposed for solving the problem when the sufficient condition is not satisfied. A numerical experiment is presented showing the efficiency of the algorithm proposed.     

The inviscid limit for the Landau-Lifshitz-Gilbert equation in the critical Besov space

We prove that in dimensions three and higher the Landau-Lifshitz-Gilbert equation with small initial data in the critical Besov space is globally well-posed in a uniform way with respect to the Gilbert damping parameter. Then we show that the global solution converges to that of the Schr¨odinger maps in the natural space as the Gilbert damping term vanishes. The proof is based on some studies on the derivative Ginzburg-Landau equations.     

A problem without initial conditions for a nonlinear ultraparabolic equation with degeneration

We consider the mixed problem for a second-order nonlinear degenerate ultraparabolic equation. We investigate the existence of generalized solutions of this problem in a bounded domain as well as of weak solutions (in the sense of a limit of sequences) of the problem without initial conditions for this equation.