Symmetries and Critical Phenomena in Fluids

We study two-dimensional active scalar systems arising in fluid dynamics in critical spaces in the whole plane. We prove an optimal well-posedness result that allows for the data and solutions to be scale-invariant. These scale-invariant solutions are new and their study seems to have far-reaching consequences. More specifically, we first show that the class of bounded vorticities satisfying a discrete rotational symmetry is a global existence and uniqueness class for the two-dimensional Euler squation. That is, in the well-known L¹∩L^∞ theory of Yudovich, the L¹-assumption can be dropped upon having an appropriate symmetry condition. We also show via explicit examples the necessity of discrete symmetry for the uniqueness. This already answers problems raised by Lions in 1996 and Bendetto, Marchioro, and Pulvirenti in 1993. Next, we note that merely bounded vorticity allows for one to look at solutions that are invariant under scaling—the class of vorticities that are 0-homo-geneous in space. Such vorticity is shown to satisfy a new one-dimensional evolution equation on 𝕊¹. Solutions are also shown to exhibit a number of interesting properties. In particular, using this framework, we construct time quasi-periodic solutions to the two-dimensional Euler equation exhibiting pendulum-like behavior. Finally, using the analysis of the one-dimensional equation, we exhibit strong solutions to the two-dimensional Euler equation with compact support for which angular derivatives grow at least (almost) quadratically in time (in particular, superlinear) or exponential in time (the latter being in the presence of a boundary). A similar study can be done for the surface quasi-geostrophic (SQG) equation. Using the same symmetry condition, we prove local existence and uniqueness of solutions that are merely Lipschitz continuous near the origin—though, without the symmetry, Lipschitz initial data is expected to lose its Lipschitz continuity immediately. Once more, a special class of radially homogeneous solutions is considered, and we extract a one-dimensional model that bears great resemblance to the so-called De Gregorio model. We then show that finite-time singularity formation for the one-dimensional model implies finite-time singularity formation in the class of Lipschitz solutions to the SQG equation that are compactly support. While the study of special infinite energy (i.e., nondecaying) solutions to fluid models is classical, this appears to be the first case where these special solutions can be embedded into a natural existence/uniqueness class for the equation. Moreover, these special solutions approximate finite-energy solutions for long time and have direct bearing on the global regularity problem for finite-energy solutions. © 2019 Wiley Periodicals, Inc.     

Error analysis of a fully discrete projection method for magnetohydrodynamic system

In this paper, we develop and analyze a finite element projection method for magnetohydrodynamics equations in Lipschitz domain. A fully discrete scheme based on Euler semi-implicit method is proposed, in which continuous elements are used to approximate the Navier–Stokes equations and H ( curl ) conforming Nédélec edge elements are used to approximate the magnetic equation. One key point of the projection method is to be compatible with two different spaces for calculating velocity, which leads one to obtain the pressure by solving a Poisson equation. The results show that the proposed projection scheme meets a discrete energy stability. In addition, with the help of a proper regularity hypothesis for the exact solution, this paper provides a rigorous optimal error analysis of velocity, pressure and magnetic induction. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme.     

The Pathwise Numerical Approximation of Stationary Solutions of Semilinear Stochastic Evolution Equations

Under a one-sided dissipative Lipschitz condition on its drift, a stochastic evolution equation with additive noise of the reaction-diffusion type is shown to have a unique stochastic stationary solution which pathwise attracts all other solutions. A similar situation holds for each Galerkin approximation and each implicit Euler scheme applied to these Galerkin approximations. Moreover, the stationary solution of the Euler scheme converges pathwise to that of the Galerkin system as the stepsize tends to zero and the stationary solutions of the Galerkin systems converge pathwise to that of the evolution equation as the dimension increases. The analysis is carried out on random partial and ordinary differential equations obtained from their stochastic counterparts by subtraction of appropriate Ornstein-Uhlenbeck stationary solutions.     

Role of fundamental solutions for optimal Lipschitz extensions on hyperbolic space

In this paper we consider the problem of finding the relation between absolutely minimizing Lipschitz extension of a given function defined over a subset of the hyperbolic space and the viscosity solution of the PDE that appears from the associated variational problem. Here we have shown that the absolute minimizers can be fully characterized by a comparison principle (comparison with cones) with the fundamental solutions of the associated PDE. We have finally proved that the three properties, (i) comparison with cones, (ii) absolutely minimizing Lipschitz extension and (iii) viscosity solution of associated PDE, are equivalent.     

Uniqueness theorem for a Cauchy problem with hysteresis

The Cauchy problem for an ordinary differential equation coupled with a hysteresis operator is studied. Under physically reasonable assumptions on the forcing term, uniqueness of solutions is shown without assuming Lipschitz continuity of the hysteresis curves. The result is true for any kind of hysteresis operators with monotone curves of motion.

     

Approximation of optimal feedback control: a dynamic programming approach

We consider the general continuous time finite-dimensional deterministic system under a finite horizon cost functional. Our aim is to calculate approximate solutions to the optimal feedback control. First we apply the dynamic programming principle to obtain the evolutive Hamilton–Jacobi–Bellman (HJB) equation satisfied by the value function of the optimal control problem. We then propose two schemes to solve the equation numerically. One is in terms of the time difference approximation and the other the time-space approximation. For each scheme, we prove that (a) the algorithm is convergent, that is, the solution of the discrete scheme converges to the viscosity solution of the HJB equation, and (b) the optimal control of the discrete system determined by the corresponding dynamic programming is a minimizing sequence of the optimal feedback control of the continuous counterpart. An example is presented for the time-space algorithm; the results illustrate that the scheme is effective.     

Quotients of little Lipschitz algebras

We prove a Tietze type theorem which provides extensions of little Lipschitz functions defined on closed subsets. As a consequence, we get that the quotient of any little Lipschitz algebra by any norm-closed ideal is another little Lipschitz algebra.

     

Symmetry Analysis of Abel's Equation

A solution algorithm for Abel's equation and some generalizations based on a nontrivial Lie symmetry of a particular kind, i.e., so-called structure-preserving symmetry, is described. For the existence of such a symmetry a criterion in terms of the coefficients of the so-called rational normal form of the given equation is derived. If it is affirmative, solving Abel's equation is reduced to a well-defined integration problem. It is shown that almost all known ad hoc methods for obtaining closed form solutions are consequences of this type of symmetry. Possible extensions of this scheme to more general classes of first-order ordinary differential equations are pointed out.     

The construction of solutions to the heat equation backward in time

Solutions to the backwards heat equation are approximated by solutions of a pseudo-heat equation. Solutions to this modified equation are constructed by means of a fundamental solution and potential theory, and it is shown that the fundamental solution can be approximated by various expansions in special functions.     

Existence and uniqueness of traveling waves and error estimates for Godunov schemes of conservation laws

The existence and uniqueness of the Lipschitz continuous traveling wave of Godunov's scheme for scalar conservation laws are proved. The structure of the traveling waves is studied. The approximation error of Godunov's scheme on single shock solutions is shown to be .

     

Stochastic Differential Equations with Non-Lipschitz Coefficients in Hilbert Spaces

Abstract

In this article, we discuss the successive approximations problem for the solutions of the semilinear stochastic differential equations in Hilbert spaces with cylindrical Wiener processes under some conditions which are weaker than the Lipschitz one. We establish the existence and the uniqueness of the solution and additionally, in our framework we consider a limiting problem for the mild solution. It is shown that the mild solution tends to the solution of the stochastic differential equation of Itô type in finite dimensional space.     

Analysis of new conservative difference scheme for two-dimensional Rosenau-RLW equation

In this article, numerical solution for the Rosenau-RLW equation in 2D is considered and a conservative Crank–Nicolson finite difference scheme is proposed. Existence of the numerical solutions for the difference scheme has been shown by Browder fixed point theorem. A priori bound and uniqueness as well as conservation of discrete mass and discrete energy for the finite difference solutions are discussed. Unconditional stability and a second-order accuracy on both space and time of the difference scheme are proved. Numerical experiments are given to support our theoretical results.     

Fractional order iterative functional differential equations with parameter

In this paper, we study a fractional order iterative functional differential equation with parameter. Some theorems to prove existence of the iterative series solutions are presented under some natural conditions. Unfortunately, uniqueness results can not be obtained since the solution operator is not Lipschitz continuous but only Hölder continuous. Meanwhile, data dependence results of solutions and parameters provide possible way to describe the error estimates between explicit and approximative solutions for such problems. We also make some examples to illustrate our results. Finally, we conclude with some possible extensions to general parametrized iterative fractional functional differential equations.     

On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming

An approximation of the Hamilton-Jacobi-Bellman equation connected with the infinite horizon optimal control problem with discount is proposed. The approximate solutions are shown to converge uniformly to the viscosity solution, in the sense of Crandall-Lions, of the original problem. Moreover, the approximate solutions are interpreted as value functions of some discrete time control problem. This allows to construct by dynamic programming a minimizing sequence of piecewise constant controls.     

Numerical viscosity and the entropy condition

Weak solutions of hyperbolic conservation laws are not uniquely determined by their initial values; an entropy condition is needed to pick out the physically relevant solutions. The question arises whether finite difference approximations converge to this particular solution. It is known that this is not always the case with the standard Lax-Wendroff (L-W) difference scheme. In this paper a simple variant of the L-W scheme is devised which retains its desirable computational features—conservation form, three point scheme, second-order accuracy on smooth solutions, but which has the additional property that limit solutions satisfy the entropy condition. This variant is constructed by adding a simple nonlinear artificial viscosity to the usual L-W operator. The nature of the viscosity is deduced by first analyzing a model differential equation derived from the truncation error for the L-W operator, keeping only terms of order (Δx)². Furthermore, this viscosity is “switched on” only when sufficiently steep discrete gradients develop in the approximate solution: The full L-W scheme is then shown to have the desired property provided that the Courant-Friedrichs-Lewy restriction |λf′(u)|≤0.14 is satisfied.     

A characterisation of oscillations in the discrete two-dimensional convection-diffusion equation

It is well known that discrete solutions to the convection-diffusion equation contain nonphysical oscillations when boundary layers are present but not resolved by the discretisation. However, except for one-dimensional problems, there is little analysis of this phenomenon. In this paper, we present an analysis of the two-dimensional problem with constant flow aligned with the grid, based on a Fourier decomposition of the discrete solution. For Galerkin bilinear finite element discretisations, we derive closed form expressions for the Fourier coefficients, showing them to be weighted sums of certain functions which are oscillatory when the mesh Péclet number is large. The oscillatory functions are determined as solutions to a set of three-term recurrences, and the weights are determined by the boundary conditions. These expressions are then used to characterise the oscillations of the discrete solution in terms of the mesh Péclet number and boundary conditions of the problem.

     

Finite difference discretization of the Benjamin‐Bona‐Mahony‐Burgers equation

Numerical solutions of the Benjamin‐Bona‐Mahony‐Burgers equation in one space dimension are considered using Crank‐Nicolson‐type finite difference method. Existence of solutions is shown by using the Brower's fixed point theorem. The stability and uniqueness of the corresponding methods are proved by the means of the discrete energy method. The convergence in L^∞‐norm of the difference solution is obtained. A conservative difference scheme is presented for the Benjamin‐Bona‐Mahony equation. Some numerical experiments have been conducted in order to validate the theoretical results.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Blowup in a mass-conserving convection-diffusion equation with superquadratic nonlinearity

A nonlinear convection-diffusion equation with boundary conditions that conserve the spatial integral of the solution is considered. Previous results on finite-time blowup of solutions and on decay of solutions to the corresponding Cauchy problem were based on the assumption that the nonlinearity obeyed a power law. In this paper, it is shown that assumptions on the growth rate of the nonlinearity, which take the form of weak superquadraticity and strong superlinearity criteria, are sufficient to imply that a large class of nonnegative solutions blow up in finite time.

     

Numerical analysis of an explicit approximation scheme for the Landau-Lifshitz-Gilbert equation

The Landau-Lifshitz-Gilbert equation describes magnetic behavior in ferromagnetic materials. Construction of numerical strategies to approximate weak solutions for this equation is made difficult by its top order nonlinearity and nonconvex constraint. In this paper, we discuss necessary scaling of numerical parameters and provide a refined convergence result for the scheme first proposed by Alouges and Jaisson (2006). As an application, we numerically study discrete finite time blowup in two dimensions.

     

Uniform Convergence Analysis of Finite Difference Scheme for Singularly Perturbed Delay Differential Equation on an Adaptively Generated Grid

<正>Adaptive grid methods are established as valuable computational technique in approximating effectively the solutions of problems with boundary or interior layers. In this paper,we present the analysis of an upwind scheme for singularly perturbed differential-difference equation on a grid which is formed by equidistributing arc-length monitor function.It is shown that the discrete solution obtained converges uniformly with respect to the perturbation parameter.Numerical experiments illustrate in practice the result of convergence proved theoretically.