Regularising discretizations of the Birkhoff-Rott equation

A thin shear layer moving from the trailing edge of a two-dimensional aerofoil section downstream can be interpreted as a curve of discontinuity for the tangential velocity and may be approximated by a vortex sheet in inviscid, incompressible fluid flow. It is well known that vortex sheets are subject to instabilities of Kelvin-Helmholtz type which lead to roll-up phenomena in the wake. The motion of such sheets is governed by the Birkhoff-Rott equation. In the case of Kelvin-Helmholtz instability it seems clear that a curvature singularity occurs at a certain critical time and that consistent discretizations of the Birkhoff-Rott equation may fail to yield reliable results even before the time of occurrence of a singularity. We discuss the modification of the Biot-Savart kernel in the sense of Krasny who regularized the kernel by means of a global parameter. Using discrete Fourier transform we show the damping influence of this regularization technique. We modify the kernel carefully by introducing a regularization found in ordinary vortex methods and show that reliable results may be obtained up to and slightly after the singularity formation without increasing the accuracy of the computation.     

Minimal Blow-Up Solutions to the Mass-Critical Inhomogeneous NLS Equation

We consider the mass-critical focusing nonlinear Schrödinger equation in the presence of an external potential, when the nonlinearity is inhomogeneous. We show that if the inhomogeneous factor in front of the nonlinearity is sufficiently flat at a critical point, then there exists a solution which blows up in finite time with the maximal (unstable) rate at this point. In the case where the critical point is a maximum, this solution has minimal mass among the blow-up solutions. As a corollary, we also obtain unstable blow-up solutions of the mass-critical Schrödinger equation on some surfaces. The proof is based on properties of the linearized operator around the ground state, and on a full use of the invariances of the equation with an homogeneous nonlinearity and no potential, via time-dependent modulations.     

Periodic solutions of nonlinear differential equations involving the method of scalar nonlinearities

Summary The recently developed method of scalar nonlinearities is applied to establish a new type of existence proof for periodic solutions of nonlinear differential equations. It is proved that given a periodic solution of a certain linear differential equation whose coefficients are subject to some nonlinear constraint, a nonlinear differential equation, which is closely related to the linear one, has a periodic solution (of the same period) as well. While, in general, the nonlinear equation will not be explicitly resolvable, the linear equation (with constraint) will allow for explicitly given solutions.The proof is carried out by constructing a homotopy (between appropriately chosen integral operators) and is based on Leray-Schauder theory. Thus, an essential hypothesis is the a-priori boundedness of certain intermediate problems. The very definition of the homotopy, which seems to be unprecedented in the literature, bears resemblance with the introduction of Dirac's-function.The theory is applied to Duffing's equation, resulting in an abstract existence statement as well as the explicit construction of numerically tractable intermediate problems.     

Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms

For the Cauchy problem for the nonlinear wave equation with nonlinear damping and source terms we define stable and unstable sets for the initial data. We prove that, if during the evolution the solution enters into the stable set, the solution is global and we are able to estimate the decay rate of the energy. If during the evolution the solution enters into the unstable set, the solution blows up in finite time.     

The Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off

In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cut-off.This equation is partially elliptic in the velocity direction and degenerates in the spatial variable.We consider the nonlinear Cauchy problem for the fluctuation around the Maxwellian distribution and prove that any solution with mild regularity will become smooth in the Gevrey class at positive time with the Gevrey index depending on the angular singularity.Our proof relies on the symbolic calculus for the collision operator and the global subelliptic estimate for the Cauchy problem of the linearized Boltzmann operator.     

On the blowing up of solutions to one-dimensional quantum Navier-Stokes equations

The blow-up in finite time for the solutions to the initial-boundary value problem associated to the one-dimensional quantum Navier-Stokes equations in a bounded domain is proved. The model consists of the mass conservation equation and a momentum balance equation, including a nonlinear third-order differen- tial operator, with the quantum Bohm potential, and a density-dependent viscosity. It is shown that, under suitable boundary conditions and assumptions on the initial data, the solution blows up after a finite time, if the viscosity constant is not bigger than the scaled Planck constant. The proof is inspired by an observable constructed by Gamba, Gualdani and Zhang, which has been used to study the blowing up of solutions to quantum hydrodynamic models.     

Dynamics of a class of ODEs more general than almost periodic

A temporally global solution, if it exists, of a nonautonomous ordinary differential equation need not be periodic, almost periodic or almost automorphic when the forcing term is periodic, almost periodic or almost automorphic, respectively. An alternative class of functions extending periodic and almost periodic functions which has the property that a bounded temporally global solution solution of a nonautonomous ordinary differential equation belongs to this class when the forcing term does is introduced here. Specifically, the class of functions consists of uniformly continuous functions, defined on the real line and taking values in a Banach space, which have pre-compact ranges. Besides periodic and almost periodic functions, this class also includes many nonrecurrent functions. Assuming a hyperbolic structure for the unperturbed linear equation and certain properties for the linear and nonlinear parts, the existence of a special bounded entire solution, as well the existence of stable and unstable manifolds of this solution are established. Moreover, it is shown that this solution and these manifolds inherit the temporal behaviour of the vector field equation. In the stable case it is shown that this special solution is the pullback attractor of the system. A class of infinite dimensional examples involving a linear operator consisting of a time independent part which generates a C₀-semigroup plus a small time dependent part is presented and applied to systems of coupled heat and beam equations.     

Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation

We consider a version of the stationary phase method in one dimension of A. Erdélyi, allowing the phase to have stationary points of non‐integer order and the amplitude to have integrable singularities. After having completed the original proof and improved the error estimate in the case of regular amplitude, we consider a modification of the method by replacing the smooth cut‐off function employed in the source by a characteristic function, leading to more precise remainder estimates. We exploit this refinement to study the time‐asymptotic behaviour of the solution of the free Schrödinger equation on the line, where the Fourier transform of the initial data is compactly supported and has a singularity. We obtain asymptotic expansions with respect to time in certain space‐time cones as well as uniform and optimal estimates in curved regions, which are asymptotically larger than any space‐time cone. These results show the influence of the frequency band and of the singularity on the propagation and on the decay of the wave packets. Copyright © 2016 John Wiley & Sons, Ltd.     

INSTABILITY OF TRAVELING WAVES OF THE KURAMOTO－SIVASHINSKY EQUATION

Iatroduct     

Finite element solution of nonlinear diffusion problems

In this paper we describe the Rothe-finite element numerical scheme to find an approximate solution of a nonlinear diffusion problem modeled as a parabolic partial differential equation of even order. This scheme is based on the Rothe’s approximation in time and on the finite element method (FEM) approximation in the spatial discretization. A proof of convergence of the approximate solution is given and error estimates are shown.     

一类非线性扩散方程解的blow up速率估计

利用F riedm an-M cleod方法和变动尺度方法研究了一类具有非线性边界条件的非线性扩散方程解的b low up问题,证明了解在有限时间b low up,并且得到了b low up速率估计.     

Superconvergence Analysis of a BDF-Galerkin FEM for the Nonlinear Klein-Gordon-Schrödinger Equations with Damping Mechanism

The focus of this paper is on a linearized backward differential formula (BDF) scheme with Galerkin FEM for the nonlinear Klein-Gordon-Schrödinger equations (KGSEs) with damping mechanism. Optimal error estimates and superconvergence results are proved without any time-step restriction condition for the proposed scheme. The proof consists of three ingredients. First, a temporal-spatial error splitting argument is employed to bound the numerical solution in certain strong norms. Second, optimal error estimates are derived through a novel splitting technique to deal with the time derivative and some sharp estimates to cope with the nonlinear terms. Third, by virtue of the relationship between the Ritz projection and the interpolation, as well as a so-called "lifting'' technique, the superconvergence behavior of order $O(h^2+\tau^2)$ in $H^1$-norm for the original variables are deduced. Finally, a numerical experiment is conducted to confirm our theoretical analysis. Here, $h$ is the spatial subdivision parameter, and $\tau$ is the time step.     

Inverse problem for a quasilinear system of partial differential equations with a nonlocal boundary condition

An initial boundary-value problem for a quasilinear system of partial differential equations with a nonlocal boundary condition involving a delayed argument is considered. The existence of a unique solution to this problem is proved by reducing it to a system of nonlinear integral-functional equations. The inverse problem of finding a solution-dependent coefficient of the system from additional information on a solution component specified at a fixed point of space as a function of time is formulated. The uniqueness of the solution of the inverse problem is proved. The proof is based on the derivation and analysis of an integral-functional equation for the difference between two solutions of the inverse problem.     

Perturbation of Solutions with Moving Singularities

We study the Cauchy problem for a nonlinear second-order differential equation with a small parameter in the case where the exact solution has a power singularity depending on a small parameter. We propose an asymptotic method similar to the Krylov–Bogoliubov method for localizing the singularity up to the accuracy of any order and construct an asymptotic expansion of the solution in the domain of regular behavior.     

Convergence Analysis of a Numerical Scheme for the Porous Medium Equation by an Energetic Variational Approach

The porous medium equation (PME)is a typical nonlinear degenerate parabolic equation. We have studied numerical methods for PME by an energetic variational approach in [C. Duan et al., J. Comput. Phys., 385 (2019), pp. 13–32], where the trajectory equation can be obtained and two numerical schemes have been developed based on different dissipative energy laws. It is also proved that the nonlinear scheme, based on $f$log $f$ as the total energy form of the dissipative law, is uniquely solvable on an admissible convex set and preserves the corresponding discrete dissipation law. Moreover, under certain smoothness assumption, we have also obtained the second order convergence in space and the first order convergence in time for the scheme. In this paper, we provide a rigorous proof of the error estimate by a careful higher order asymptotic expansion and two step error estimates. The latter technique contains a rough estimate to control the highly nonlinear term in a discrete $W$^1,∞norm and a refined estimate is applied to derive the optimal error order.     

Solution of nonlinear weakly singular Volterra integral equations using the fractional‐order Legendre functions and pseudospectral method

In this article, our main goal is to render an idea to convert a nonlinear weakly singular Volterra integral equation to a non‐singular one by new fractional‐order Legendre functions. The fractional‐order Legendre functions are generated by change of variable on well‐known shifted Legendre polynomials. We consider a general form of singular Volterra integral equation of the second kind. Then the fractional Legendre–Gauss–Lobatto quadratures formula eliminates the singularity of the kernel of the integral equation. Finally, the Legendre pseudospectral method reduces the solution of this problem to the solution of a system of algebraic equations. This method also can be utilized on fractional differential equations as well. The comparison of results of the presented method and other numerical solutions shows the efficiency and accuracy of this method. Also, the obtained maximum error between the results and exact solutions shows that using the present method leads to accurate results and fast convergence for solving nonlinear weakly singular Volterra integral equations. Copyright © 2015 John Wiley & Sons, Ltd.     

On the solution of a mixed nonlinear integral equation

In this paper, we consider a mixed nonlinear integral equation of the second kind in position and time. The existence of a unique solution of this equation is discussed and proved. A numerical method is used to obtain a system of Harmmerstein integral equations of the second kind in position. Then the modified Toeplitz matrix method, as a numerical method, is used to obtain a nonlinear algebraic system. Many important theorems related to the existence and uniqueness solution to the produced nonlinear algebraic system are derived. The rate of convergence of the total error is discussed. Finally, numerical examples when the kernel of position takes a logarithmic and Carleman forms, are presented and the error estimate, in each case, is calculated.     

A fully Galerkin method for the damped generalized regularized long‐wave (DGRLW) equation

In this article, a fully discrete Galerkin scheme based on a nonlinear Crank–Nicolson method to approximate the solution of the DGRLW equation is constructed. Some a priori bounds are proved as well as error estimates. Then, a linearized modification scheme by an extrapolation method is discussed. The two schemes are time second order convergence. The last part is devoted to some numerical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

On the solvability of nonlocal nonstationary problems with double degeneration

We study the solvability in Sobolev spaces of the first boundary value problem for a nonlinear evolution equation degenerating both on the solution and on the solution gradient. We consider the case in which the spatial operator can depend on a nonlocal characteristic of a solution, for example, on an integral characteristic. The theorem is proved with the use of the time discretization method. To study the solvability of the spatial problems arising in the course of the proof, we use the Galerkin method.