Stability and Uniqueness for the Spatially Homogeneous Boltzmann Equation with Long-Range Interactions

In this paper, we prove some a priori stability estimates (in weighted Sobolev spaces) for the spatially homogeneous Boltzmann equation without angular cutoff (covering all physical collision kernels). These estimates are conditional on some regularity estimates on the solutions, and therefore reduce the stability and uniqueness issue to one of proving suitable regularity bounds on the solutions. We then prove such regularity bounds for a class of interactions including the so-called (non-cutoff and non-mollified) hard potentials and moderately soft potentials. In particular, we obtain the first result of global existence and uniqueness for these long-range interactions.     

Lack of Uniqueness for Weak Solutions of the Incompressible Porous Media Equation

In this work we consider weak solutions of the incompressible two-dimensional porous media (IPM) equation. By using the approach of De Lellis–Székelyhidi, we prove non-uniqueness for solutions in L ^∞ in space and time.     

Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

We study a mathematical model describing dislocation dynamics in crystals. We consider a single dislocation line moving in its slip plane. The normal velocity is given by the Peach-Koehler force created by the dislocation line itself. The mathematical model is an eikonal equation with a velocity which is a non-local quantity depending on the whole shape of the dislocation line. We study the special case where the dislocation line is assumed to be a graph or a closed loop. In the framework of discontinuous viscosity solutions for Hamilton–Jacobi equations, we prove the existence and uniqueness of a solution for small time. We also give physical explanations and a formal derivation of the mathematical model. Finally, we present numerical results based on a level-sets formulation of the problem. These results illustrate in particular the fact that there is no general inclusion principle for this model.     

Multiplicity of Radial Ground States for the Scalar Curvature Equation Without Reciprocal Symmetry

Journal of Dynamics and Differential Equations - We study existence and multiplicity of positive ground states for the scalar curvature equation $$begin{aligned} varDelta u+ K(|x|),...     

The FENE Dumbbell Polymer Model: Existence and Uniqueness of Solutions for the Momentum Balance Equation

We consider the FENE dumbbell polymer model which is the coupling of the incompressible Navier-Stokes equations with the corresponding Fokker–Planck–Smoluchowski diffusion equation. We show global well-posedness in the case of a 2D bounded domain. We assume in the general case that the initial velocity is sufficiently small and the initial probability density is sufficiently close to the equilibrium solution; moreover an additional condition on the coefficients is imposed. In the corotational case, we only assume that the initial probability density is sufficiently close to the equilibrium solution.     

Analytical Solution of Boundary-Value Problems for the Ellipsoidal Statistical Equation

This paper describes an analytical method for solving semispatial boundary-value problems for the ellipsoidal statistical equation with a frequency proportional to the molecular velocity. The classical Smoluchowski problem of a temperature jump in a rarefied gas and weak vaporization (condensation) is solved. Numerical calculations of the obtained expressions are performed. A comparison is made with previous results.     

A General Solution of the Duffing Equation

In this paper, we describe the application of the elliptic balance method (EBM) to obtain a general solution of the forced, damped Duffing equation by assuming that the modulus of the Jacobian elliptic functions are slowly varying as a function of time. From this solution, the maximum transient and steady-state amplitudes will be determined for large nonlinearities and positive damping. The amplitude–time response curves obtained from our elliptic balance approximate solution are in good agreement with those obtained from the numerical integration solution over the selected time interval.     

Hertz接触问题的解的唯一性条件

讨论了接触面为圆面的Ｈｅｒｔｚ接触问题。若压力分布是轴对称的，则该接触问题的解必是唯一的。且在上述条件下，该接触问题的积分方程可化为两个推广的Ａｂｅｌ积分方程组，此方程组的解便给出此接触问题的解。     

Approximate Analytical Solution of the Nonlinear Diffusion Equation for Arbitrary Boundary Conditions

A general approximation for the solution of the one-dimensional nonlinear diffusion equation is presented. It applies to arbitrary soil properties and boundary conditions. The approximation becomes more accurate when the soil-water diffusivity approaches a delta function, yet the result is still very accurate for constant diffusivity suggesting that the present formulation is a reliable one. Three examples are given where the method is applied, for a constant water content at the surface, when a saturated zone exists and for a time-dependent surface flux.     

Existence of Multi-Pulses of the Regularized Short-Pulse and Ostrovsky Equations

The existence of multi-pulse solutions near orbit-flip bifurcations of a primary single-humped pulse is shown in reversible, conservative, singularly perturbed vector fields. Similar to the non-singular case, the sign of a geometric condition that involves the first integral decides whether multi-pulses exist or not. The proof utilizes a combination of geometric singular perturbation theory and Lyapunov–Schmidt reduction through Lin’s method. The motivation for considering orbit flips in singularly perturbed systems comes from the regularized short-pulse equation and the Ostrovsky equation, which both fit into this framework and are shown here to support multi-pulses.     

On an Exact Analytical Solution of the Boussinesq Equation

A useful exact analytical solution of the Boussinesq equation is discussed and is the most general solution presently available, and in particular yields a solution for a finite aquifer. It provides insight into the physical processes arising during the exchange of water between an aquifer and a free body of water of varying height as an application and extension of Barenblatt's solution. We also illustrate the value of such a solution to check numerical and approximate schemes.     

Irrotational Blowup of the Solution to Compressible Euler Equation

Compressible Euler equation is studied. First, we examine the validity of physical laws such as the conservations of total mass and energy and also the decay of total pressure. Then we show the non-existence of global-in-time irrotational solution with positive mass.     

多肢墙的线性方程解

本文去掉了R.Rossman方法中连梁连续化这一基本假定,给出了一种更符合实际的解法,该方法可以直接处理多肢墙,进而解决了R.Rossman方法中弯矩分配系数的确定问题。     

A 1/3 Pure Subharmonic Solution and Transient Process for the Duffing's Equation

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｍｏｄｅｌｉｎｇｆｏｒｄｙｎａｍｉｃｓｙｓｔｅｍ ,ｗｅｏｆｔｅｎｇｅｔｎｏｎｌｉｎｅａｒｉｔｙｍａｔｈｅｍａｔｉｃａｌｅｑｕａｔｉｏｎ .Ｉｔｉｓｄｉｆｆｉｃｕｌｔｔｏｇｉｖｅｏｕｔｐｕｒｅａｎａｌｙｔｉｃｒｅｓｏｌｕｔｉｏｎｔｏｔｈｅｅｑｕａｔｉｏｎ ,ｍｏｓｔａｎｓｗｅｒｓｗｏｒｋｅｄｏｕｔａｒｅｄｅｐｅｎｄｅｎｔｕｐｏｎｎｕｍｅｒｉｃａｌｓｏｌｕｔｉｏｎ .Ｈｅｎｃｅ,ｉｔｉｓｅｘｔｒｅｍｅｌｙｉｍｐｏｒｔａｎｔｔｏｆｉｎｄｏｕｔａｓｕｉｔａｂｌｅａｐｐｒｏｘｉｍａｔ…     

A Note on the Legendre Series Solution of the Biharmonic Equation for Cylindrical Problems

The solution of cylindrical problems is addressed. A series solution is considered of the biharmonic equation, in which the series terms of the stress function Φ are expressions based upon Legendre polynomials and logarithmically singular functions. An explicit form of a polynomial supplementing each logarithmically singular part of the series solution is obtained.     

Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations

We consider the initial-value problem for a system of coupled Boussinesq equations on the infinite line for localised or sufficiently rapidly decaying initial data, generating sufficiently rapidly decaying right- and left-propagating waves. We study the dynamics of weakly nonlinear waves, and using asymptotic multiple-scale expansions and averaging with respect to the fast time, we obtain a hierarchy of asymptotically exact coupled and uncoupled Ostrovsky equations for unidirectional waves. We then construct a weakly nonlinear solution of the initial-value problem in terms of solutions of the derived Ostrovsky equations within the accuracy of the governing equations, and show that there are no secular terms. When coupling parameters are equal to zero, our results yield a weakly nonlinear solution of the initial-value problem for the Boussinesq equation in terms of solutions of the initial-value problems for two Korteweg-de Vries equations, integrable by the Inverse Scattering Transform. We also perform relevant numerical simulations of the original unapproximated system of Boussinesq equations to illustrate the difference in the behaviour of its solutions for different asymptotic regimes.     

An Elastic-viscoplastic Quasistatic Contact Problem: Existence and Uniqueness of a Weak Solution

In this paper, the contact between an elastic-viscoplastic body and a deformable obstacle is studied. The effect of the damage, due to internal tension or compression and caused by the opening and growth of micro-cracks and micro-cavities, is also considered. The variational formulation leads to a coupled system of evolutionary equations. An existence and uniqueness result is established by using approximate problems, the pseudomonotone operators theory, Schauder fixed-point theorem and a comparison result.     

Existence,Uniqueness and Asymptotic Stability of Traveling Wavefronts in A Non-Local Delayed Diffusion Equation

In this paper, we study the existence, uniqueness, and global asymptotic stability of traveling wave fronts in a non-local reaction–diffusion model for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment. Under realistic assumptions on the birth function, we construct various pairs of super and sub solutions and utilize the comparison and squeezing technique to prove that the equation has exactly one non-decreasing traveling wavefront (up to a translation) which is monotonically increasing and globally asymptotic stable with phase shift.