一类大气演化方程组的Cauchy问题

在分层理论的框架下讨论一类大气演化方程组的Cauchy问题,证明了:1) 惯性力对一类大气演化方程组Cauchy问题的适定性判别标准没有影响;2) 可压缩性对粘性大气方程组Cauchy问题的适定性判别标准没有影响,但对无粘大气方程组,可压缩性改变Cauchy问题适定性判别标准;3) 所论方程组在t=0超平面上的Cauchy问题均是不适定的,并不受粘性和可压缩性的影响;4) 可压无粘大气方程与运动静止初始条件构成的Cauchy问题是不适定的．     

Solvability of the Cauchy problem for linear operator-differential equations with variable operator coefficients

We study the Cauchy problem for linear operator-differential equations with unbounded, nondensely defined, variable operator coefficients in a Banach space. We single out new classes of evolution equations of first and second order for which the Cauchy problem is solvable.     

The analytic smoothing effect of solutions for the nonlinear spatially homogeneous Landau equation with hard potentials

In this work, we study the Cauchy problem of the nonlinear spatially homogeneous Landau equation with hard potentials in a close-to-equilibrium framework. We prove that the solution to the Cauchy problem with the initial datum in L² enjoys an analytic regularizing effect, and the evolution of the analytic radius is the same as that of heat equations.

     

带线性坍塌项和竞争势的非线性波动方程柯西问题的稳定集和不稳定集(英文)

本文考虑带线性坍塌项和竞争势的非线性波动方程柯西问题,定义了新的稳定集和不稳定集,证明了如果初值进入不稳定集,则解在有限时间爆破;如果初值进入稳定集,则整体解存在.运用势井讨论,回答了当初值在多么小的时候,该柯西问题的整体解存在.     

Gelfand–Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff

We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules enjoys the same Gelfand–Shilov regularizing effect as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator.     

带势的非线性Klein-Gordon方程柯西问题的稳定和不稳定集

对带势的非线性Klein-Gordon方程柯西问题,我们定义了新的对于初值的稳定和不稳定集.我们证明了如果发展进入了不稳定集,解在有限时间内爆破;如果发展进入了稳定集,解整体存在.运用势并讨论,我们回答了当初值为多少时,柯西问题的整体解存在.     

Loss of derivatives in evolution Cauchy problems

Abstract We study relations between modulus of continuity of the coefficients and loss of derivatives in the Cauchy problem for evolution operators with real characteristics in the Petrovsky sense. We also provide counterexamples to show that the obtained classification is sharp. Keywords: Cauchy problem, Evolution equations, Loss of regularity of the solution     

Nonexistence of global solutions of nonlinear evolution equations

We consider general nonlinear evolution equations of arbitrary order. For these equations, we find conditions under which the Cauchy problem has no solutions global in t > 0. We also estimate the time beyond which the solution of the considered Cauchy problem necessarily does not exist.     

Well-Posedness of the Cauchy Problem for Stochastic Evolution Functional Equations

We prove the existence, uniqueness, and continuous dependence on the initial data of the solutions of the Cauchy problem for stochastic evolution functional equations with random coefficients in Hilbert spaces. We propose a method for constructing an approximating sequence for the solution of the Cauchy problem and obtain an estimate for the rate of convergence to the exact solution.     

On the Cauchy problem for the evolution p-Laplacian equations with gradient term and source and measures as initial data

In this paper we study the Cauchy problem for the evolution $p$-Laplacian equations with gradient term and source on the assumption that measures as initial conditions.     

The Cauchy problem of generalized Landau-Lifshitz equation into S n

In this paper,we study the Cauchy problem of an integrable evolution system,i.e.,the n-dimensional generalization of third-order symmetry of the well-known Landau-Lifshitz equation.By rewriting this equation in a geometric form and applying the geometric energy method with a forth-order perturbation,we show the global well-posedness of the Cauchy problem in suitable Sobolev spaces.     

The cauchy problem for a class of evolution systems of the parabolic type with unbounded coefficients

For a class of evolution systems of the parabolic type with unbounded coefficients, we study the properties of the fundamental solution matrices and establish the well-posed solvability of the Cauchy problem for these systems in spaces of distributions similar to Gevrey ultradistributions. For a subclass of such systems, we describe the maximal classes of well-posed solvability of the Cauchy problem.     

Necessary conditions for the well-posedness of Schrödinger type equations in Gevrey spaces

We discuss evolution operators of Schrödinger type which have a non-self-adjoint lower order term and give a necessary condition for the Cauchy problem to such operators to be well-posed in Gevrey spaces. Under an additional assumption, this necessary condition is sharp.     

On the Collapse of Solutions of the Cauchy Problem for the Cubic Schrödinger Evolution Equation

It is proved that, for some initial data, the solutions of the Cauchy problem for the cubic Schrödinger evolution equation blow up in finite time whose exact value is estimated from above. In addition, lower bounds for the blow-up rate of the solution in certain norms are obtained.

     

The evolution operator solution of the Cauchy problem for the Hamilton-Jacobi equation

The theory of nonlinear evolution equations in a Banach space is used to prove the existence of global weak solutions of the Cauchy problem for the general time and space-dependent Hamilton-Jacobi equation.     

A Probability Method for the Solution of the Telegraph Equation with Real-Analytic Initial Conditions

We propose a method for the construction of an analytic solution of the Cauchy problem for the telegraph equation that is based on its simulation by a one-dimensional Markov random evolution.     

Numerical Solution of the Cauchy Problem for a Second-Order Integro-Differential Equation

Differential Equations - In a finite-dimensional Hilbert space, we consider the Cauchy problem for a second-order integro-differential evolution equation with memory where the integrand is the...     

Algorithm for Solving the Cauchy Problem for One Infinite-Dimensional System of Nonlinear Differential Equations

Computational Mathematics and Mathematical Physics - The Cauchy problem for an infinite-dimensional system of nonlinear evolution equations, which is a generalization of the Langmuir chain, is...     

Cauchy problem for evolution equations with pseudo-Bessel operators: II

We prove that the Cauchy problem for an evolution equation with a pseudo-Bessel operator with variable symbol is solvable in the class of bounded continuous functions on ℝ.     

Existence Theorems for the Initial Value Problem for the Bogolyubov Chain of Equations in the Space of Sequences of Bounded Functions

We prove that the Cauchy problem for a nonsymmetric Bogolyubov chain of equations has a solution representable as an expansion in particle groups (clusters) whose evolution is governed by the cumulant (semi-invariant) of the evolution operator for this particle group in the space of sequences of summable and bounded functions.