Existence and selection of steady bubbles in a Hele–Shaw cell

This paper concerns the existence of a steadily translating bubble in a Hele–Shaw cell for small but non-zero surface tension ². We rigorously conclude that for bubble velocity U relative to the fluid velocity at infinity in the interval (1,2), analytic symmetric solutions exist in the asymptotic limit of surface tension ²→0 if and only if the Stokes constant for a relatively simple nonlinear differential equation is zero. This Stokes constant S depends on the parameter α(0,1) corresponding to bubble size and . Earlier calculations have shown S to be zero for a discrete set of values of a.     

Uniqueness of weak solutions of the Navier-Stokes equations

Consider the Navier-Stokes equation with the initial data a ∈ L _σ ²(ℝ ^d ). Let u and v be two weak solutions with the same initial value a. If u satisfies the usual energy inequality and if ∇v ∈ L ²((0, T); (ℝ ^d ) ^d ) where (ℝ ^d ) is the multiplier space, then we have u = v.     

Existence and uniqueness results for boundary value problems of higher order fractional integro-differential equations involving gronwall's inequality in banach spaces

We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α ∈ (n-1,n) in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder's inequality, a suitable singular Gronwall's inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.     

A nullstellensatz and a positivstellensatz for ordered differential fields

We use the model completeness and axiomatisation of the theory of closed ordered differential fields to give a differential version of Dubois, Krivine and Risler's nullstellensatz and Stengle's positivstellensatz for ordered fields.     

Morrey-Campanato空间中三维 Navier—Stokes方程的正则性准则

利用能量不等式和一些临界空间中的不等式,在Morrey—Campanato空间获得了两个只涉及水平速度场的正则性准则,改进了一些已有的结果．     

Global existence and nonexistence results for a class of semilinear hyperbolic systems

In this paper, we investigate the Cauchy problem for a class of the system of semilinear hyperbolic equations with damping. With the use of the L_p → L_q type estimation for the corresponding linear problem and the method of comparison of functional, the existence and nonexistence criteria of global solutions are found. Copyright © 2012 John Wiley & Sons, Ltd.     

Existence and uniqueness of the solution,separation for certain second order elliptic differential equation

In this paper, we study conditions under which Schrodinger type operators with a matrix potential is separated and Schrodinger equation has a unique solution in the weighted space L_2,k(Rⁿ)^l, where l is any natural number and k ε C¹(Rⁿ) is a positive function     

A non-stationary problem coupling PDEs and ODEs modelizing an automotive catalytic converter

In this article we prove the existence and uniqueness of the solution of a non-stationary problem that modelizes the behavior of the concentrations and the temperature of gases going through a cylindrical passage of an automotive catalytic converter. This problem couples parabolic partial differential equations (PDEs) in a domain with one parabolic partial differential equation and some ordinary differential equations (ODEs) on a part of its boundary.     

Mathematical aspects of the Maxwell problem

We study the large-time behaviour of global smooth solutions to the Cauchy problem for hyperbolic regularization of conservation laws. An attracting manifold of special smooth global solutions is determined by the Chapman projection onto the phase space of consolidated variables. For small initial data we construct the Chapman projection and describe its properties in the case of the Cauchy problem for moment approximations of kinetic equations. The existence conditions for the Chapman projection are expressed in terms of the solvability of the Riccati matrix equations with parameter.