LOCAL WELL-POSEDNESS OF STRONG SOLUTIONS FOR THE NONHOMOGENEOUS MHD EQUATIONS WITH A SLIP BOUNDARY CONDITIONS

This article is concerned with the 3 D nonhomogeneous incompressible magnetohydrodynamics equations with a slip boundary conditions in bounded domain.We obtain weighted estimates of the velocity and magnetic field,and address the issue of local existence and uniqueness of strong solutions with the weaker initial data which contains vacuum states.

GLOBAL STRONG SOLUTION AND EXPONENTIAL DECAY OF 3D NONHOMOGENEOUS ASYMMETRIC FLUID EQUATIONS WITH VACUUM

We prove the global existence and exponential decay of strong solutions to the three-dimensional nonhomogeneous asymmetric fluid equations with nonnegative density provided that the initial total energy is suitably small.Note that although the system degenerates near vacuum,there is no need to require compatibility conditions for the initial data via time-weighted techniques.

THE COMBINED INVISCID AND NON-RESISTIVE LIMIT FOR THE NONHOMOGENEOUS INCOMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS WITH NAVIER BOUNDARY CONDITIONS

In this paper, we establish the existence of the global weak solutions for the non-homogeneous incompressible magnetohydrodynamic equations with Navier boundary condi-tions for the velocity field and the magnetic field in a bounded domain Ω R^3. Furthermore,we prove that as the viscosity and resistivity coefficients go to zero simultaneously, these weaksolutions converge to the strong one of the ideal nonhomogeneous incompressible magneto-hydrodynamic equations in energy space.

LOCAL EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS TO THE TWO DIMENSIONAL NONHOMOGENEOUS INCOMPRESSIBLE PRIMITIVE EQUATIONS

In this article,we study the initial boundary value problem of the two-dimensional nonhomogeneous incompressible primitive equations and obtain the local existence and uniqueness of strong solutions.The initial vacuum is allowed.

Large time asymptotics for solutions to a nonhomogeneous Burgers equation

In this article, the solutions to a nonhomogeneous Burgers equation subject to bounded and compactly supported initial profiles are constructed. In an interesting study, Kloosterziel （Journal of Engineering Mathematics 24, 213-236 （1990）） represented a solution to an initial value problem （IVP） for the heat equation, with an initial data in a class of rapidly decaying functions, as a series of self-similar solutions to the heat equation. This approach quickly revealed the large time behaviour for the solution to the IVP. Inspired by Kloosterziel＇s approach, the solution to the nonhomogeneous Burgers equation is expressed in terms of the self-similar solutions to the heat equation. The large time behaviour of the solutions to the nonhomogeneous Burgers equation is obtained.

Existence of <i>T</i>-<i>ν</i>-<i>p</i>(<i>x</i>)-Solution of a Nonhomogeneous Elliptic Problem with Right Hand Side Measure

Using the theory of weighted Sobolev spaces with variable exponent and the L1-version on Minty’s lemma, we investigate the existence of solutions for some nonhomogeneous Dirichlet problems generated by the Leray-Lions operator of divergence form, with right-hand side measure. Among the interest of this article is the given of a very important approach to ensure the existence of a weak solution of this type of problem and of generalization to a system with the minimum of conditions.

Theoretically Expected Phase Portrait of an Initial Value Problem with Periodic Piecewise Continuous Functions

This work predicts theoretically the phase portrait including the existence and uniqueness of ω-limit cycle for an initial value problem y(0)=y0with an ordinary differential equation y′+py=fin which p and f are L-periodic piecewise continuous functions.

Compactly Supported Distributional Solutions of Nonstationary Nonhomogeneous Refinement Equations

Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Z0 be a subset of Z such that n∈Z0 implies n+1∈Z0. Denote the space of all compactly supported distributions by D’, and the usual convolution between two compactly supported distributions f and g by f*g. For any bounded sequence Gn and Hn, n∈Z0, in D’, define the corresponding nonstationary nonhomogeneous refinement equation Φn=Hn*Φn+1（A.）+Gn for all n∈Z0, （*） where Φn, n∈Z0, is in a bounded set of D’. The nonstationary nonhomogeneous refinement equation （*） arises in the construction of wavelets on bounded domain, multiwavelets, and of biorthogonal wavelets on nonuniform meshes. In this paper, we study the existence problem of compactly supported distributional solutions Φn, n∈Z0, of the equation （*）. In fact, we reduce the existence problem to finding a bounded solution Fn of the linear equations n-Sn n+1= n for all n∈Z0, where the matrices Sn and the vectors n, n∈Z0, can be constructed explicitly from Hn and Gn respectively. The results above are still new even for stationary nonhomogeneous refinement equations.

VISCOSITY METHOD OF A NON-HOMOGENEOUS BURGERS EQUATION

In [1], Ding et al. studied the nonhomogeneous Burgers equationut + uuX = μ + 4x. (1.1)This paper will prove that when μ→0 the solution of (1.1) will approach the generalized solution ofut + uux = 4z. (1.2)The authors notice that the equation (1.2) is beyond the scope of investigations by OleinikO. in [2]. The solutions here are unbounded in general.The paper also studies the δ-wave phenomenon when (1.2) is jointed with some otherequation.

Global L^2 Stability of the Nonhomogeneous Incompressible Navier–Stokes Equations

In this paper, the problem of the global L^2 stability for large solutions to the nonhomogeneous incompressible Navier-Stokes equations in 3D bounded or unbounded domains is studied. By delicate energy estimates and under the suitable condition of the large solutions, it shows that if the initial data are small perturbation on those of the known strong solutions, the large solutions are stable.

Nonlinear instability for nonhomogeneous incompressible viscous fluids

We investigate the nonlinear instability of a smooth steady density profile solution to the threedimensional nonhomogeneous incompressible Navier-Stokes equations in the presence of a uniform gravitational field,including a Rayleigh-Taylor steady-state solution with heavier density with increasing height(referred to the Rayleigh-Taylor instability).We first analyze the equations obtained from linearization around the steady density profile solution.Then we construct solutions to the linearized problem that grow in time in the Sobolev space H k,thus leading to a global instability result for the linearized problem.With the help of the constructed unstable solutions and an existence theorem of classical solutions to the original nonlinear equations,we can then demonstrate the instability of the nonlinear problem in some sense.Our analysis shows that the third component of the velocity already induces the instability,which is different from the previous known results.

On a nonhomogeneous Burgers' equation

In this paper the Cauchy problem for the following nonhomogeneous Burgers' equation is considered: (1) ut + uux = μuxx - kx, x∈R, t > 0, where μ and k are positive constants. Since the nonhomogeneous term kx does not belong to any Lp(R) space, this type of equation is beyond usual Sobolev framework in some sense. By Hopf-Cole transformation, (1) takes the form (2) (?)t - (?)xx = - x2(?). With the help of the Hermite polynomials and their properties, (1) and (2) are solved exactly. Moreover, the large time behavior of the solutions is also considered, similar to the discussion in Hopf's paper. Especially, we observe that the nonhomogeneous Burgers' equation (1) is nonlinearly unstable.