58 research outputs found
Some qualitative properties of the solutions of the Magnetohydrodynamic equations for nonlinear bipolar fluids
In this article we study the long-time behaviour of a system of nonlinear
Partial Differential Equations (PDEs) modelling the motion of incompressible,
isothermal and conducting modified bipolar fluids in presence of magnetic
field. We mainly prove the existence of a global attractor denoted by \A for
the nonlinear semigroup associated to the aforementioned systems of nonlinear
PDEs. We also show that this nonlinear semigroup is uniformly differentiable on
\A. This fact enables us to go further and prove that the attractor \A is
of finite-dimensional and we give an explicit bounds for its Hausdorff and
fractal dimensions.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/s10440-014-9964-
Some results on blow up for semilinear parabolic problems
The authors describe the asymptotic behavior of blow-up for the semilinear heat equation ut=uxx+f(u) in R×(0,T), with initial data u0(x)>0 in R, where f(u)=up, p>1, or f(u)=eu. A complete description of the types of blow-up patterns and of the corresponding blow-up final-time profiles is given. In the rescaled variables, both are governed by the structure of the Hermite polynomials H2m(y). The H2-behavior is shown to be stable and generic. The existence of H4-behavior is proved. A nontrivial blow-up pattern with a blow-up set of nonzero measure is constructed. Similar results for the absorption equation ut=uxx−up, 0<p<1, are discussed
The Inviscid Limit and Boundary Layers for Navier-Stokes Flows
The validity of the vanishing viscosity limit, that is, whether solutions of
the Navier-Stokes equations modeling viscous incompressible flows converge to
solutions of the Euler equations modeling inviscid incompressible flows as
viscosity approaches zero, is one of the most fundamental issues in
mathematical fluid mechanics. The problem is classified into two categories:
the case when the physical boundary is absent, and the case when the physical
boundary is present and the effect of the boundary layer becomes significant.
The aim of this article is to review recent progress on the mathematical
analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of
Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final
publication is available at http://www.springerlink.co
Non-existence and uniqueness results for supercritical semilinear elliptic equations
Non-existence and uniqueness results are proved for several local and
non-local supercritical bifurcation problems involving a semilinear elliptic
equation depending on a parameter. The domain is star-shaped but no other
symmetry assumption is required. Uniqueness holds when the bifurcation
parameter is in a certain range. Our approach can be seen, in some cases, as an
extension of non-existence results for non-trivial solutions. It is based on
Rellich-Pohozaev type estimates. Semilinear elliptic equations naturally arise
in many applications, for instance in astrophysics, hydrodynamics or
thermodynamics. We simplify the proof of earlier results by K. Schmitt and R.
Schaaf in the so-called local multiplicative case, extend them to the case of a
non-local dependence on the bifurcation parameter and to the additive case,
both in local and non-local settings.Comment: Annales Henri Poincar\'e (2009) to appea
Similarity solutions for the generalized equation of steady transonic gas flow with a singular source
In this brief paper, we consider the generalized equation of steady transonic gas flow with the addition of a singular source term. While the addition of a source term often destroys self-similarity of such flows, we demonstrate that a self-similar solution can still exist in the case of a singular source. We first reduce the governing nonlinear partial differential equation into an ordinary differential equation for a class of similarity solutions. Then, we study the existence of solutions for this similarity equation. After that, several explicit solution forms are given. In constructing exact solutions analytically, we demonstrate that dual solution branches may exist for some parameter regimes. For those parameter regimes where exact or analytical solutions are not possible, we obtain numerical solutions. The results demonstrate interesting properties of the solutions which warrant further study
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