4,060 research outputs found
Incompressible viscous fluid flows in a thin spherical shell
Linearized stability of incompressible viscous fluid flows in a thin
spherical shell is studied by using the two-dimensional Navier--Stokes
equations on a sphere. The stationary flow on the sphere has two singularities
(a sink and a source) at the North and South poles of the sphere. We prove
analytically for the linearized Navier--Stokes equations that the stationary
flow is asymptotically stable. When the spherical layer is truncated between
two symmetrical rings, we study eigenvalues of the linearized equations
numerically by using power series solutions and show that the stationary flow
remains asymptotically stable for all Reynolds numbers.Comment: 28 pages, 10 figure
Conservation laws for the Maxwell-Dirac equations with a dual Ohm's law
Using a general theorem on conservation laws for arbitrary differential
equations proved by Ibragimov, we have derived conservation laws for Dirac's
symmetrized Maxwell-Lorentz equations under the assumption that both the
electric and magnetic charges obey linear conductivity laws (dual Ohm's law).
We find that this linear system allows for conservation laws which are
non-local in time
Nonlinear self-adjointness and conservation laws
The general concept of nonlinear self-adjointness of differential equations
is introduced. It includes the linear self-adjointness as a particular case.
Moreover, it embraces the strict self-adjointness and quasi self-adjointness
introduced earlier by the author. It is shown that the equations possessing the
nonlinear self-adjointness can be written equivalently in a strictly
self-adjoint form by using appropriate multipliers. All linear equations
possess the property of nonlinear self-adjointness, and hence can be rewritten
in a nonlinear strictly self-adjoint. For example, the heat equation becomes strictly self-adjoint after multiplying by
Conservation laws associated with symmetries can be constructed for all
differential equations and systems having the property of nonlinear
self-adjointness
Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations
A complete group classification of a class of variable coefficient
(1+1)-dimensional telegraph equations , is
given, by using a compatibility method and additional equivalence
transformations. A number of new interesting nonlinear invariant models which
have non-trivial invariance algebras are obtained. Furthermore, the possible
additional equivalence transformations between equations from the class under
consideration are investigated. Exact solutions of special forms of these
equations are also constructed via classical Lie method and generalized
conditional transformations. Local conservation laws with characteristics of
order 0 of the class under consideration are classified with respect to the
group of equivalence transformations.Comment: 23 page
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