661 research outputs found
Existence of solutions for fractional differential equations with three-point boundary conditions at resonance in
In this paper, by applying the coincidence degree theory which was first introduced by Mawhin, we obtain an existence result for the fractional three-point boundary value problems in , where the dimension of the kernel of fractional differential operator with the boundary conditions can take any value in . This is our novelty. Several examples are presented to illustrate the result
A note on the Dancer-Fucik spectra of the fractional p-Laplacian and Laplacian operators
We study the Dancer-Fucik spectrum of the fractional p-Laplacian operator. We
construct an unbounded sequence of decreasing curves in the spectrum using a
suitable minimax scheme. For p=2, we present a very accurate local analysis. We
construct the minimal and maximal curves of the spectrum locally near the
points where it intersects the main diagonal of the plane. We give a sufficient
condition for the region between them to be nonempty, and show that it is free
of the spectrum in the case of a simple eigenvalue. Finally we compute the
critical groups in various regions separated by these curves. We compute them
precisely in certain regions, and prove a shifting theorem that gives a
finite-dimensional reduction in certain other regions. This allows us to obtain
nontrivial solutions of perturbed problems with nonlinearities crossing a curve
of the spectrum via a comparison of the critical groups at zero and infinity.Comment: 13 pages, typos correcte
Libration driven multipolar instabilities
We consider rotating flows in non-axisymmetric enclosures that are driven by
libration, i.e. by a small periodic modulation of the rotation rate. Thanks to
its simplicity, this model is relevant to various contexts, from industrial
containers (with small oscillations of the rotation rate) to fluid layers of
terrestial planets (with length-of-day variations). Assuming a multipolar
-fold boundary deformation, we first obtain the two-dimensional basic flow.
We then perform a short-wavelength local stability analysis of the basic flow,
showing that an instability may occur in three dimensions. We christen it the
Libration Driven Multipolar Instability (LDMI). The growth rates of the LDMI
are computed by a Floquet analysis in a systematic way, and compared to
analytical expressions obtained by perturbation methods. We then focus on the
simplest geometry allowing the LDMI, a librating deformed cylinder. To take
into account viscous and confinement effects, we perform a global stability
analysis, which shows that the LDMI results from a parametric resonance of
inertial modes. Performing numerical simulations of this librating cylinder, we
confirm that the basic flow is indeed established and report the first
numerical evidence of the LDMI. Numerical results, in excellent agreement with
the stability results, are used to explore the non-linear regime of the
instability (amplitude and viscous dissipation of the driven flow). We finally
provide an example of LDMI in a deformed spherical container to show that the
instability mechanism is generic. Our results show that the previously studied
libration driven elliptical instability simply corresponds to the particular
case of a wider class of instabilities. Summarizing, this work shows that
any oscillating non-axisymmetric container in rotation may excite intermittent,
space-filling LDMI flows, and this instability should thus be easy to observe
experimentally
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