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  • 1
    Electronic Resource
    Electronic Resource
    On the evolution of nearly circular vortex patches (1988)
    Springer
    Communications in mathematical physics 119 (1988), S. 177-198 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Recently, the classical problem of the evolution of patches of constant vorticity was reformulated as an evolution equation for the boundary of the patch. We study this equation in the neighborhood of the circular vortex patch and introduce a hierarchy of area-preserving nonlinear approximate equations. The first of these equations is shown to have a rich rigid structure: it possesses an exhaustive increasing sequence of linear invariant manifolds of arbitrarily large finite dimensions. On each of these manifolds the equation can be written as an explicit finite system of ordinary differential equations. Solutions of these ODEs, starting from arbitrarily small neighborhoods of the circular vortex patch, are shown to blow up.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01217737
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  • 2
    Electronic Resource
    Electronic Resource
    Global stability of large solutions to the 3D Navier-Stokes equations (1994)
    Springer
    Communications in mathematical physics 159 (1994), S. 329-341 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We prove the stability of mildly decaying global strong solutions to the Navier-Stokes equations in three space dimensions. Combined with previous results on the global existence of large solutions with various symmetries, this gives the first global existence theorem for large solutions with approximately symmetric initial data. The stability of unforced 2D flow under 3D perturbations is also obtained.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02102642
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  • 3
    Electronic Resource
    Electronic Resource
    Global Gevrey Regularity for the Bénard Convection in a Porous Medium with Zero Darcy-Prandtl Number (1999)
    Springer
    Journal of nonlinear science 9 (1999), S. 333-362 
    Details
    ISSN: 1432-1467
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Summary. In this paper, we prove the existence and uniqueness for the threedimensional B{é}nard convection model in a porous medium with zero Darcy-Prandtl number using the Galerkin procedure. In addition, we show that the solutions to this problem are analytic in time with values in a Gevrey class regularity. We also prove that the solution of the standard Galerkin method converges exponentially fast, in the wave number, to the exact solution. This gives an analytical justification to the two-dimensional computational results of Graham, Steen, and Titi [{J. Nonlin. Sci.} 3 (1993), 153—167].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s003329900073
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  • 4
    Electronic Resource
    Electronic Resource
    Computational efficiency and approximate inertial manifolds for a Bénard convection system (1993)
    Springer
    Journal of nonlinear science 3 (1993), S. 153-167 
    Details
    ISSN: 1432-1467
    Keywords: inertial manifolds ; Bénard convection ; nonlinear Galerkin
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Summary A computational comparison between classical Galerkin and approximate inertial manifold (AIM) methods is performed for the case of two-dimensional natural convection in a saturated porous material. For prediction of Hopf and torus bifurcations far from convection onset, the improvements of the AIM method over the classical one are small or negligible. Two reasons are given for the lack of distinct improvement. First, the small boundary layer length scale is the source of the instabilities, so it cannot be modeled as a “slave” to the larger scales, as the AIM attempts to do. Second, estimates based on the Gevrey class regularity of solutions to the governing equations show that the classical and AIM methods may be virtually equivalent. It is argued that these two reasons are physical and mathematical reflections of one another.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02429862
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  • 5
    Electronic Resource
    Electronic Resource
    Preserving dissipation in approximate inertial forms for the Kuramoto-Sivashinsky equation (1991)
    Springer
    Journal of dynamics and differential equations 3 (1991), S. 179-197 
    Details
    ISSN: 1572-9222
    Keywords: Inertial manifold ; dissipation ; Kuramoto-Sivashinsky ; non-linear Galerkin method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It has been observed, in earlier computations of bifurcation diagrams for dissipative partial differential equations, that the use of certain explicit approximate inertial forms can give rise to numerical artifacts such as spurious turning points and inaccurate solution branches. These shortcomings were attributed to a lack of dissipation in the forms used. We show analytically and verify numerically that with an appropriate adjustment we can eliminate these numerical artifacts. The motivation for this adjustment is to enforce dissipation, while maintaining the same order of approximation. We demonstrate with computations that the most natural remedy, namely, preparation of the equation, can be highly sensitive to assumptions on the size of the absorbing ball. In addition, we show that certain implicit forms are dissipative without any adjustment. As an illustrative example we use here the Kuramoto-Sivashinsky equation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01047708
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  • 6
    Electronic Resource
    Electronic Resource
    Invariant helical subspaces for the Navier-Stokes equations (1990)
    Springer
    Archive for rational mechanics and analysis 112 (1990), S. 193-222 
    Details
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Three-dimensional solutions with helical symmetry are shown to form an invariant subspace for the Navier-Stokes equations. Uniqueness of weak helical solutions in the sense of Leray is proved, and these weak solutions are shown to be regular (strong) solutions existing for arbitrary time t. The global universal attractor for the infinite-dimensional dynamical system generated by the corresponding semi-group of helical flows is shown to be compact and finite-dimensional. The Hausdorff and fractal dimensions of the global attractors are estimated in terms of the governing physical parameters and in terms of the helical parameters for several problems in the class, with the most detailed results obtained for rotating Hagen-Poiseuille (pipe) flow. In this case, the dimension, either Hausdorff or fractal, up to an absolute constant is bounded from above by $$\frac{{\operatorname{Re} }}{{\sqrt {\alpha ^2 + n^2 } }}$$ , where α is the axial wavenumber, n is the azimuthal wavenumber and Re is the Reynolds number based on the radius of the pipe. These upper bounds are independent of the rotation rate.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00381234
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  • 7
    Electronic Resource
    Electronic Resource
    The Navier-Stokes equations on the rotating 2-D sphere: Gevrey regularity and asymptotic degrees of freedom (1999)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 50 (1999), S. 341-360 
    Details
    ISSN: 0044-2275
    Keywords: Key words. Gevrey regularity, Navier-Stokes equations on the sphere, geophysical flows, determining degrees of freedom.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract. In this article we prove a Gevrey class global regularity to the Navier-Stokes equations on the rotating two dimensional sphere, S 2 - a fundamental model that arises naturally in large scale atmospheric dynamics. As a result one concludes the exponential convergence of the spectral Galerkin numerical method, based on spherical harmonic functions. Moreover, we provide an upper bound for the number of asymptotic degrees of freedom for this system.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/PL00001493
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