 Publication Date:
 NSFPAR ID:
 10286493
 Journal Name:
 Communications in mathematical physics
 Volume:
 383
 Issue:
 3
 Page Range or eLocationID:
 15591667
 ISSN:
 14320916
 Sponsoring Org:
 National Science Foundation
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We present a novel method of analysis and prove finite time asymptotically self similar blowup of the De Gregorio model [13,14] for some smooth initial data on the real line with compact support. We also prove selfsimilar blowup results for the generalized De Gregorio model [41] for the entire range of parameter on R or $S^1$ for Holder continuous initial data with compact support. Our strategy is to reformulate the problem of proving finite time asymptotically selfsimilar singularity into the problem of establishing the nonlinear stability of an approximate selfsimilar profile with a small residual error using the dynamic rescaling equation. We use the energy method with appropriate singular weight functions to extract the damping effect from the linearized operator around the approximate selfsimilar profile and take into account cancellation among various nonlocal terms to establish stability analysis. We remark that our analysis does not rule out the possibility that the original De Gregorio model is well posed for smooth initial data on a circle. The method of analysis presented in this paper provides a promising new framework to analyze finite time singularity of nonlinear nonlocal systems of partial differential equations.

We present a novel method of analysis and prove finite time asymptotically self similar blowup of the De Gregorio model [13,14] for some smooth initial data on the real line with compact support. We also prove selfsimilar blowup results for the generalized De Gregorio model [41] for the entire range of parameter on R or $S^1$ for Holder continuous initial data with compact support. Our strategy is to reformulate the problem of proving finite time asymptotically selfsimilar singularity into the problem of establishing the nonlinear stability of an approximate selfsimilar profile with a small residual error using the dynamic rescaling equation. We use the energy method with appropriate singular weight functions to extract the damping effect from the linearized operator around the approximate selfsimilar profile and take into account cancellation among various nonlocal terms to establish stability analysis. We remark that our analysis does not rule out the possibility that the original De Gregorio model is well posed for smooth initial data on a circle. The method of analysis presented in this paper provides a promising new framework to analyze finite time singularity of nonlinear nonlocal systems of partial differential equations

This work concerns the asymptotic behavior of solutions to a (strictly) subcritical fluid model for a data communication network, where file sizes are generally distributed and the network operates under a fair bandwidthsharing policy. Here we consider fair bandwidthsharing policies that are a slight generalization of the [Formula: see text]fair policies introduced by Mo and Walrand [Mo J, Walrand J (2000) Fair endtoend windowbased congestion control. IEEE/ACM Trans. Networks 8(5):556–567.]. Since the year 2000, it has been a standing problem to prove stability of the data communications network model of Massoulié and Roberts [Massoulié L, Roberts J (2000) Bandwidth sharing and admission control for elastic traffic. Telecommunication Systems 15(1):185–201.], with general file sizes and operating under fair bandwidth sharing policies, when the offered load is less than capacity (subcritical conditions). A crucial step in an approach to this problem is to prove stability of subcritical fluid model solutions. In 2012, Paganini et al. [Paganini F, Tang A, Ferragut A, Andrew LLH (2012) Network stability under alpha fair bandwidth allocation with general file size distribution. IEEE Trans. Automatic Control 57(3):579–591.] introduced a Lyapunov function for this purpose and gave an argument, assuming that fluid model solutions are sufficiently smooth in timemore »

Abstract We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiplevalued velocity potentials but singlevalued stream functions. We prove that the resulting secondkind Fredholm integral equations are invertible, possibly after a physically motivated finiterank correction. In an anglearclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravitycapillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually selfintersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid,more »

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