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"title": "Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric Euler Equations",
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"abstract": "Global regularity of the Euler equations in the three-dimensional (3D) setting is regarded as one of the most important open questions in mathematical fluid mechanics. In this work we consider two one-dimensional (1D) models approximating the dynamics of the 3D axisymmetric Euler equations on the solid boundary of a periodic cylinder, which are motivated by a potential finite-time singularity formation scenario proposed recently by Luo and Hou (PNAS 111(36):12968\u201312973, 2014), and numerically investigate the stability of the self-similar profiles in their singular solutions. We first review some recent existence results about the self-similar profiles for one model, and then derive the evolution equations of the spatial profiles in the singular solutions for both models through a dynamic rescaling formulation. We demonstrate the stability of the self-similar profiles by analyzing their discretized dynamics using linearization, and it is hoped that these computations can help to understand the potential singularity formation mechanism of the 3D Euler equations.",
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"1. G.I. Barenblatt, Y.B. Zel\u2019Dovich, Self-similar solutions as intermediate asymptotics. Annu.\r\nRev. Fluid Mech. 4(1), 285\u2013312 (1972)\r\n2. J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D\r\nEuler equations. Commun. Math. Phys. 94(1), 61\u201366 (1984)\r\n3. A.J. Bernoff, A.L. Bertozzi, T.P. Witelski, Axisymmetric surface diffusion: dynamics and\r\nstability of self-similar pinchoff. J. Stat. Phys. 93(3-4), 725\u2013776( 1998)\r\n4. M.D. Bustamante, R.M. Kerr, 3D Euler about a 2D symmetry plane. Phys. D: Nonlinear\r\nPhenom. 237(14), 1912\u20131920 (2008)\r\n5. K. Choi, T.Y. Hou, A. Kiselev, G. Luo, V. Sverak, Y. Yao, On the fiinite-time blowup of a 1D\r\nmodel for the 3D axisymmetric Euler equations (2014). arXiv preprint arXiv:1407.4776\r\n6. K. Choi, A. Kiselev, Y. Yao, Finite time blow up for a 1D model of 2D Boussinesq system.\r\nCommun. Math. Phys. 334(3), 1667\u20131679 (2015)\r\n7. P. Constantin, C. Fefferman, A.J. Majda, Geometric constraints on potentially singular\r\nsolutions for the 3-D Euler equations. Commun. Partial Differ. Equ. 21(3\u20134), 559\u2013571 (1996)\r\n8. J. Deng, T.Y. Hou, X. Yu, Geometric properties and nonblowup of 3D incompressible Euler\r\nflow. Commun. Partial Differ. Equ. 30(1-2), 225\u2013243 (2005)\r\n9. J. Deng, T.Y. Hou, X. Yu, Improved geometric conditions for non-blowup of the 3D\r\nincompressible Euler equation. Commun. Partial Differ. Equ. 31(2), 293\u2013306 (2006)\r\n10. E. Weinan, C.W. Shu, Small-scale structures in boussinesq convection. Phys. Fluids 6(1),\r\n49\u201358 (1994)\r\n11. J. Eggers, M.A. Fontelos, The role of self-similarity in singularities of partial differential\r\nequations. Nonlinearity 22(1), R1 (2009)\r\n12. A.B. Ferrari, On the blow-up of solutions of the 3-D Euler equations in a bounded domain.\r\nCommun. Math. Phys. 155(2), 277\u2013294 (1993)\r\n13. Y. Giga, R.V. Kohn, Nondegeneracy of blowup for semilinear heat equations. Commun. Pure\r\nAppl. Math. 42(6), 845\u2013884 (1989)\r\n14. Y. Giga, R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations.\r\nCommun. Pure Appl. Math. 38(3), 297\u2013319 (1985)\r\n15. R. Grauer, T.C. Sideris, Numerical computation of 3D incompressible ideal fluids with swirl.\r\nPhys. Rev. Lett. 67(25), 3511 (1991)\r\n16. T.Y. Hou, Z. Lei, On the partial regularity of a 3D model of the Navier-Stokes equations.\r\nCommun. Math. Phys. 287(2), 589\u2013612 (2009)\r\n17. T.Y. Hou, Z. Lei, G. Luo, S.Wang, C. Zou, On finite time singularity and global regularity of an\r\naxisymmetric model for the 3D Euler equations. Arch. Ration. Mech. Anal. 212(2), 683\u2013706\r\n(2014)\r\n18. T.Y. Hou, C. Li, Dynamic stability of the three-dimensional axisymmetric Navier-Stokes\r\nequations with swirl. Commun. Pure Appl. Math. 61(5), 661\u2013697 (2008)\r\n19. T.Y. Hou, R. Li, Dynamic depletion of vortex stretching and non-blowup of the 3-D\r\nincompressible Euler equations. J. Nonlinear Sci. 16(6), 639\u2013664 (2006)\r\n20. T.Y. Hou, P. Liu, Self-similar singularity of a 1D model for the 3D axisymmetric Euler\r\nequations. Res. Math. Sci. 2(1), 1\u201326 (2015)\r\n21. T.Y. Hou, G. Luo, On the finite-time blowup of a 1D model for the 3D incompressible Euler\r\nequations (2013). arXiv preprint arXiv:1311.2613\r\n22. T.Y. Hou, Z. Shi, S.Wang, On singularity formation of a 3D model for incompressible Navier-\r\nStokes equations. Adv. Math. 230(2), 607\u2013641 (2012)\r\n23. Y. Huang, T.P.Witelski, A.L. Bertozzi, Anomalous exponents of self-similar blow-up solutions\r\nto an aggregation equation in odd dimensions. Appl. Math. Lett. 25(12), 2317\u20132321 (2012)\r\n24. R.M. Kerr, Evidence for a singularity of the three-dimensional, incompressible euler equations.\r\nPhys. Fluids A: Fluid Dyn. (1989-1993) 5(7), 1725\u20131746 (1993)\r\n25. R.M. Kerr, Bounds for Euler from vorticity moments and line divergence. J. Fluid Mech. 729,\r\nR2 (2013)\r\n26. M. Landman, G.C. Papanicolaou, C. Sulem, P.L. Sulem, X.P. Wang, Stability of isotropic\r\nself-similar dynamics for scalar-wave collapse. Phys. Rev. A 46(12), 7869 (1992)\r\n27. M.J. Landman, G.C. Papanicolaou, C. Sulem, P.L. Sulem, Rate of blowup for solutions of the\r\nnonlinear schr\u00f6dinger equation at critical dimension. Phys. Rev. A 38(8), 3837 (1988)\r\n28. Z. Lei, T.Y. Hou, On the stabilizing effect of convection in three-dimensional incompressible\r\nflows. Commun. Pure Appl. Math. 62(4), 501\u2013564 (2009)\r\n29. B.J. LeMesurier, G. Papanicolaou, C. Sulem, P.L. Sulem. Focusing and multi-focusing\r\nsolutions of the nonlinear schr\u00f6dinger equation. Phys. D: Nonlinear Phenom. 31(1), 78\u2013102\r\n(1988)\r\n30. G. Luo, T.Y. Hou, Potentially singular solutions of the 3D incompressible Euler equations.\r\nPNAS 111(36), 12968\u201312973 (2014)\r\n31. D.W. McLaughlin, G.C. Papanicolaou, C. Sulem, P.L. Sulem, Focusing singularity of the cubic\r\nschr\u00f6dinger equation. Phys. Rev. A 34(2), 1200 (1986)\r\n32. F. Merle, Y. Tsutsumi, L 2 concentration of blow-up solutions for the nonlinear schr\u00f6dinger\r\nequation with critical power nonlinearity. J. Differ. Equ. 84(2), 205\u2013214 (1990)\r\n33. R.E. Moore, R.E. Moore, Methods and Applications of Interval Analysis, vol. 2 (SIAM,\r\nPhiladelphia, 1979)\r\n34. G.C. Papanicolaou, C. Sulem, P.L. Sulem, X.P. Wang, The focusing singularity of the daveystewartson\r\nequations for gravity-capillary surface waves. Phys. D: Nonlinear Phenom. 72(1),\r\n61\u201386 (1994)\r\n35. G. Perelman, On the formation of singularities in solutions of the critical nonlinear Schr\u00f6dinger\r\nequation. in Annales Henri Poincar\u00e9, vol. 2 (Springer, 2001), pp. 605\u2013673\r\n36. L. Perko, Differential Equations and Dynamical Systems, vol. 7 (Springer, New York, 2001)\r\n37. A. Pumir, E.D. Siggia, Development of singular solutions to the axisymmetric Euler equations.\r\nPhys. Fluids A: Fluid Dyn. (1989\u20131993) 4(7), 1472\u20131491 (1992)\r\n38. L.I. Sedov, Similarity and Dimensional Methods in Mechanics (CRC Press, Boca Raton, 1993)\r\n39. C. Sulem and P.L. Sulem, The Nonlinear Schr\u00f6dinger Equation: Self-Focusing and Wave\r\nCollapse, vol. 139 (Springer, New York, 1999)"
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"abstract": "In this chapter, we consider the 3D incompressible Euler equations. We present classical and recent results on the issue of global existence/finite time singularity. We also introduce the theories of lower dimensional model equations of the 3D Euler equations and the vortex patch problem.",
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"abstract": "Many problems of fundamental and practical importance have multiple scale solutions. The direct numerical solution of multiple scale problems is difficult to obtain even with modern supercomputers. The major difficulty of direct solutions is the scale of computation. The ratio between the largest scale and the smallest scale could be as large as 10^5 in each space dimension. From an engineering perspective, it is often sufficient to predict the macroscopic properties of the multiple-scale systems, such as the effective conductivity, elastic moduli, permeability, and eddy diffusivity. Therefore, it is desirable to develop a method that captures the small scale features. This paper reviews some of the recent advances in developing systematic multiscale methods with particular emphasis on multiscale finite element methods with applications to flow and transport in heterogeneous porous media. This manuscript is not intended to be a general survey paper on this topic. The discussion is limited by the scope of the lectures and expertise of the author.",
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"note": "\u00a9 2008 Springer. \r\n\r\nThe authors gratefully acknowledge financial support from US DOE under grant DE-FG02-06ER25727. T. Hou would like also to acknowledge a partial support from NSF grant ITR Grant ACI-0204932.",
"abstract": "Many problems of fundamental and practical importance have multiple scale solutions. The direct numerical solution of multiple scale problems is difficult to obtain even with modern supercomputers. The major difficulty of direct solutions is due to disparity of scales. From an engineering perspective, it is often sufficient to predict macroscopic properties of the multiple-scale systems, such as the effective conductivity, elastic moduli, permeability, and eddy diffusivity. Therefore, it is desirable to develop a method that captures the small scale effect on the large scales, but does not require resolving all the small scale features. The purpose of this lecture note is to review some recent advances in developing multiscale finite element (finite volume) methods for flow and transport in strongly heterogeneous porous media. Extra effort is made in developing a multiscale computational method that can be potentially used for practical multiscale for problems with a large range of nonseparable scales. Some recent theoretical and computational developments in designing global upscaling methods will be reviewed. The lectures can be roughly divided into four parts. In part 1, we review some homogenization theory for elliptic and hyperbolic equations. This homogenization theory provides a guideline for designing effective multiscale methods. In part 2, we review some recent developments of multiscale finite element (finite volume) methods. We also discuss the issue of upscaling one-phase, two-phase flows through heterogeneous porous media and the use of limited global information in multiscale finite element (volume) methods. In part 4, we will consider multiscale simulations of two-phase flow immiscible flows using a flow-based adaptive coordinate, and introduce a theoretical framework which enables us to perform global upscaling for heterogeneous media with long range connectivity.",
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"title": "Numerical Study of Nearly Singular Solutions of the 3-D Incompressible Euler Equations",
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"note": "\u00a9 2008 Springer-Verlag Berlin Heidelberg.\r\n\r\nWe would like to thank Prof. Lin-Bo Zhang from the Institute of Computational\r\nMathematics in Chinese Academy of Sciences for providing us with the\r\ncomputing resource to perform this large scale computational project. Additional\r\ncomputing resource was provided by the Center of Super Computing\r\nCenter of Chinese Academy of Sciences. We also thank Prof. Robert Kerr for\r\nproviding us with his Fortran subroutine that generates his initial data. This\r\nwork was in part supported by NSF under the NSF FRG grant DMS-0353838\r\nand DMS-0713670. Part of this work was done while Hou visited the Academy\r\nof Systems and Mathematical Sciences of CAS in the summer of 2005 as a\r\nmember of the Oversea Outstanding Research Team for Complex Systems.",
"abstract": "In this paper, we perform a careful numerical study of nearly singular solutions of the 3D incompressible Euler equations with smooth initial data. We consider the interaction of two perturbed antiparallel vortex tubes which was previously investigated by Kerr in [16, 19]. In our numerical study, we use both the pseudo-spectral method with the 2/3 dealiasing rule and the pseudo-spectral method with a high order Fourier smoothing. Moreover, we perform a careful resolution study with grid points as large as 1,536 \u00d7 1,024 \u00d7 3,072 to demonstrate the convergence of both numerical methods. Our computational results show that the maximum vorticity does not grow faster than doubly exponential in time while the velocity field remains bounded up to T = 19, beyond the singularity time T = 18.7 reported by Kerr in [16, 19]. The local geometric regularity of vortex lines near the region of maximum vorticity seems to play an important role in depleting the nonlinear vortex stretching dynamically.",
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"title": "A Relay-Zone Technique for Computing Dynamic Dislocations",
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"abstract": "We propose a multiscale method for simulating solids with moving dislocations. Away from atomistic subdomains where the atomistic dynamics are fully resolved, a dislocation is represented by a localized jump profile, superposed on a defect-free field. We assign a thin relay zone around an atomistic subdomain to detect the dislocation profile and its propagation speed at a selected relay time. The detection technique utilizes a lattice time history integral treatment. After the relay, an atomistic computation is performed only for the defect-free field. The method allows one to effectively absorb the fine scale fluctuations and the dynamic dislocations at the interface between the atomistic and continuum domains. In the surrounding region, a coarse grid computation is adequate. \r\n\r\nWe illustrate the algorithm for a 1D Frenkel-Kontorova model at finite temperature. By comparison of the numerical results in the following figure, the reflection is absorbed by the proposed relay-zone technique.",
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"title": "Multiscale computation of isotropic homogeneous turbulent\r flow",
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"keywords": "Multiscale analysis, Euler equations, turbulence",
"note": "\u00a9 2006 American Mathematical Society. \r\n\r\nThe first and the third author was supported in part by an NSF grant DMS-0073916 and an NSF ITR grant ACI-0204932. \r\n\r\nThe second author was supported in part by an NSF grant DMS-0073916, the National Basic Research Program of P. R. China under the grant 2005CB321703, and Natural Science Foundation of China under the Grant 10441005, 10571108.",
"abstract": "In this article we perform a systematic multi-scale analysis and\r\ncomputation for incompressible Euler equations and Navier-Stokes Equations\r\nin both 2D and 3D. The initial condition for velocity field has multiple length\r\nscales. By reparameterizing them in the Fourier space, we can formally organize\r\nthe initial condition into two scales with the fast scale component being\r\nperiodic. By making an appropriate multiscale expansion for the velocity field,\r\nwe show that the two-scale structure is preserved dynamically. Moreover, we\r\nderive a well-posed homogenized equation for the incompressible Euler equations\r\nin the Eulerian formulations. Numerical experiments are presented to\r\ndemonstrate that the homogenized equations indeed capture the correct averaged\r\nsolution of the incompressible Euler and Navier Stokes equations. Moreover,\r\nour multiscale analysis reveals some interesting structure for the Reynolds\r\nstress terms, which provides a theoretical base for establishing an effective LES\r\ntype of model for incompressible fluid flows.",
"date": "2006",
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"items": [
"[1] A. Bensoussan, J. L. Lions, and G. Papanicolaou. Asymptotic Analysis for Periodic Structure.\r\nvolume 5 of Studies in Mathematics and Its Applications. North-Holland Pub!., 1978.\r\n[2] A. Chorin and J. Marsden, A Mathematical Introduction to Fluid Mechanics. Springer, 1983.\r\n[3] U. Frisch, Turbulence: the legacy of A. N. Kolmogorov. Cambridge University Press, New\r\nYork,1995.\r\n[4] T. Y. Hou and X. H. Wu, A Multiscale Finite Element Method for Elliptic Problems in\r\nComposite Materials and Porous Media, J. Comput. Phys., 134 (1997), 169-189.\r\n[5] A. N. Kolmogorov, Local structure of turbulence in an incompressible fluid at a very high\r\nReynolds number. Dokl. Akad. Nauk SSSR, 30 (1941), 299-302.\r\n[6] D. W. McLaughlin, G. C. Papanicolaou, and O. Pironneau, Convection of Microstructure\r\nand Related Problems. SIAM J. Applied Math, 45 (1985), 780-797.\r\n[7] J. Smogorinsky. General circulation experiments with the primitive equations. I. The basic\r\nexperiment. Monthly Weather Review, 91 (1963), 99-164.\r\n[8] T. Y. Hou, D. P. Yang and K. Wang Homogenization of incompressible Euler equation, J.\r\nComput. Math, 22 (2004), 220-229.\r\n[9] T. Y. Hou, D. P. Yang and H. Ran Multiscale analysis for the 3D incompressible Euler\r\nequations, preprint, 2005.\r\n[10] T. Y. Hou, D. P. Yang and H. Ran Multiscale Analysis in Lagrangian Formulation for the\r\n2-D Incompressible Euler Equation, Discrete and Continuous Dynamical Systems, 13, No.5\r\n, 1153-1186, 2005.\r\n[11] W. D. Henshaw, H. O. Kreiss and L. G. Reyna Smallest Scale Estimates for the Navier-Stokes\r\nEquations for Incompressible Fluids, Arch. Ration. Mech. An., 112 (1990), 21-44.\r\n[12] K. G. Batchelor Computation of the Energy Spectrum in Homogeneous Two-Dimensional\r\nTurbulence, Phys. Fluids. Supp!. II, 12 (1969), 233-239\r\n[13] R. Kraichnan Inertial Ranges in Two Dimensional Turbulence, Phys. Fluids, 10 (1967),\r\n1417-1423.\r\n[14] T. C. Rebollo and D. F. Coronil Derivation of the k - E model for locally homogeneous\r\nturbulence by homogenization techniques, C.R.Acad. Sci. Paris, Ser. 1337 (2003), 431-436."
]
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"classification_code": "1991 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20."
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"id": "Zhang-Jin-E",
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"name": {
"family": "Wu",
"given": "Theodore Y."
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"id": "Hou-T-Y",
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"title": "Coastal hydrodynamics of ocean waves on beach",
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"note": "\u00a9 2001 Elsevier.",
"abstract": "This chapter describes the coastal hydrodynamics of ocean waves on beach. A comprehensive study on modeling three-dimensional ocean waves coming from an open ocean of uniform depth and obliquely incident on beach with arbitrary offshore slope distribution, while evolving under balanced effects of nonlinearity and dispersion is presented. A family of beach configurations that is uniform in the long-shore direction as a first approximation for beaches with negligible long-shore curvature is considered. The beach slope variation is assumed to have such distributions that the ocean waves will evolve on beach without breaking. The overall approach adopted begins with development of a three-dimensional linear shallow-water wave theory, followed by taking, step by step, the nonlinear and dispersive effects into account. The linear theory is shown to provide a fundamental solution involving a central function, called the beach-wave function that delineates the evolution of the incoming train of simple waves during interaction with any beach belonging to this broad family of beach configurations. This linear theory can easily afford to cover such factors as oblique wave incidence, arbitrary distribution of offshore beach slope, and wavelength variations with respect to beach breadth.",
"date": "2001",
"date_type": "published",
"publication": "Advances in Applied Mechanics",
"volume": "37",
"publisher": "Elsevier",
"place_of_pub": "New York, NY",
"pagerange": "89-165",
"id_number": "CaltechAUTHORS:20200226-133732456",
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"isbn": "9780120020379",
"issn": "0065-2156",
"book_title": "Advances in Applied Mechanics",
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"items": [
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"id": "van-der-Giesen-E",
"name": {
"family": "van der Giesen",
"given": "Erik"
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"id": "Wu-T-Y-T",
"name": {
"family": "Wu",
"given": "Theodore Y."
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"url": "https://doi.org/10.1016/s0065-2156(00)80005-8"
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"rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.",
"collection": "CaltechAUTHORS",
"reviewer": "Tony Diaz",
"official_cit": "Jin E. Zhang, Theodore Y. Wu, Thomas Y. Hou,\r\nCoastal hydrodynamics of ocean waves on beach,\r\nEditor(s): Erik van der Giesen, Theodore Y. Wu,\r\nAdvances in Applied Mechanics,\r\nElsevier,\r\nVolume 37,\r\n2001,\r\nPages 89-165,\r\nISSN 0065-2156,\r\nISBN 9780120020379,\r\nhttps://doi.org/10.1016/S0065-2156(00)80005-8.",
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