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U Frisch - One of the best experts on this subject based on the ideXlab platform.

  • geometric formulation of the cauchy invariants for incompressible Euler Flow in flat and curved spaces
    Journal of Fluid Mechanics, 2017
    Co-Authors: Nicolas Besse, U Frisch
    Abstract:

    Cauchy invariants are now viewed as a powerful tool for investigating the Lagrangian structure of three-dimensional (3D) ideal Flow (Frisch & Zheligovsky, Commun. Math. Phys., vol. 326, 2014, pp. 499–505; Podvigina et al., J. Comput. Phys., vol. 306, 2016, pp. 320–342). Looking at such invariants with the modern tools of differential geometry and of geodesic Flow on the space SDiff of volume-preserving transformations (Arnold, Ann. Inst. Fourier, vol. 16, 1966, pp. 319–361), all manners of generalisations are here derived. The Cauchy invariants equation and the Cauchy formula, relating the vorticity and the Jacobian of the Lagrangian map, are shown to be two expressions of this Lie-advection invariance, which are duals of each other (specifically, Hodge dual). Actually, this is shown to be an instance of a general result which holds for Flow both in flat (Euclidean) space and in a curved Riemannian space: any Lie-advection invariant -form which is exact (i.e. is a differential of a -form) has an associated Cauchy invariants equation and a Cauchy formula. This constitutes a new fundamental result in linear transport theory, providing a Lagrangian formulation of Lie advection for some classes of differential forms. The result has a broad applicability: examples include the magnetohydrodynamics (MHD) equations and various extensions thereof, discussed by Lingam et al. (Phys. Lett. A, vol. 380, 2016, pp. 2400–2406), and include also the equations of Tao (2016, arXiv:1606.08481 [math.AP]), Euler equations with modified Biot–Savart law, displaying finite-time blow-up. Our main result is also used for new derivations, and several new results, concerning local helicity-type invariants for fluids and MHD Flow in flat or curved spaces of arbitrary dimension.

  • a constructive approach to regularity of lagrangian trajectories for incompressible Euler Flow in a bounded domain
    Communications in Mathematical Physics, 2017
    Co-Authors: Nicolas Besse, U Frisch
    Abstract:

    The 3D incompressible Euler equations are an important research topic in the mathematical study of fluid dynamics. Not only is the global regularity for smooth initial data an open issue, but the behaviour may also depend on the presence or absence of boundaries. For a good understanding, it is crucial to carry out, besides mathematical studies, high-accuracy and well-resolved numerical exploration. Such studies can be very demanding in computational resources, but recently it has been shown that very substantial gains can be achieved first, by using Cauchy’s Lagrangian formulation of the Euler equations and second, by taking advantage of analyticity results of the Lagrangian trajectories for Flows whose initial vorticity is Holder-continuous. The latter has been known for about 20 years (Serfati in J Math Pures Appl 74:95–104, 1995), but the combination of the two, which makes use of recursion relations among time-Taylor coefficients to obtain constructively the time-Taylor series of the Lagrangian map, has been achieved only recently (Frisch and Zheligovsky in Commun Math Phys 326:499–505, 2014; Podvigina et al. in J Comput Phys 306:320–342, 2016 and references therein). Here we extend this methodology to incompressible Euler Flow in an impermeable bounded domain whose boundary may be either analytic or have a regularity between indefinite differentiability and analyticity. Non-constructive regularity results for these cases have already been obtained by Glass et al. (Ann Sci Ec Norm Sup 45:1–51, 2012). Using the invariance of the boundary under the Lagrangian Flow, we establish novel recursion relations that include contributions from the boundary. This leads to a constructive proof of time-analyticity of the Lagrangian trajectories with analytic boundaries, which can then be used subsequently for the design of a very high-order Cauchy–Lagrangian method.

  • cauchy s almost forgotten lagrangian formulation of the Euler equation for 3d incompressible Flow
    European Physical Journal H, 2014
    Co-Authors: U Frisch, Barbara Villone
    Abstract:

    Two prized papers, one by Augustin Cauchy in 1815, presented to the French Academy and the other by Hermann Hankel in 1861, presented to Gottingen University, contain major discoveries on vorticity dynamics whose impact is now quickly increasing. Cauchy found a Lagrangian formulation of 3D ideal incompressible Flow in terms of three invariants that generalize to three dimensions the now well-known law of conservation of vorticity along fluid particle trajectories for two-dimensional Flow. This has very recently been used to prove analyticity in time of fluid particle trajectories for 3D incompressible Euler Flow and can be extended to compressible Flow, in particular to cosmological dark matter. Hankel showed that Cauchy’s formulation gives a very simple Lagrangian derivation of the Helmholtz vorticity-flux invariants and, in the middle of the proof, derived an intermediate result which is the conservation of the circulation of the velocity around a closed contour moving with the fluid. This circulation theorem was to be rediscovered independently by William Thomson (Kelvin) in 1869. Cauchy’s invariants were only occasionally cited in the 19th century – besides Hankel, foremost by George Stokes and Maurice Levy – and even less so in the 20th until they were rediscovered via Emmy Noether’s theorem in the late 1960, but reattributed to Cauchy only at the end of the 20th century by Russian scientists.

  • complex space singularities of 2d Euler Flow in lagrangian coordinates
    Physica D: Nonlinear Phenomena, 2008
    Co-Authors: Takeshi Matsumoto, U Frisch
    Abstract:

    We show that, for 2D space-periodic incompressible Flow, the solution can be evaluated numerically in Lagrangian coordinates with the same accuracy that is achieved in standard Eulerian spectral methods. This allows the determination of complex-space Lagrangian singularities. Lagrangian singularities are found to be closer to the real domain than Eulerian singularities and seem to correspond to fluid particles which escape to (complex) infinity by the current time. Various mathematical conjectures regarding Eulerian/Lagrangian singularities are presented.

  • the analytic structure of 2d Euler Flow at short times
    Fluid Dynamics Research, 2005
    Co-Authors: U Frisch, Takeshi Matsumoto, Jeremie Bec
    Abstract:

    Abstract Using a very high precision spectral calculation applied to the incompressible and inviscid Flow with initial condition ψ 0 ( x 1 , x 2 ) = cos x 1 + cos 2 x 2 , we find that the width δ ( t ) of its analyticity strip follows a ln ( 1 / t ) law at short times over eight decades. The asymptotic equation governing the structure of spatial complex-space singularities at short times [Frisch, U., Matsumoto, T., Bec, J., 2003. J. Stat. Phys. 113, 761–781] is solved by a high-precision expansion method. Strong numerical evidence is obtained that singularities have infinite vorticity and lie on a complex manifold which is constructed explicitly as an envelope of analyticity disks.

Takeshi Matsumoto - One of the best experts on this subject based on the ideXlab platform.

  • complex space singularities of 2d Euler Flow in lagrangian coordinates
    Physica D: Nonlinear Phenomena, 2008
    Co-Authors: Takeshi Matsumoto, U Frisch
    Abstract:

    We show that, for 2D space-periodic incompressible Flow, the solution can be evaluated numerically in Lagrangian coordinates with the same accuracy that is achieved in standard Eulerian spectral methods. This allows the determination of complex-space Lagrangian singularities. Lagrangian singularities are found to be closer to the real domain than Eulerian singularities and seem to correspond to fluid particles which escape to (complex) infinity by the current time. Various mathematical conjectures regarding Eulerian/Lagrangian singularities are presented.

  • the analytic structure of 2d Euler Flow at short times
    Fluid Dynamics Research, 2005
    Co-Authors: U Frisch, Takeshi Matsumoto, Jeremie Bec
    Abstract:

    Abstract Using a very high precision spectral calculation applied to the incompressible and inviscid Flow with initial condition ψ 0 ( x 1 , x 2 ) = cos x 1 + cos 2 x 2 , we find that the width δ ( t ) of its analyticity strip follows a ln ( 1 / t ) law at short times over eight decades. The asymptotic equation governing the structure of spatial complex-space singularities at short times [Frisch, U., Matsumoto, T., Bec, J., 2003. J. Stat. Phys. 113, 761–781] is solved by a high-precision expansion method. Strong numerical evidence is obtained that singularities have infinite vorticity and lie on a complex manifold which is constructed explicitly as an envelope of analyticity disks.

  • singularities of Euler Flow not out of the blue
    Journal of Statistical Physics, 2003
    Co-Authors: U Frisch, Takeshi Matsumoto, Jeremie Bec
    Abstract:

    Does three-dimensional incompressible Euler Flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time? This blowup problem is still open. After briefly reviewing what is known and pointing out some of the difficulties, we propose to tackle this issue for the class of Flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations. We present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup.

  • the analytic structure of 2d Euler Flow at short times
    arXiv: Chaotic Dynamics, 2003
    Co-Authors: U Frisch, Takeshi Matsumoto, Jeremie Bec
    Abstract:

    Using a very high precision spectral calculation applied to the incompressible and inviscid Flow with initial condition $\psi_0(x_1, x_2) = \cos x_1+\cos 2x_2$, we find that the width $\delta(t)$ of its analyticity strip follows a $\ln(1/t)$ law at short times over eight decades. The asymptotic equation governing the structure of spatial complex-space singularities at short times (Frisch, Matsumoto and Bec 2003, J.Stat.Phys. 113, 761--781) is solved by a high-precision expansion method. Strong numerical evidence is obtained that singularities have infinite vorticity and lie on a complex manifold which is constructed explicitly as an envelope of analyticity disks.

  • singularities of Euler Flow not out of the blue
    arXiv: Chaotic Dynamics, 2002
    Co-Authors: U Frisch, Takeshi Matsumoto, Jeremie Bec
    Abstract:

    Does three-dimensional incompressible Euler Flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time? This blowup problem is still open. After briefly reviewing what is known and pointing out some of the difficulties, we propose to tackle this issue for the class of Flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations. We present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup.(A version of the paper with higher-quality figures is available at this http URL)

Jeremie Bec - One of the best experts on this subject based on the ideXlab platform.

  • the analytic structure of 2d Euler Flow at short times
    Fluid Dynamics Research, 2005
    Co-Authors: U Frisch, Takeshi Matsumoto, Jeremie Bec
    Abstract:

    Abstract Using a very high precision spectral calculation applied to the incompressible and inviscid Flow with initial condition ψ 0 ( x 1 , x 2 ) = cos x 1 + cos 2 x 2 , we find that the width δ ( t ) of its analyticity strip follows a ln ( 1 / t ) law at short times over eight decades. The asymptotic equation governing the structure of spatial complex-space singularities at short times [Frisch, U., Matsumoto, T., Bec, J., 2003. J. Stat. Phys. 113, 761–781] is solved by a high-precision expansion method. Strong numerical evidence is obtained that singularities have infinite vorticity and lie on a complex manifold which is constructed explicitly as an envelope of analyticity disks.

  • singularities of Euler Flow not out of the blue
    Journal of Statistical Physics, 2003
    Co-Authors: U Frisch, Takeshi Matsumoto, Jeremie Bec
    Abstract:

    Does three-dimensional incompressible Euler Flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time? This blowup problem is still open. After briefly reviewing what is known and pointing out some of the difficulties, we propose to tackle this issue for the class of Flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations. We present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup.

  • the analytic structure of 2d Euler Flow at short times
    arXiv: Chaotic Dynamics, 2003
    Co-Authors: U Frisch, Takeshi Matsumoto, Jeremie Bec
    Abstract:

    Using a very high precision spectral calculation applied to the incompressible and inviscid Flow with initial condition $\psi_0(x_1, x_2) = \cos x_1+\cos 2x_2$, we find that the width $\delta(t)$ of its analyticity strip follows a $\ln(1/t)$ law at short times over eight decades. The asymptotic equation governing the structure of spatial complex-space singularities at short times (Frisch, Matsumoto and Bec 2003, J.Stat.Phys. 113, 761--781) is solved by a high-precision expansion method. Strong numerical evidence is obtained that singularities have infinite vorticity and lie on a complex manifold which is constructed explicitly as an envelope of analyticity disks.

  • singularities of Euler Flow not out of the blue
    arXiv: Chaotic Dynamics, 2002
    Co-Authors: U Frisch, Takeshi Matsumoto, Jeremie Bec
    Abstract:

    Does three-dimensional incompressible Euler Flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time? This blowup problem is still open. After briefly reviewing what is known and pointing out some of the difficulties, we propose to tackle this issue for the class of Flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations. We present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup.(A version of the paper with higher-quality figures is available at this http URL)

Yongqian Zhang - One of the best experts on this subject based on the ideXlab platform.

  • Two-Dimensional Steady Supersonic Exothermically Reacting Euler Flow past Lipschitz Bending Walls
    SIAM Journal on Mathematical Analysis, 2017
    Co-Authors: Gui-qiang G. Chen, Jie Kuang, Yongqian Zhang
    Abstract:

    We are concerned with the two-dimensional steady supersonic reacting Euler Flow past Lipschitz bending walls that are small perturbations of a convex one, and establish the existence of global entropy solutions when the total variation of both the initial data and the slope of the boundary is sufficiently small. The Flow is governed by an ideal polytropic gas and undergoes a one-step exothermic chemical reaction under the reaction rate function that is Lipschtiz and has a positive lower bound. The heat released by the reaction may cause the total variation of the solution to increase along the Flow direction. We employ the modified wavefront tracking scheme to construct approximate solutions and develop a Glimm-type functional by incorporating the approximate strong rarefaction waves and Lipschitz bending walls to obtain the uniform bound on the total variation of the approximate solutions. Then we employ this bound to prove the convergence of the approximate solutions to a global entropy solution that co...

  • Two-Dimensional Steady Supersonic Exothermically Reacting Euler Flow past Lipschitz Bending Walls
    arXiv: Analysis of PDEs, 2015
    Co-Authors: Gui-qiang G. Chen, Jie Kuang, Yongqian Zhang
    Abstract:

    We are concerned with the two-dimensional steady supersonic reacting Euler Flow past Lipschitz bending walls that are small perturbations of a convex one, and establish the existence of global entropy solutions when the total variation of both the initial data and the slope of the boundary is sufficiently small. The Flow is governed by an ideal polytropic gas and undergoes a one-step exothermic chemical reaction under the reaction rate function that is Lipschtiz and has a positive lower bound. The heat released by the reaction may cause the total variation of the solution to increase along the Flow direction. We employ the modified wave-front tracking scheme to construct approximate solutions and develop a Glimm-type functional by incorporating the approximate strong rarefaction waves and Lipschitz bending walls to obtain the uniform bound on the total variation of the approximate solutions. Then we employ this bound to prove the convergence of the approximate solutions to a global entropy solution that contains a strong rarefaction wave generated by the Lipschitz bending wall. In addition, the asymptotic behavior of the entropy solution in the Flow direction is also analyzed.

Hairong Yuan - One of the best experts on this subject based on the ideXlab platform.

  • stability of transonic characteristic discontinuities in two dimensional steady compressible Euler Flows
    Journal of Mathematical Physics, 2013
    Co-Authors: Guiqiang Chen, Vaibhav Kukreja, Hairong Yuan
    Abstract:

    For a two-dimensional steady supersonic Euler Flow past a convex cornered wall with right angle, a characteristic discontinuity (vortex sheet and/or entropy wave) is generated, which separates the supersonic Flow from the quiescent gas (hence subsonic). We proved that such a transonic characteristic discontinuity is structurally stable under small perturbations of the upstream supersonic Flow in BV. The existence of a weak entropy solution and Lipschitz continuous free boundary (i.e., characteristic discontinuity) is established. To achieve this, the problem is formulated as a free boundary problem for a nonstrictly hyperbolic system of conservation laws; and the free boundary problem is then solved by analyzing nonlinear wave interactions and employing the front tracking method.

  • stability of transonic characteristic discontinuities in two dimensional steady compressible Euler Flows
    arXiv: Analysis of PDEs, 2012
    Co-Authors: Guiqiang Chen, Vaibhav Kukreja, Hairong Yuan
    Abstract:

    For a two-dimensional steady supersonic Euler Flow past a convex cornered wall with right angle, a characteristic discontinuity (vortex sheet and/or entropy wave) is generated, which separates the supersonic Flow from the gas at rest (hence subsonic). We proved that such a transonic characteristic discontinuity is structurally stable under small perturbations of the upstream supersonic Flow in $BV$. The existence of a weak entropy solution and Lipschitz continuous free boundary (i.e. characteristic discontinuity) is established. To achieve this, the problem is formulated as a free boundary problem for a nonstrictly hyperbolic system of conservation laws; and the free boundary problem is then solved by analyzing nonlinear wave interactions and employing the front tracking method.