The 3D incompressible Euler equations with a passive scalar : a road to blow-up? (original) (raw)
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Title The 3 D incompressible euler equations with a passive scalar : A road to blow-up ?
2013
The 3D incompressible Euler equations with a passive scalar θ are considered in a smooth domain Ω ⊂ R 3 with no-normal-flow boundary conditions u •n| ∂Ω = 0. It is shown that smooth solutions blow up in a finite time if a null (zero) point develops in the vector B = ∇q × ∇θ, provided B has no null points initially : ω = curl u is the vorticity and q = ω • ∇θ is a potential vorticity. The presence of the passive scalar concentration θ is an essential component of this criterion in detecting the formation of a singularity. The problem is discussed in the light of a kinematic result by Graham and Henyey (2000) on the non-existence of Clebsch potentials in the neighbourhood of null points.
Potential Singularity of the 3D Euler Equations in the Interior Domain
Foundations of Computational Mathematics
A. Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D axisymmetric incompressible Euler equations with smooth initial data of finite energy develop a potential finite time singularity at the origin. This potential singularity is different from the blowup scenario revealed by Luo-Hou in [ , ], which occurs on the boundary. Our initial condition has a simple form and shares several attractive features of a more sophisticated initial condition constructed by Hou-Huang in [ ]. One important difference between these two blowup scenarios is that the solution for our initial data has a one-scale structure instead of a two-scale structure reported in [ ]. More importantly, the solution seems to develop nearly self-similar scaling properties that are compatible with those of the 3D Navier-Stokes equations. We will present strong numerical evidence that the 3D Euler equations seem to develop a potential finite time singularity. Moreover, the nearly self-similar profile seems to be very stable to the small perturbation of the initial data. Finally, we present some preliminary results to demonstrate that the 3D Navier-Stokes equations using the same initial condition develop nearly singular behavior with maximum vorticity increased by a factor of 10 7 .
Singular solutions to the 3D axisymmetric incompressible Euler equations
Physica D: Nonlinear Phenomena, 1992
The problem of development of singular solutions of the 3D Euler equations is considered in the particular case of flows with an axis of symmetry. There is a strong analogy between the physics of such flows, and Boussinesq convection in two dimensions, the buoyancy being replaced by the centripetal acceleration. A hot bubble, initially at rest in cold fluid, tends to rise. During the evolution, strong gradients of temperature develop near the cap of the bubble, while on the sides, vortex sheets tend to roll up. When the cap of the bubble starts folding, a rapid growth of the gradients is observed, suggesting a singularity of the vorticity in the 3D axisymmetric flows like ItO[m~x-I/(t*-t) z. Analytic estimates for the rate of stretching, consistent with our numerical observations, are provided.
Physica D: Nonlinear Phenomena, 2005
The solutions of time-dependent PDEs may show a bewildering variety of behaviors. The example of the Kuramoto-Sivashinsky (KS) equation illustrates how simple an equation can be with extremely complex solutions. The equations for incompressible fluids are also notorious for the extremely complex behavior of their solutions. But in the latter case, the situation is in a sense far less satisfactory than for the KS equations, since one still ignores if their solution remains smooth as time goes on, with smooth initial data in 3D. This leaves in a quite uncertain state all theories of 3D turbulence, which may be either a way of coping with our ignorance of the true dynamics of real fluids, or a way of analyzing the behavior of the smooth solutions of the Navier-Stokes equations. Below, in this homage to Professor Kuramoto, we present some remarks on this question, limited to the 3D Euler equations without viscosity. The existence of solutions of the 3D incompressible Euler equations for fluids blowing-up in finite time remains an open question if the initial energy is finite. The consensus seems to be that the singularity, if there is any, is local in space and time, although it is difficult to find a straight statement supporting such a claim in the literature. Based on the properties of self-similar equations, we claim that this is not possible at least if one assumes such a self-similar blow-up, including some refinements in the time dependence that extend the class of self-similar solutions. Actually, the possible blow-up solutions are very much constrained by the conservation of energy and of circulation. Based on scaling transformations, we argue that the singularity cannot be a simple self-similar blow-up, but requires some sort of oscillations on a logarithmic timescale. The space dependence cannot be simple either. The various constraints lead to a self-similar collapse towards a line (at least for an axisymmetric flow with swirl), with a wavelength along the axis decreasing by steps. Our final result is a set of explicit equations such that, if they have a smooth solution satisfying certain conditions, the original Euler equations have a finite-time singularity.
Improved Geometric Conditions for Non-Blowup of the 3D Incompressible Euler Equation
Communications in Partial Differential Equations, 2006
This is a follow-up of our recent article Deng et al. (2004). In Deng et al. (2004), we derive some local geometric conditions on vortex filaments which can prevent finite time blowup of the 3D incompressible Euler equation. In this article, we derive improved geometric conditions which can be applied to the scenario when velocity blows up at the same time as vorticity and the rate of blowup of velocity is proportional to the square root of vorticity. This scenario is in some sense the worst possible blow-up scenario for velocity field due to Kelvin's circulation theorem. The improved conditions can be checked by numerical computations. This provides a sharper local geometric constraint on the finite time blowup of the 3D incompressible Euler equation.
OPEN PROBLEM The three-dimensional Euler equations: singular or non-singular
One of the outstanding open questions in modern applied mathematics is whether solutions of the incompressible Euler equations develop a singularity in the vorticity field in a finite time. This paper briefly reviews some of the issues concerning this problem, together with some observations that may suggest that it may be more subtle than first thought.
The three-dimensional Euler equations: singular or non-singular?
Nonlinearity, 2008
One of the outstanding open questions in modern applied mathematics is whether solutions of the incompressible Euler equations develop a singularity in the vorticity field in a finite time. This paper briefly reviews some of the issues concerning this problem, together with some observations that may suggest that it may be more subtle than first thought.
Methods and Applications of Analysis, 2004
Non blow-up of the 3D incompressible Euler Equations is proven for a class of threedimensional initial data characterized by uniformly large vorticity in bounded cylindrical domains. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach of proving regularity is based on investigation of fast singular oscillating limits and nonlinear averaging methods in the context of almost periodic functions. We establish the global regularity of the 3D limit resonant Euler equations without any restriction on the size of 3D initial data. After establishing strong convergence to the limit resonant equations, we bootstrap this into the regularity on arbitrary large time intervals of the solutions of 3D Euler Equations with weakly aligned uniformly large vorticity at t = 0.
On the Finite-Time Blowup of a 1D Model for the 3D Incompressible Euler Equations
arXiv (Cornell University), 2013
We study a 1D model for the 3D incompressible Euler equations in axisymmetric geometries, which can be viewed as a local approximation to the Euler equations near the solid boundary of a cylindrical domain. We prove the local well-posedness of the model in spaces of zero-mean functions, and study the potential formation of a finite-time singularity under certain convexity conditions for the velocity field. It is hoped that the results obtained on the 1D model will be useful in the analysis of the full 3D problem, whose loss of regularity in finite time has been observed in a recent numerical study (Luo and Hou, 2013).
Potentially singular solutions of the 3D axisymmetric Euler equations
Proceedings of the National Academy of Sciences of the United States of America, 2014
The question of finite-time blowup of the 3D incompressible Euler equations is numerically investigated in a periodic cylinder with solid boundaries. Using rotational symmetry, the equations are discretized in the (2D) meridian plane on an adaptive (moving) mesh and is integrated in time with adaptively chosen time steps. The vorticity is observed to develop a ring-singularity on the solid boundary with a growth proportional to ∼(ts - t)(-2.46), where ts ∼ 0.0035056 is the estimated singularity time. A local analysis also suggests the existence of a self-similar blowup. The simulations stop at τ(2) = 0.003505 at which time the vorticity amplifies by more than (3 × 10(8))-fold and the maximum mesh resolution exceeds (3 × 10(12))(2). The vorticity vector is observed to maintain four significant digits throughout the computations.