Blow-Up Solutions to 3D Euler are Hydrodynamically Unstable

Whether the three-dimensional incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and a no-flow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over (3 x 10^(12))^2 near the point of the singularity, we are able to advance the solution up to tau_2 = 0.003505 and predict a singularity time of t(s) approximate to 0.0035056, while achieving a pointwise relative error of O(10^(-4)) in the vorticity vector. and observing a (3 x 10^8)-fold increase in the maximum vorticity parallel to omega parallel to(infinity). The numerical data are checked against all major blowup/non-blowup criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A local analysis near the point of the singularity also suggests the existence of a self-similar blowup in the meridian plane.

Chapter 7 On the Global Weak Solutions to a Variational Wave Equation

Substitution of the flow field U (x,y,z,t)={u (x, y, t), v (x, y, t), zγ (x, y, t)} into the three-dimensional incompressible Euler equations generates a closed system of evolution equations, for the strain rate γ (x, y, t) and the two-dimensional vorticity ω (x, y, t), which are uniform in the z-direction. The system models a class of dynamical, stretched three-dimensional vortex flows that include Burgers' vortices. Recent numerical simulations by Ohkitani & Gibbon have revealed that the strain rate γ (x, y, t) appears to develop a finite-time singularity, from smooth initial data, in the region where gamma is negative. Here, we prove that, for a large class of initial data, the support of γ- := max{0, -γ} necessarily collapses to zero in a finite time, while at the same time, the L1 norm of γ- remains non-zero. Hence, γ- must necessarily become singular before or at the time of collapse. Our vortex flow represents one of a subclass of Euler solutions that have infinite energy. The fundamental question of finite-time singularity formation from smooth initial data for finite-energy three-dimensional Euler solutions remains the important open question.

Whether the 3D incompressible Navier–Stokes 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 incompressible axisymmetric Navier–Stokes equations with smooth initial data of finite energy seem to develop potentially singular behavior at the origin. This potentially singular behavior is induced by a potential finite time singularity of the 3D Euler equations that we reported in a companion paper published in the same issue, see also Hou (Potential singularity of the 3D Euler equations in the interior domain. arXiv:2107.05870 [math.AP], 2021). We present numerical evidence that the 3D Navier–Stokes equations develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of \(10^7\). We have applied several blow-up criteria to study the potentially singular behavior of the Navier–Stokes equations. The Beale–Kato–Majda blow-up criterion and the blow-up criteria based on the growth of enstrophy and negative pressure seem to imply that the Navier–Stokes equations using our initial data develop a potential finite time singularity. We have also examined the Ladyzhenskaya–Prodi–Serrin regularity criteria (Kiselev and Ladyzhenskaya in Izv Akad Nauk SSSR Ser Mat 21(5):655–690, 1957; Prodi in Ann Math Pura Appl 4(48):173–182, 1959; Serrin in Arch Ration Mech Anal 9:187–191, 1962) that are based on the growth rate of \(L_t^q L_x^p\) norm of the velocity with \(3/p + 2/q \le 1\). Our numerical results for the cases of \((p,q) = (4,8),\; (6,4),\; (9,3)\) and \((p,q)=(\infty ,2)\) provide strong evidence for the potentially singular behavior of the Navier–Stokes equations. The critical case of \((p,q)=(3,\infty )\) is more difficult to verify numerically due to the extremely slow growth rate in the \(L^3\) norm of the velocity field and the significant contribution from the far field where we have a relatively coarse grid. Our numerical study shows that while the global \(L^3\) norm of the velocity grows very slowly, the localized version of the \(L^3\) norm of the velocity experiences rapid dynamic growth relative to the localized \(L^3\) norm of the initial velocity. This provides further evidence for the potentially singular behavior of the Navier–Stokes equations.

This paper provides a new general method for establishing a finite-time singularity formation for moving interface problems involving the incompressible Euler equations in the plane. This methodology is applied to two different problems. The first problem considered is the two-phase vortex sheet problem with surface tension, for which, under suitable assumptions of smallness of the initial height of the heaviest phase and velocity fields, is proved the finite-time singularity of the natural norm of the problem. This is in striking contrast with the case of finite-time splash and splat singularity formation for the one-phase Euler equations of [4] and [8], for which the natural norm (in the one-phase fluid) stays finite all the way until contact. The second problem considered involves the presence of a heavier rigid body moving in the inviscid fluid. For a very general set of geometries (essentially the contact zone being a graph) we first establish that the rigid body will hit the bottom of the fluid domain in finite time. Compared to the previous paper [20] for the rigid body case, the present paper allows for small square integrable vorticity and provides a characterization of acceleration at contact. A surface energy is shown to blow up and acceleration at contact is shown to oppose the motion: it is either strictly positive and finite if the contact zone is of non zero length, or infinite otherwise.

Nonuniqueness for the Yang–Mills heat flow

The primitive equations (PEs) model large-scale dynamics of the oceans and the atmosphere. While it is by now well known that the three-dimensional viscous PEs are globally well posed in Sobolev spaces, and that there are solutions to the inviscid PEs (also called the hydrostatic Euler equations) that develop singularities in finite time, the qualitative description of the blowup still remains undiscovered. In this paper, we provide a full description of two blow-up mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and pressure effects are subleading, but in a critical way: they localize the singularity closer and closer to the boundary near the blow-up time (with a logarithmic-in-time law). This first mechanism involves a smooth blow-up profile and is stable among smooth enough solutions. In the second one the pressure effects are fully negligible; this dynamics involves a two-parameter family of nonsmooth profiles, and is stable only by smoother perturbations.

3D Euler equations and ideal MHD mapped to regular systems: Probing the finite-time blowup hypothesis

We study here the initial value problem for a two‐dimensional Korteweg–de Vries (KdV) equation, first derived by Calogero and Bogoyavlenskii, by means of the inverse scattering transform. The dynamics of the discrete spectrum of an associated Schrödinger operator is far richer than that of KdV equation. Even for optimal eigenvalues, generic smooth solutions may develop shocks with multiple branches and/or cusp singularities in finite time. However, evolution may move poles of the transmission coefficient off the imaginary axis, destroy or even create them. We characterize conditions to prevent these pathologies before explosion time and describe ample classes of solutions, corresponding to both continuous and discrete spectrum. We also find that in certain conditions new eigenvalues might be created; in these cases a minimal set of initial spectral data must incorporate additionally the transmission coefficient on the entire plane. The previous results are applied to describe the Cauchy problem corresponding to initial data combinations of delta terms and derivatives and show that for long time the delta singularity may persist or be smoothed to a cusp‐discontinuity. Finally, we give conditions under which the evolution is reduced to the classical KdV.

The primitive equations (PEs) model planetary large-scale oceanic and atmospheric dynamics. While it has been shown that there are smooth solutions to the inviscid PEs (also called the hydrostatic Euler equations) with constant temperature (isothermal) that develop stable singularities in finite time, the effect of non-constant temperature on the singularity formation has not been established yet. This paper studies the stability of singularity formation for non-constant temperature in two scenarios: when there is no diffusion in the temperature, or when a vertical diffusivity is added to the temperature dynamics. For both scenarios, our results indicate that the variation of temperature affects neither the formation of singularity, nor its stability, in the velocity field, respectively.

The theorems in Chapter 1 are results for small data. The necessity for dealing with small perturbations of the linearized equations is underlined in the sequel by examples which show that, in general, one has to expect the development of singularities in finite time. In particular neither the smallness of the initial data nor the smoothness of data including the coefficients can prevent a solution from blowing up. We shall not go into the details here but we just present an illustration of the typical hyperbolic phenomenon that the solution and/or derivatives of the solution become singular after some time. This will mean in general that norms like the L∞-norm of the local regular solution or of its derivatives become infinite. The only way to avoid a blow-up are smallness of the data in connection with a sufficiently strong vanishing of the nonlinearity near zero and a sufficiently high space dimension. This is the message of the Theorems 1.1 and 1.2. Moreover we have learned from Theorem 1.2 that a solution of the nonlinear wave equation with a quadratic nonlinearity in ℝ3 lives at least exponentially long, although the examples below show that in general a blow-up occurs. Nevertheless this result justifies the notion of “almost global existence” in this case (cf. the paper of John & Klainerman [75]). We mention that for large data a blow-up may occur also in the cases where one has global existence for small data, see Example 1 in Chapter 1 and the remarks below.

A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow with initial data an isoparametric submanifold in Euclidean space and sphere. We show that the mean curvature flow preserves the isoparametric condition, develops singularities in finite time, and converges in finite time to a smooth submanifold of lower dimension. We also give a precise description of the collapsing.

OF THE DISSERTATION On the Questions of Local and Global Well-Posedness for the Hyperbolic PDEs Occurring in Some Relativistic Theories of Gravity and Electromagnetism by Jared R. Speck Dissertation Directors: Michael K.-H. Kiessling and A. Shadi Tahvildar-Zadeh The two hyperbolic systems of PDEs we consider in this work are the source-free Maxwell-Born-Infeld (MBI) field equations and the Euler-Nordstrom system for gravitationally self-interacting fluids. The former system plays a central role in Kiessling’s recently proposed self-consistent model of classical electrodynamics with point charges, a model that does not suffer from the infinities found in the classical Maxwell-Maxwell model with point charges. The latter system is a scalar gravity caricature of the incredibly more complex Euler-Einstein system. The primary original contributions of the thesis can be summarized as follows: • We give a sharp non-local criterion for the formation of singularities in planesymmetric solutions to the source-free MBI field equations. We also use a domain of dependence argument to show that 3-d initial data agreeing with certain planesymmetric data on a large enough ball lead to solutions that form singularities in finite time. This work is an extension of a theorem of Brenier, who studied singularity formation in periodic plane-symmetric solutions.

We consider a free boundary problem where the velocity depends on the mean curvature and on some non-local term. This problem arises as the singular limit of a reaction-diffusion system which describes the microphase separation of diblock copolymers. The interface may present singularities in finite time. This leads us to consider weak solutions on an arbitrary time interval and to prove the global-in-time convergence of solutions of the reaction-diffusion system.

In this thesis I present three separate studies on three different topics in General Relativity. The first study investigates the accuracy with which the mass and angular momentum of a black hole can be determined by measurements of gravitational waves from the hole, using a laser-interferometer gravitational-wave detector. The black hole is assumed to have been strongly perturbed, perhaps by coalescence with a binary companion, and the detector measures the waves produced by its resulting vibration and ring-down. The uncertainties in the measured mass and angular momentum arise from the unavoidable presence of noise in the detector. It is found that the faster the hole rotates, the more accurate the measurements will be, with the uncertainty in the angular momentum decreasing rapidly with increasing rotation speed. It is also found that the errors in the mass and angular momentum are highly correlated. The second study is an analysis of the gravitational collapse of an infinitely long, cylindrical dust shell. This analysis is expected to be helpful in understanding the behavior during collapse of more realistic, finite-length bodies. It is found that the collapse evolves into a naked singularity in finite time, as measured by a distant observer or by one riding on the shell. Analytical expressions for the variables describing the collapse are found at late times, near the singularity. The picture is completed with a numerical simulation that follows the collapse from the start until very close to the singularity. The singularity is found to be strong, in the sense that an observer riding on the shell will be infinitely stretched in the direction parallel to the symmetry axis, and infinitely compressed in the azimuthal direction. The gravitational waves emitted from the collapse are also analyzed. The last study focuses on a different kind of phenomenon, namely, the consequences of the existence of closed timelike curves in a spacetime that contains a wormhole. One might expect that the closed timelike curves would cause difficulty for the initial value problem for systems that evolve in such a spacetime: a system with apparently well-posed initial conditions might have no self-consistent solutions to its evolution equations. We study the simple case of a macroscopic, classical particle with a hard-sphere potential (a billiard ball), and we focus attention on initial conditions for which the evolution, if followed naively, is self- inconsistent: the ball enters one mouth of the wormhole and then comes out of the other mouth at an earlier time, then collides with its younger self, preventing itself from ever entering the first mouth. We find, surprisingly, that for all such dangerous initial conditions, there are an infinite number of self-consistent evolutionary solutions, involving a glancing collision and any number of wormhole traversals. We also find that for many non-dangerous initial conditions, there also exist an infinity of possible evolutions.