Blow-up for the incompressible 3D-Euler equations with uniform \( C^{1,\frac{1}{2}-ε}\cap L^2 \) force
Diego Córdoba, Luis Martínez-Zoroa
https://arxiv.org/abs/2309.08495
This paper presents a novel approach to establish a blow-up mechanism for the forced 3D incompressible Euler equations, with a specific focus on non-axisymmetric solutions. We construct solutions on \( \mathbb{R}^3 \) within the function space \( C^{3,\frac12}\cap L^2 \) for the time interval \( \left[ 0, T \right) \), where \( T > 0 \) is finite, subject to a uniform force in \( C^{1,\frac12 -\epsilon}\cap L^2 \). Remarkably, our methodology results in a blow-up: as the time t approaches the blow-up moment T, the integral \( \int_0^t |\nabla u| ds \) tends to infinity, all while preserving the solution’s smoothness throughout, except at the origin. In the process of our blow-up construction, self-similar coordinates are not utilized and we are able to treat solutions beyond the \( C^{1,\frac13+} \) threshold regularity of axy-symmetric solutions without swirl.
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Finite time blowup for an averaged three-dimensional Navier-Stokes equation
The Navier-Stokes equation on the Euclidean space \( \mathbf{R}^3 \) can be expressed in the form \( \partial_t u = \Delta u + B(u,u) \), where B is a certain bilinear operator on divergence-free vector fields u obeying the cancellation property \( \langle B(u,u), u\rangle=0 \) (which is equivalent to the energy identity for the Navier-Stokes equation). In this paper, we consider a modification \( \partial_t u = \Delta u + \tilde B(u,u) \) of this equation, where \( \tilde B \) is an averaged version of the bilinear operator \( B \) (where the average involves rotations and Fourier multipliers of order zero), and which also obeys the cancellation condition \( \langle \tilde B(u,u), u \rangle = 0 \) (so that it obeys the usual energy identity). By analysing a system of ODE related to (but more complicated than) a dyadic Navier-Stokes model of Katz and Pavlovic, we construct an example of a smooth solution to such a averaged Navier-Stokes equation which blows up in finite time. This demonstrates that any attempt to positively resolve the Navier-Stokes global regularity problem in three dimensions has to use finer structure on the nonlinear portion \( B(u,u) \) of the equation than is provided by harmonic analysis estimates and the energy identity. We also propose a program for adapting these blowup results to the true Navier-Stokes equations.
https://arxiv.org/abs/1402.0290
- Finite time blowup for an averaged three-dimensional Navier-Stokes equation
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Theory of Fluids Enters the 21st Century
From the 1800s to the early 2000s, physicists used the same theory to understand the behavior of fluids. Now, using a modern insight, they have redefined fluids from the bottom up.
Hydrodynamics as the effective field theory of strong-to-weak spontaneous symmetry breaking
Xiaoyang Huang, Marvin Qi, Jian-Hao Zhang, Andrew Lucas
https://arxiv.org/abs/2407.08760
Inspired by the hunt for new phases of matter in quantum mixed states, it has recently been proposed that the equivalence of microcanonical and canonical ensembles in statistical mechanics is a manifestation of strong-to-weak spontaneous symmetry breaking (SWSSB) in an underlying many-body quantum description. Here, we build an effective field theory for SWSSB of a global \( U(1) \) symmetry; the answer exactly reproduces the Schwinger-Keldysh effective field theory of diffusion for the conserved charge. We conclude that hydrodynamics can be understood as a theory of “superfluidity” for the broken strong symmetry: a non-vanishing susceptibility is a measurable order parameter for SWSSB, the diffusion mode is the Goldstone boson of the spontaneously broken continuous symmetry, and a generalization of Goldstone’s Theorem implies that the diffusion mode is always long-lived. This perspective provides a transparent physical explanation for the unusual “reparameterization” symmetries which are a necessary ingredient of Schwinger-Keldysh effective field theories for “normal fluids”.
Strong-to-Weak Spontaneous Symmetry Breaking in Mixed Quantum States
Leonardo A. Lessa, Ruochen Ma, Jian-Hao Zhang, Zhen Bi, Meng Cheng, Chong Wang
https://arxiv.org/abs/2405.03639
Symmetry in mixed quantum states can manifest in two distinct forms: strong symmetry, where each individual pure state in the quantum ensemble is symmetric with the same charge, and weak symmetry, which applies only to the entire ensemble. This paper explores a novel type of spontaneous symmetry breaking (SSB) where a strong symmetry is broken to a weak one. While the SSB of a weak symmetry is measured by the long-ranged two-point correlation function, the strong-to-weak SSB (SW-SSB) is measured by the fidelity correlator. We prove that SW-SSB is a universal property of mixed-state quantum phases, in the sense that the phenomenon of SW-SSB is robust against symmetric low-depth local quantum channels. We also show that the symmetry breaking is “spontaneous” in the sense that the effect of a local symmetry-breaking measurement cannot be recovered locally. We argue that a thermal state at a nonzero temperature in the canonical ensemble (with fixed symmetry charge) should have spontaneously broken strong symmetry. Additionally, we study non-thermal scenarios where decoherence induces SW-SSB, leading to phase transitions described by classical statistical models with bond randomness. In particular, the SW-SSB transition of a decohered Ising model can be viewed as the “ungauged” version of the celebrated toric code decodability transition. We confirm that, in the decohered Ising model, the SW-SSB transition defined by the fidelity correlator is the only physical transition in terms of channel recoverability. We also comment on other (inequivalent) definitions of SW-SSB, through correlation functions with higher Renyi indices.
Fracton hydrodynamics
Andrey Gromov, Andrew Lucas, Rahul M. Nandkishore
https://arxiv.org/abs/2003.09429
We introduce new classes of hydrodynamic theories inspired by the recently discovered fracton phases of quantum matter. Fracton phases are characterized by elementary excitations (fractons) with restricted mobility. The hydrodynamic theories we introduce describe thermalization in systems with fracton-like mobility constraints, including fluids where charge and dipole moment are both locally conserved, and fluids where charge is conserved along every line or plane of a lattice. Each of these fluids is subdiffusive, and constitutes a new universality class of hydrodynamic behavior. There are infinitely many such classes, each with distinct subdiffusive exponents, all of which are captured by our formalism. Our framework naturally explains recent results on dynamics with constrained quantum circuits, as well as recent experiments with ultracold atoms in tilted optical lattices. We identify crisp experimental signatures of these novel hydrodynamics, and explain how they may be realized in near term ultracold atom experiments.
Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics
Paolo Glorioso, Hong Liu
https://arxiv.org/abs/1805.09331
We review recent progress in developing effective field theories (EFTs) for non-equilibrium processes at finite temperature, including a new formulation of fluctuating hydrodynamics, and a new proof of the second law of thermodynamics. There are a number of new elements in formulating EFTs for such systems. Firstly, the nature of IR variables is very different from those of a system in equilibrium or near the vacuum. Secondly, while all static properties of an equilibrium system can in principle be extracted from the partition function, there appears no such quantity which can capture all non-equilibrium properties. Thirdly, non-equilibrium processes often involve dissipation, which is notoriously difficult to deal with using an action principle. The purpose of the review is to explain how to address these issues in a pedagogic manner, with fluctuating hydrodynamics as a main example.
Effective field theory of dissipative fluids
Michael Crossley, Paolo Glorioso, Hong Liu
https://arxiv.org/abs/1511.03646
We develop an effective field theory for dissipative fluids which governs the dynamics of long-lived gapless modes associated with conserved quantities. The resulting theory gives a path integral formulation of fluctuating hydrodynamics which systematically incorporates nonlinear interactions of noises. The dynamical variables are mappings between a “fluid spacetime” and the physical spacetime and an essential aspect of our formulation is to identify the appropriate symmetries in the fluid spacetime. The theory applies to nonlinear disturbances around a general density matrix. For a thermal density matrix, we require an additional \( Z_2 \) symmetry, to which we refer as the local KMS condition. This leads to the standard constraints of hydrodynamics, as well as a nonlinear generalization of the Onsager relations. It also leads to an emergent supersymmetry in the classical statistical regime, and a higher derivative deformation of supersymmetry in the full quantum regime.
Null energy condition and superluminal propagation
S. Dubovsky, T. Gregoire, A. Nicolis, R. Rattazzi
https://arxiv.org/abs/hep-th/0512260
We study whether a violation of the null energy condition necessarily implies the presence of instabilities. We prove that this is the case in a large class of situations, including isotropic solids and fluids relevant for cosmology. On the other hand we present several counter-examples of consistent effective field theories possessing a stable background where the null energy condition is violated. Two necessary features of these counter-examples are the lack of isotropy of the background and the presence of superluminal modes. We argue that many of the properties of massive gravity can be understood by associating it to a solid at the edge of violating the null energy condition. We briefly analyze the difficulties of mimicking \( \dot H>0 \) in scalar tensor theories of gravity.
The Four Color Theorem with Linearly Many Reducible Configurations and Near-Linear Time Coloring
Yuta Inoue, Ken-ichi Kawarabayashi, Atsuyuki Miyashita, Bojan Mohar, Carsten Thomassen, Mikkel Thorup
https://arxiv.org/abs/2603.24880
We give a near-linear time 4-coloring algorithm for planar graphs, improving on the previous quadratic time algorithm by Robertson et al. from 1996. Such an algorithm cannot be achieved by the known proofs of the Four Color Theorem (4CT). Technically speaking, we show the following significant generalization of the 4CT: every planar triangulation contains linearly many pairwise non-touching reducible configurations or pairwise non-crossing obstructing cycles of length at most 5 (which all allow for making effective 4-coloring reductions).
The known proofs of the 4CT only show the existence of a single reducible configuration or obstructing cycle in the above statement. The existence is proved using the discharging method based on combinatorial curvature. It identifies reducible configurations in parts where the local neighborhood has positive combinatorial curvature. Our result significantly strengthens the known proofs of 4CT, showing that we can also find reductions in large ``flat” parts where the curvature is zero, and moreover, we can make reductions almost anywhere in a given planar graph. This also opens possibilities for extensions to higher surfaces since we can find such flat parts in any large-width triangulation of any fixed surface.
From a computational perspective, the old proofs allowed us to apply induction on a problem that is smaller by some additive constant. The inductive step took linear time, resulting in a quadratic total time. With our linear number of reducible configurations or obstructing cycles, we can reduce the problem size by a constant factor. Our inductive step takes \( O(n\log n) \) time, yielding a 4-coloring in \( O(n\log n) \) total time. To efficiently handle a linear number of reducible configurations, we need them to be sufficiently robust to be useful in other applications. All our reducible configurations are what is known as D-reducible.
