Degenerate Vacuum Research Articles

On regular solutions and singularity formation for Vlasov/Navier-Stokes equations with degenerate viscosities and vacuum

<p style='text-indent:20px;'>We analyze the Vlasov equation coupled with the compressible Navier–Stokes equations with degenerate viscosities and vacuum. These two equations are coupled through the drag force which depends on the fluid density and the relative velocity between particle and fluid. We first establish the existence and uniqueness of local-in-time regular solutions with arbitrarily large initial data and a vacuum. We then present sufficient conditions on the initial data leading to the finite-time blowup of regular solutions. In particular, our study makes the result on the finite-time singularity formation for Vlasov/Navier–Stokes equations discussed by Choi [J. Math. Pures Appl., 108, (2017), 991–1021] completely rigorous.</p>

Global well-posedness of regular solutions to the three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities and vacuum

In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations with degenerate viscosities is considered. By introducing some new variables and making use of the “quasi-symmetric hyperbolic”–“degenerate elliptic” coupled structure to control the behavior of the fluid velocity, we prove the global-in-time well-posedness of regular solutions with vacuum for a class of smooth initial data that are of small density but possibly large velocities. Here the initial mass density is required to decay to zero in the far field, and the spectrum of the Jacobi matrix of the initial velocity are all positive. The result here applies to a class of degenerate density-dependent viscosity coefficients, is independent of the BD-entropy, and seems to be the first on the global existence of smooth solutions which have large velocities and contain vacuum state for such degenerate system in three space dimensions.

Non-existence of global classical solutions to barotropic compressible Navier-Stokes equations with degenerate viscosity and vacuum

We are concerned about the barotropic compressible Navier-Stokes equations with density-dependent viscosities which may degenerate in vacuum. We show that any classical solution to barotropic compressible Navier-Stokes equations in bounded domains will blow up, when the initial density admits an isolated mass group and the viscosity coefficients satisfy some conditions. A new condition on viscosities is first put forward in this paper.

Vortex sheet turbulence as solvable string theory

We study steady vortex sheet solutions of the Navier–Stokes in the limit of vanishing viscosity at fixed energy flow. We refer to this as the turbulent limit. These steady flows correspond to a minimum of the Euler Hamiltonian as a functional of the tangent discontinuity of the local velocity parametrized as [Formula: see text]. This observation means that the steady flow represents the low-temperature limit of the Gibbs distribution for vortex sheet dynamics with the normal displacement [Formula: see text] of the vortex sheet as a Hamiltonian coordinate and [Formula: see text] as a conjugate momentum. An infinite number of Euler conservation laws lead to a degenerate vacuum of this system, which explains the complexity of turbulence statistics and provides the relevant degrees of freedom (random surfaces). The simplest example of a steady solution of the Navier–Stokes equation in the turbulent limit is a spherical vortex sheet whose flow outside is equivalent to a potential flow past a sphere, while the velocity is constant inside the sphere. Potential flow past other bodies provide other steady solutions. The new ingredient we add is a calculable gap in tangent velocity, leading to anomalous dissipation. This family of steady solutions provides an example of the Euler instanton advocated in our recent work, which is supposed to be responsible for the dissipation of the Navier–Stokes equation in the turbulent limit. We further conclude that one can obtain turbulent statistics from the Gibbs statistics of vortex sheets by adding Lagrange multipliers for the conserved volume inside closed surfaces, the rate of energy pumping, and energy dissipation. The effective temperature in our Gibbs distribution goes to zero as [Formula: see text] with Reynolds number [Formula: see text] in the turbulent limit. The Gibbs statistics in this limit reduces to the solvable string theory in two dimensions (so-called [Formula: see text] critical matrix model). This opens the way for nonperturbative calculations in the Vortex Sheet Turbulence, some of which we report here.

Extended metastable dark energy

The metastable dark energy scenario is revisited by assuming that the current false vacuum energy density is the remnant from a primeval inflationary stage. The zero temperature scalar field potential is here described by an even power series up to order six which depends on 3 free parameters: the mass of the scalar field ($m$), the dimensionless ($\lambda$) specifying the standard self-interaction term, and a free cutoff mass scale ($M$) quantifying all possible deviations from the degenerate false vacuum state. The current $\Lambda$CDM model is a consequence of the very long decay time of the false vacuum which although finite is much greater than the current age of the Universe. This result remains valid for arbitrary combinations of the $m/M$ ratio which can analytically be determined in the thin-wall approximation and numerically calculated outside this limit. Unlike many claims in the literature the vacuum dominance may be temporary. The finiteness of the decay time suggests that the ultimate stage of the observed Universe in such a scenario will not be driven by a de Sitter type cosmology.

String monopoles, string walls, vortex skyrmions, and nexus objects in the polar distorted B phase of He3

The composite cosmological objects -- Kibble-Lazarides-Shafi (KLS) walls bounded by strings and cosmic strings terminated by Nambu monopoles -- could be produced during the phase transitions in the early Universe. Recent experiments in superfluid $^3$He reproduced the formation of the KLS domain walls, which opened the new arena for the detailed study of those objects in human controlled system with different characteristic lengths. These composite defects are formed by two successive symmetry breaking phase transitions. In the first transition the strings are formed, then in the second transition the string becomes the termination line of the KLS wall. In the same manner, in the first transition monopoles are formed, and then in the second transition these monopoles become the termination points of strings. Here we show that in the vicinity of the second transition the composite defects can be described by relative homotopy groups. This is because there are two well separated length scales involved, which give rise to two different classes of the degenerate vacuum states, $R_1$ and $R_2$, and the composite objects correspond to the nontrivial elements of the group $\pi_n(R_1,R_2)$. We discuss this on example of the so-called polar distorted B phase, which is formed in the two-step phase transition in liquid $^3$He distorted by aerogel. In this system the string monopoles terminate spin vortices with even winding number, while KLS string walls terminate on half quantum vortices. In the presence of magnetic field, vortex-skyrmions are formed, and the string monopole transforms to the nexus. We also discuss the integer-valued topological invariants of those objects. Our consideration can be applied to the composite defects in other condensed matter and cosmological systems.

Spontaneous symmetry breaking in pure 2D Yang-Mills theory

We consider purely topological 2D Yang-Mills theory on a torus with the second Stiefel-Whitney class added to the Lagrangian in the form of a $\ensuremath{\theta}$ term. It will be shown that at $\ensuremath{\theta}=\ensuremath{\pi}$ there exists a class of $SU(2N)/{\mathbb{Z}}_{2}$ ($N>1$) gauge theories with a twofold degenerate vacuum, which spontaneously breaks the time reversal and charge conjugation symmetries. The corresponding order parameter is given by the generator $\mathcal{O}$ of the ${\mathbb{Z}}_{N}$ one-form symmetry.

Thermal representations in group field theory: squeezed vacua and quantum gravity condensates

We apply the formalism of thermofield dynamics to group field theory quantum gravity and construct thermal representations associated with generalised equilibrium Gibbs states using Bogoliubov transformations. The newly constructed class of thermal vacua are entangled, two-mode squeezed, thermofield double states. The corresponding finite temperature representations are inequivalent to the standard zero temperature one based on a degenerate vacuum. An interesting class of states, coherent thermal states, are defined and understood as thermal quantum gravity condensates.

Metastable vacua in large-N QCD3

We reexamine the vacuum structure of three-dimensional quantum chromodynamics (QCD3) with gauge group SU(N), Nf fundamental quark flavors, and a level-k Chern-Simons term. This analysis can be reliably carried out in the large-N, fixed Nf, k limit of the theory, up to certain assumptions that we spell out explicitly. At leading order in the large-N expansion we find Nf + 1 distinct, exactly degenerate vacuum superselection sectors with different patterns of flavor-symmetry breaking. The associated massless Nambu-Goldstone bosons are generically accompanied by topological Chern-Simons theories. This set of vacua explicitly realizes many candidate phases previously proposed for QCD3. At subleading order in the large-N expansion, the exact degeneracy between the different superselection sectors is lifted, leading to a multitude of metastable vacua. If we dial the quark masses, different metastable vacua can become the true vacuum of the theory, leading to a sequence of first-order phase transitions. We show that this intricate large-N dynamics can be captured by the previously proposed bosonic dual theories for QCD3, provided these bosonic duals are furnished with a suitable scalar potential. Interestingly, this potential must include terms beyond quartic order in the scalar fields.

On Classical Solutions for Viscous Polytropic Fluids with Degenerate Viscosities and Vacuum

In this paper, we consider the three-dimensional isentropic Navier–Stokes equations for compressible fluids allowing initial vacuum when viscosities depend on density in a superlinear power law. We introduce the notion of regular solutions and prove the local-in-time well-posedness of solutions with arbitrarily large initial data and a vacuum in this class, which is a long-standing open problem due to the very high degeneracy caused by a vacuum. Moreover, for certain classes of initial data with a local vacuum, we show that the regular solution that we obtained will break down in finite time, no matter how small and smooth the initial data are.

Existence results for chemotaxis-shallow water system with degenerate viscosity coefficients and initial vacuum

This paper is concerned with the Cauchy problem of the chemotaxis-shallow water system with degenerate viscosity coefficients. A local existence of the unique regular solution is established when the initial data are arbitrarily large and include vacuum. Moreover, a Beal–Kate–Majda type blow-up criterion is provided.

The Quantum Yang-Baxter Conditions: The Fundamental Relations behind the Nambu-Goldstone Theorem

We demonstrate that when there is spontaneous symmetry breaking in any system, relativistic or non-relativistic, the dynamic of the Nambu-Goldstone bosons is governed by the Quantum Yang-Baxter equations. These equations describe the triangular dynamical relations between pairs of Nambu-Goldstone bosons and the degenerate vacuum. We then formulate a theorem and a corollary showing that these relations guarantee the appropriate dispersion relation and the appropriate counting for the Nambu-Goldstone bosons.

Optimal decay rates on compressible Navier–Stokes equations with degenerate viscosity and vacuum

This is a continuation of the paper [16, Math. Models Methods Appl. Sci. 28 (2018) 337–386] on the study of the large time behavior of the weak solution to the free boundary problem for one-dimensional isentropic compressible Navier–Stokes equations with degenerate viscosity and vacuum. Under appropriate smallness conditions on the initial data (initial energy), we extend the results in [16, Math. Models Methods Appl. Sci. 28 (2018) 337–386] to the case γ>1 and θ<min⁡{1,γ−12}. Clearly, the optimal decay rate of the density function along with its behavior near the interfaces is studied. In the meanwhile, we obtain also sharper decay rates for the norms in terms of the velocity function. The proof is based on the standard line method. The key is to establish some new global-in-time weighted (both in time and space) estimates uniformly up to the vacuum boundary, which ensures the uniform convergence of the approximate solutions.

Existence results for the radiation hydrodynamic equations with degenerate viscosity coefficients and vacuum

In this paper, we consider the compressible isentropic radiation hydrodynamic equations with density-dependent viscosity coefficients when the initial data are arbitrarily large and include vacuum at least appearing in the far field. Based on some reasonable assumptions for the radiation coefficients, we firstly establish the existence of a unique local regular solution, which implies the existence of the local strong solution. Moreover, we show a blow-up criterion for the regular solution that we obtained.

The origin of the mass of the Nambu–Goldstone bosons

We explain the origin of the mass for the Nambu–Goldstone bosons when there is a chemical potential in the action which explicitly breaks the symmetry. The method is based on the number of independent histories for the interaction of the pair of Nambu–Goldstone bosons with the degenerate vacuum (triangle relations). The analysis suggests that under some circumstances, pairs of massive Nambu–Goldstone bosons can become a single degree of freedom with an effective mass defined by the superposition of the individual masses of each boson. Possible mass oscillations for the Nambu–Goldstone bosons are discussed.

Weak-strong uniqueness for compressible Navier–Stokes system with degenerate viscosity coefficient and vacuum in one dimension

We prove weak-strong uniqueness results for the compressible Navier-Stokes sys-tem with degenerate viscosity coefficient and with vacuum in one dimension. In other words, we give conditions on the weak solution constructed in [20] so that it is unique. The main novelty consists in dealing with initial density ρ 0 which contains vacuum, indeed the most of the results exclude this situation in order to use the parabolic-ity of the momentum equation (see [25]). To do this we use the notion of relative entropy developed recently by Germain and Feireisl et al (see [9, 8]) combined with a new formulation of the compressible system ([10, 12, 11, 13]), more precisely we introduce a new effective velocity which makes the system parabolic on the density and hyperbolic on this velocity.

Recent progress on classical solutions for compressible isentropic Navier-Stokes equations with degenerate viscosities and vacuum

The purpose of this paper is to present our recent progress on the local theory of classical solutions for compressible isentropic Navier-Stokes equations with density dependent viscosities in a power law and vacuum, which is a long-standing open problem due to the very high degeneracy caused by vacuum for this system. We will also mention some related open problems of high mathematical interest for this system.

On the smallness of the cosmological constant

In N=1 supergravity the scalar potential of the hidden sector may have degenerate supersymmetric (SUSY) and non-supersymmetric Minkowski vacua. In this case local SUSY in the second supersymmetric Minkowski phase can be broken dynamically. Assuming that such a second phase and the phase associated with the physical vacuum are exactly degenerate, we estimate the value of the cosmological constant. We argue that the observed value of the dark energy density can be reproduced if in the second vacuum local SUSY breaking is induced by gaugino condensation at a scale which is just slightly lower than ΛQCD in the physical vacuum. The presence of a third degenerate vacuum, in which local SUSY and electroweak (EW) symmetry are broken near the Planck scale, may lead to small values of the quartic Higgs self–coupling and the corresponding beta function at the Planck scale in the phase in which we live.

Spontaneous symmetry breaking, and strings defects in hypercomplex gauge field theories

Inspired by the appearance of split-complex structures in the dimensional reduction of string theory, and in the theories emerging as byproducts, we study the hyper-complex formulation of Abelian gauge field theories, by incorporating a new complex unit to the usual complex one. The hypercomplex version of the traditional Mexican hat potential associated with the $U(1)$ gauge field theory, corresponds to a {\it hybrid} potential with two real components, and with $U(1)\times SO(1,1)$ as symmetry group. Each component corresponds to a deformation of the hat potential, with the appearance of a new degenerate vacuum. Hypercomplex electrodynamics will show novel properties, such as the spontaneous symmetry breaking scenarios with running masses for the vectorial and scalar Higgs fields, and the Aharonov-Bohm type strings defects as exact solutions; these topological defects may be detected only by quantum interference of charged particles through gauge invariant loop integrals. In a particular limit, the {\it hyperbolic} electrodynamics does not admit topological defects associated with continuous symmetries

Existence results for viscous polytropic fluids with degenerate viscosity coefficients and vacuum

In this paper, we considered the isentropic Navier–Stokes equations for compressible fluids with density-dependent viscosities in R3. These systems come from the Boltzmann equations through the Chapman–Enskog expansion to the second order, cf. [17], and are degenerate when vacuum appears. We firstly establish the existence of the unique local regular solution (see Definition 1.1 or [11]) when the initial data are arbitrarily large with vacuum at least appearing in the far field. Moreover it is interesting to show that we couldn't obtain any global regular solution satisfying that the L∞ norm of u decays to zero as time t goes to infinity.