Null Structures Research Articles

Rough solutions of the relativistic Euler equations

We prove that the time of classical existence of smooth solutions to the relativistic Euler equations can be bounded from below in terms of norms that measure the “(sound) wave-part” of the data in Sobolev space and “transport-part” in higher regularity Sobolev space and Hölder spaces. The solutions are allowed to have nontrivial vorticity and entropy. We use the geometric framework from [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270], where the relativistic Euler flow is decomposed into a “wave-part”, that is, geometric wave equations for the velocity components, density and enthalpy, and a “transport-part”, that is, transport-div-curl systems for the vorticity and entropy gradient. Our main result is that the Sobolev norm [Formula: see text] of the variables in the “wave-part” and the Hölder norm [Formula: see text] of the variables in the “transport-part” can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption [Formula: see text] is the optimal result for the variables in the “wave-part”. Compared to low-regularity results for quasilinear wave equations and the three-dimensional (3D) non-relativistic compressible Euler equations, the main new challenge of the paper is that when controlling the acoustic geometry and bounding the wave equation energies, we must deal with the difficulty that the vorticity and entropy gradient are four-dimensional space-time vectors satisfying a space-time div-curl-transport system, where the space-time div-curl part is not elliptic. Due to lack of ellipticity, one cannot immediately rely on the approach taken in [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270] to control these terms. To overcome this difficulty, we show that the space-time div-curl systems imply elliptic div-curl-transport systems on constant-time hypersurfaces plus error terms that involve favorable differentiations and contractions with respect to the four-velocity. By using these structures, we are able to adequately control the vorticity and entropy gradient with the help of energy estimates for transport equations, elliptic estimates, Schauder estimates and Littlewood–Paley theory.

The case for weak null in English

The concept of the null article in English was originally developed as a means of explaining the apparent paradox that singular nouns with the same surface form, namely the non-presence of a grammatical article, are used in quite different communicative situations. Null has been described as the most definite form in the English article system, standing in direct contrast to zero at the opposite end of a scale. Yet, while this may applicable in the case of predicates nouns denoting unique roles, certain institutions where the referent is pragmatically unique, or coordinate structures with a preceding referent, there is persuasive evidence to suggest a considerable number of other null structures do not fit such a description but, on the contrary, share common characteristics with the weak form of the definite article or occupy more of a neutral position which would permit insertion of either one of the surface articles. On this basis, a proposal is advanced to acknowledge two forms of the null article, strong and weak, according to context. Rather than a corpus-based statistical approach, this paper draws upon a smaller selection of contemporary, non-fictional texts from a variety of semantic fields to illustrate the points being made.

Парентетические конструкции в типологическом аспекте

The article deals with the structures with epistemic modals — parenthetical questions and imperatives. A comparative study of a number of languages — Russian, Polish, Bulgarian, Ukrainian, German, English, Italian, Spanish, and Portuguese — was carried out focusing on parenthetical structures that include mental verbs. In addition, the study highlights the fact that in many cases, depending on the language pair in translation, parenthetical structures in a source language match matrix structures introducing clauses in a target language, and vice versa. Building on the bilingual corpora material the author proves that the Russian language is more likely to use parenthetical structures, while their translation equivalents appear to be either null or present matrix structures controlling a subordinate clause.

Topological analysis of three-dimensional magnetic reconnection in SPERF-AREX for simulated magnetopause events

A scientific research device for geospace physics, the Space Plasma Environment Research Facility (SPERF), is going to be in operation this year at the Harbin Institute of Technology, China. One of its laboratory simulation platforms, the Asymmetric Reconnection EXperiment (AREX), is designed to investigate the asymmetric magnetic reconnection relevant to the process in the dayside magnetopause of the Earth. As a ground-based experimental platform, AREX emphasizes three-dimensional (3D) asymmetric driving features with topological similarity relevant to dayside reconnection. Thus, in this paper, we focus on topological analysis based on numerical simulation of AREX reconnection processes. 3D topological features in various scenarios relevant to magnetopause reconnection in different interplanetary magnetic field (IMF) conditions are investigated by adjusting setup parameters of plasma sources and magnetic coils. The simulation results reveal that the plasma distribution of asymmetric reconnection in SPERF-AREX is analogous to that at the magnetopause. Various types of the 3D magnetic topology relevant to typical IMF conditions, including the X-line and the A–B null chain structures, are, respectively, identified and analyzed in detail.

The Effect of Current on Magnetic Null Topology during Turbulent Reconnection

Abstract Using data from the Cluster mission and the First-Order Taylor Expansion method, we investigate the spiral magnetic nulls nested in the diffusion region of turbulent reconnection in the magnetotail. We particularly focus on the relation between the magnetic null topologies and currents, which can be decomposed into a component perpendicular to spine (j ⊥) and a component parallel to spine (j ∥). We find that (1) the currents surrounding the spiral nulls are mainly contributed by j ∥; (2) the null with large (j ⊥) and small spine-fan angle (θ), which are predicted by traditional linear theory, does not exist in the turbulent diffusion region; (3) the background current j b plays an important role in determining the direction of the currents around spiral nulls and consequently the orientation of the magnetic null structures; and (4) the spiral nulls with strong current (large magnitude j ) tend to degenerate into 2D configurations, whereas the nulls with weak currents retain the 3D features. Since the spiral magnetic nulls are crucial for the energy dissipation during the turbulent reconnection process, all of these results can provide important information for better understanding 3D turbulent reconnection.

The hidden null structure of the compressible Euler equations and a prelude to applications

We derive a new formulation of the compressible Euler equations exhibiting remarkable structures, including surprisingly good null structures. The new formulation comprises covariant wave equations for the Cartesian components of the velocity and the logarithmic density coupled to transport equations for the Cartesian components of the specific vorticity, defined to be vorticity divided by density. The equations allow one to use the full power of the geometric vectorfield method in treating the “wave part” of the system. A crucial feature of the new formulation is that all derivative-quadratic inhomogeneous terms verify the strong null condition. The latter is a nonlinear condition signifying the complete absence of nonlinear interactions involving more than one differentiation in a direction transversal to the acoustic characteristics. Moreover, the same good structures are found in the equations verified by the Euclidean divergence and curl of the specific vorticity. This is important because one needs to combine estimates for the divergence and curl with elliptic estimates to obtain sufficient regularity for the specific vorticity, whose derivatives appear as inhomogeneous terms in the wave equations. The structures described above collectively open the door for our companion results, in which we exhibit a stable regime of initially smooth solutions that develop a shock singularity. In particular, the first Cartesian coordinate partial derivatives of the velocity and density blow up, while relative to a system of geometric coordinates adapted to the acoustic characteristics, the solution (including the vorticity) remains many times differentiable, all the way up to the shock. The good null structures, which are often associated with global solutions, are in fact key to proving that the shock singularity forms. Our secondary goal in this paper is to provide an overview of the central role that the structures play in the proof.

Global regularity for the 3D finite depth capillary water waves

In this paper, we prove global regularity, scattering, and the non-existence of small traveling waves for the $3D$ finite depth capillary waves system for small initial data. The non-existence of small traveling waves shows a fundamental difference between the capillary waves ($\sigma=1, g=0$) and the gravity-capillary waves ($\sigma=1,$ $ 0< g< 3$) in the finite depth setting. As, for the later case, there exists arbitrary small $L^2$ traveling waves. Different from the water waves system in the infinite depth setting, the quadratic terms of the same system in the finite depth setting are worse due to the absence of null structure inside the Dirichlet-Neumann operator. In the finite depth setting, the capillary waves system has the worst quadratic terms among the water waves systems with all possible values of gravity effect constant and surface tension coefficient. It loses favorable cancellations not only in the High $\times$ Low type interaction but also in the High $\times $ High type interaction. In the worst scenario, the best decay rate of the nonlinear solution that one could expect is $ (1+t) ^ {-1/2} $, because the $3D$ finite depth capillary waves system lacks null structures and there exists $Q(u, \bar{u})$ type quadratic term, which causes a very large time resonance set and the definite growth of the associated profile. As a result, the problematic terms are not only the quadratic terms but also the cubic terms. To prove global regularity for the capillary waves system, we identify a good variable to prove the dispersion estimate, fully exploit the hidden structures inside the capillary waves system, and use a novel method to control the weighted norms.

New Zero Poisson's Ratio Structures

Most materials exhibit positive Poisson's ratio (PR) values, but special structures can also present negative and, even rarer, zero (or close to zero) PR. Null PR structures have received much attention due to their unusual properties and potential applications in different fields, such as aeronautics and bioengineering. Herein, a new and simple near‐zero PR 2D topological model is presented based on a structural block composed of two smooth and rigid bars connected by a soft membrane or spring. It is not based on reentrant or honeycomb‐like configurations, which have been the basis of many null or quasinull PR models. The topological model is 3D printed, and the experimentally obtained PR is −0.003 ± 0.001 which is one of the closest to zero values ever reported. The possibility to extend this model to 3D systems with compression in any direction is discussed. The advantages and disadvantages of these models are also addressed.

On the Ideal Magnetohydrodynamics in Three-Dimensional Thin Domains: Well-Posedness and Asymptotics

We consider the ideal magnetohydrodynamics (MHD) subjected to a strong magnetic field along $x_1$ direction in three-dimensional thin domains $\Omega_\delta=\mathbb{R}^2\times(-\delta,\delta)$ with slip boundary conditions. It is well-known that in this situation the system will generate Alfv\'en waves. Our results are summarized as follows: (i).\, We construct the global solutions (Alfv\'en waves) to MHD in the thin domain $\Omega_\delta$ with $\delta>0$. In addition, the uniform energy estimates are obtained with respected to the parameter $\delta$. (ii). We justify the asymptotics of the MHD equations from the thin domain $\Omega_\delta$ to the plane $\mathbb{R}^2$. More precisely, we prove that the 3D Alfv\'en waves in $\Omega_\delta$ will converge to the Alfv\'en waves in $\mathbb{R}^2$ in the limit that $\delta$ goes to zero. This shows that Alfv\'en waves propagating along the horizontal direction of the (3D) strip are stable and can be approximated by the (2D) Alfv\'en waves when $\delta$ is sufficiently small. Moreover, the control of the (2D) Alfv\'en waves can be obtained from the control of (3D) Alfv\'en waves in the thin domain $\Omega_\delta$ with aid of the uniform bounds. The proofs of main results rely on the design of the proper energy functional and the null structures of the nonlinear terms. Here the null structures means two aspects: separation of the Alfv\'en waves ($z_+$ and $z_-$) and no bad quadratic terms $Q(\partial_3 z_-^h, \partial_3 z_+^h)$ where $z_\pm=(z_\pm^h, z^3_\pm)$ and $Q(\partial_3 z_-^h,\partial_3 z_+^h)$ is the linear combination of terms $\partial^\alpha \partial_3 z_-^h\partial^\beta \partial_3 z_+^h$ with $\alpha, \beta \in (\mathbb{Z}_{\geq0})^3$.

Global Existence of Compressible Dissipative Elastodynamics Systems with Zero Shear Viscosity in Two Dimensions

In this paper, we study the two dimensional compressible dissipative elastodynamics system with zero shear viscosity, and prove global stability around a constant equilibrium. Due to the low pointwise dispersive estimate, $$\frac{1}{\sqrt{t}}$$, for the divergence-free projection of solutions in dimension two, and the lack of null structures in convective terms, energy estimates, combined with Green’s functions and dispersive estimates, for both solutions and their potentials, are obtained to control the interactions in nonlinear terms.

A New Formulation of the 3D Compressible Euler Equations with Dynamic Entropy: Remarkable Null Structures and Regularity Properties

We derive a new formulation of the 3D compressible Euler equations exhibiting remarkable null structures and regularity properties. Our results hold for an arbitrary equation of state (which yields the pressure in terms of the density and the entropy) in non-vacuum regions where the speed of sound is positive. Our work here is an extension of our prior joint work with J. Luk, in which we derived a similar new formulation in the special case of a barotropic fluid, that is, when the equation of state depends only on the density. The new formulation comprises covariant wave equations for the Cartesian components of the velocity and the logarithmic density coupled to a transport equation for the specific vorticity (defined to be vorticity divided by density), transport equations for the entropy and its gradient, and some additional transport–divergence–curl-type equations involving special combinations of the derivatives of the solution variables. The good geometric structures in the equations allow one to use the full power of the vectorfield method in treating the “wave part” of the system. In a forthcoming application, we will use the new formulation to give a sharp, constructive proof of finite-time shock formation, tied to the intersection of acoustic “wave characteristics,” for solutions with nontrivial vorticity and entropy at the singularity. In the present article, we derive the new formulation and provide an overview of the central role that it plays in the proof of shock formation. Although the equations are significantly more complicated than they are in the barotropic case, they enjoy many of same remarkable features, including: (i) all derivative-quadratic inhomogeneous terms are null forms relative to the acoustical metric, which is the Lorentzian metric driving the propagation of sound waves and (ii) the transport–divergence–curl-type equations allow one to show that the entropy is one degree more differentiable than the velocity and that the vorticity is exactly as differentiable as the velocity, assuming that the initial data enjoy the same gain in regularity. This represents a gain of one derivative compared to standard estimates. This gain of a derivative, which seems to be new for the entropy, is essential for closing the energy estimates in our forthcoming proof of shock formation, since the second derivatives of the entropy and the first derivatives of the vorticity appear as inhomogeneous terms in the wave equations.

Geometry and symmetries of null G-structures

We present a definition of null G-structures on Lorentzian manifolds and investigate their geometric properties. This definition includes the Robinson structure on 4-dimensional black holes as well as the null structures that appear in all supersymmetric solutions of supergravity theories. We also identify the induced geometry on some null hypersurfaces and that on the orbit spaces of null geodesic congruences in such Lorentzian manifolds. We give the algebra of diffeomorphisms that preserves a null G-structure and demonstrate that in some cases it interpolates between the BMS algebra of an asymptotically flat spacetime and the Lorentz symmetry algebra of a Killing horizon.

The Relativistic Euler Equations: Remarkable Null Structures and Regularity Properties

We derive a new formulation of the relativistic Euler equations that exhibits remarkable properties. This new formulation consists of a coupled system of geometric wave, transport, and elliptic equations, sourced by nonlinearities that are null forms relative to the acoustical metric. Our new formulation is well-suited for various applications, in particular for the study of stable shock formation, as it is surveyed in the paper. Moreover, using the new formulation presented here, we establish a local well-posedness result showing that the vorticity and the entropy of the fluid are one degree more differentiable compared to the regularity guaranteed by standard estimates (assuming that the initial data enjoy the extra differentiability). This gain in regularity is essential for the study of shock formation without symmetry assumptions. Our results hold for an arbitrary equation of state, not necessarily of barotropic type.

Null Structures and Degenerate Dispersion Relations in Two Space Dimensions

Abstract For a dispersive partial differential equation, the degeneracy of its dispersion relation will deteriorate dispersion of waves and strengthen nonlinear effects. Such negative effects can sometimes be mitigated by some null structure in the nonlinearity. Motivated by water-wave problems, in this paper we consider a class of nonlinear dispersive PDEs in 2D with cubic nonlinearities, whose dispersion relations are radial and have vanishing Gaussian curvature on a circle. For such a model we identify certain null structures for the cubic nonlinearity, which suffice in order to guarantee global scattering solutions for the small data problem. Our null structures in the power-type nonlinearity are weak and only eliminate the worst nonlinear interaction. Such null structures arise naturally in some water-wave problems.

Learning Task Priorities from Demonstrations

Bimanual operations in humanoids offer the possibility to carry out more than one manipulation task at the same time, which in turn introduces the problem of task prioritization. We address this problem from a learning from demonstration perspective, by extending the task-parameterized Gaussian mixture model to Jacobian and null space structures. The proposed approach is tested on bimanual skills but can be applied in any scenario where the prioritization between potentially conflicting tasks needs to be learned. We evaluate the proposed framework in: two different tasks with humanoids requiring the learning of priorities and a loco-manipulation scenario, showing that the approach can be exploited to learn the prioritization of multiple tasks in parallel.

The Cauchy problem for space-time monopole equations in Sobolev spaces

We consider the initial value problem of space-time monopole equations in one space dimension with initial data in Sobolev space Hs. Observing null structures of the system, we prove local well-posedness in almost critical space. Unconditional uniqueness and global existence are proved for s ≥ 0. Moreover, we show that the H1 Sobolev norm grows at a rate of at most c exp(ct2).

Pure spinors, intrinsic torsion and curvature in odd dimensions

We study the geometric properties of a (2m+1)-dimensional complex manifold M admitting a holomorphic reduction of the frame bundle to the structure group P⊂Spin(2m+1,C), the stabiliser of the line spanned by a pure spinor at a point. Geometrically, M is endowed with a holomorphic metric g, a holomorphic volume form, a spin structure compatible with g, and a holomorphic pure spinor field ξ up to scale. The defining property of ξ is that it determines an almost null structure, i.e. an m-plane distribution Nξ along which g is totally degenerate.We develop a spinor calculus, by means of which we encode the geometric properties of Nξ and of its rank-(m+1) orthogonal complement Nξ⊥ corresponding to the algebraic properties of the intrinsic torsion of the P-structure. This is the failure of the Levi-Civita connection ∇ of g to be compatible with the P-structure. In a similar way, we examine the algebraic properties of the curvature of ∇.Applications to spinorial differential equations are given. Notably, we relate the integrability properties of Nξ and Nξ⊥ to the existence of solutions of odd-dimensional versions of the zero-rest-mass field equation. We give necessary and sufficient conditions for the almost null structure associated to a pure conformal Killing spinor to be integrable. Finally, we conjecture a Goldberg–Sachs-type theorem on the existence of a certain class of almost null structures when (M,g) has prescribed curvature.We discuss applications of this work to the study of real pseudo-Riemannian manifolds.

Asymptotic structure of electrodynamics revisited

We point out that recently published analyses of null and timelike infinity and long-range structures in electrodynamics to large extent rediscover results present in the literature. At the same time, some of the conclusions these recent works put forward may prove controversial. In view of these facts, we find it desirable to revisit the analysis taken up more than two decades ago, starting from earlier works on null infinity by other authors.

Global smooth solutions for the nonlinear Schrödinger equation with magnetic effect

We consider the Cauchy problem of the nonlinear Schrödinger equation with magnetic effect, and prove global existence of smooth solutions and decay estimates for suitably small initial data. The key step in our analysis is to exploit the null structures for the phases, which allow us to close our argument in the framework of space-time resonance method.

Cosmic acceleration from matter–curvature coupling

We consider f(R,T) modified theory of gravity in which, in general, the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar and the trace of the energy-momentum tensor. We indicate that in this type of the theory, the coupling energy-momentum tensor is~not conserved. However, we mainly focus on a particular model that matter is minimally coupled to the geometry in the metric formalism and wherein, its coupling energy--momentum tensor is also conserved. We obtain the corresponding Raychaudhuri dynamical equation that presents the evolution of the kinematic quantities. Then for the chosen model, we derive the behavior of the deceleration parameter, and show that the coupling term can lead to an acceleration phase after the matter dominated phase. On the other hand, the curvature of the universe corresponds with the deviation from parallelism in the geodesic motion. Thus, we also scrutinize the motion of the free test particles on their geodesics, and derive the geodesic deviation equation in this modified theory to study the accelerating universe within the spatially flat FLRW background. Actually, this equation gives the relative accelerations of adjacent particles as a measurable physical quantity, and provides an elegant tool to investigate the timelike and the null structures of spacetime geometries. Then, through the null deviation vector, we find the observer area-distance as a function of the redshift for the chosen model, and compare the results with the corresponding results obtained in the literature.