Divergence Equation Research Articles

Solvability of the divergence equation implies John via Poincaré inequality

Let Ω⊂R2 be a bounded simply connected domain. We show that, for a fixed (every) p∈(1,∞), the divergence equation divv=f is solvable in W01,p(Ω)2 for every f∈L0p(Ω), if and only if Ω is a John domain, if and only if the weighted Poincaré inequality ∫Ω|u(x)−uΩ|qdx≤C∫Ω|∇u(x)|q dist (x,∂Ω)qdx holds for some (every) q∈[1,∞). This gives a positive answer to a question raised by Russ (2013) in the case of bounded simply connected domains. In higher dimensions similar results are proved under some additional assumptions on the domain in question.

Necessary and Sufficient Conditions of Stabilization of Solutions of the First Boundary-Value Problem for a Parabolic Equation

Necessary and sufficient conditions are established for the stabilization of the solution of the first boundary-value problem for a divergent second-order parabolic equation with no lower-order terms. For an arbitrary (possibly unbounded) domain Q ⊂ ℝN, N ≥ 3, with an initial function u 0(x) defined in Q for t = 0, one studies the influence of Q on the stabilization of the solution in the cylinder Z = Q × [0,∞) with the homogeneous boundary conditions on the lateral surface of Z.

Green’s formula and singularity at a triple contact line. Example of finite-displacement solution

The various equations at the surfaces and triple contact lines of a deformable body are obtained from a variational condition, by applying Green’s formula in the whole space and on the Riemannian surfaces. The surface equations are similar to the Cauchy’s equations for the volume, but involve a special definition of the ‘divergence’ (tensorial product of the covariant derivatives on the surface and the whole space). The normal component of the divergence equation generalizes the Laplace’s equation for a fluid–fluid interface. Assuming that Green’s formula remains valid at the contact line (despite the singularity), two equations are obtained at this line. The first one expresses that the fluid–fluid surface tension is equilibrated by the two surface stresses (and not by the volume stresses of the body) and suggests a finite displacement at this line (contrary to the infinite-displacement solution of classical elasticity, in which the surface properties are not taken into account). The second equation represents a strong modification of Young’s capillary equation. The validity of Green’s formula and the existence of a finite-displacement solution are justified with an explicit example of finite-displacement solution in the simple case of a half-space elastic solid bounded by a plane. The solution satisfies the contact line equations and its elastic energy is finite (whereas it is infinite for the classical elastic solution). The strain tensor components generally have different limits when approaching the contact line under different directions. Although Green’s formula cannot be directly applied, because the stress tensor components do not belong to the Sobolev space H1(V), it is shown that this formula remains valid. As a consequence, there is no contribution of the volume stresses at the contact line. The validity of Green’s formula plays a central role in the theory.

Field Synergy Principle Base on Divergence Equation for Heat and Mass Transfer of Porous Media

According to field synergy principle, analyzes the relationship of the effect for heat transfer with mass transfer in porous media. Using the divergence equation, express the degree of gas phase velocity vector and temperature gradient velocity in the process of heat and mass transfer, and the field synergy angleθplay an important role in heat and mass transfer either global or partial. In the result, the heat and mass transfer would be constrained by each other whileπθ>π/2, and be promoted whileπ/2>θ0, and be independency whileθ=π/2. The divergence equation of field synergy principle present the method of strengthen the heat and mass transfer in porous media.

ANALYSIS OF DIRECT SEGREGATED BOUNDARY-DOMAIN INTEGRAL EQUATIONS FOR VARIABLE-COEFFICIENT MIXED BVPs IN EXTERIOR DOMAINS

Direct segregated systems of boundary-domain integral equations are formulated for the mixed (Dirichlet–Neumann) boundary value problems for a scalar second-order divergent elliptic partial differential equation with a variable coefficient in an exterior three-dimensional domain. The boundary-domain integral equation system equivalence to the original boundary value problems and the Fredholm properties and invertibility of the corresponding boundary-domain integral operators are analyzed in weighted Sobolev spaces suitable for infinite domains. This analysis is based on the corresponding properties of the BVPs in weighted Sobolev spaces that are proved as well.

Wewnątrzregionalne zróżnicowanie rynku pracy w Polsce

The authors present a statistical analysis of the feature convergence and divergence processes describing the situation on the labor markets in voivodships. The analysis was conducted based on CSO data on the poviat registered unemployment rates and real gross wages. The paper discusses the theoretical aspects of the processes of α- and β-convergence and real divergence and presents an analysis of intra-regional differences in registered unemployment rates and real gross wages as well as o-process of convergence and divergence, and then estimates the parameters of ff-convergence and divergence equations of the analyzed features.

Potentials of divergent equations and new additional conservation laws for two-dimensional steady gas flows

An algorithm is constructed that compares a new divergent equation (the additional conservation law) to each of two divergent equations. Both the starting divergent equations themselves and their potentials participate in the additional conservation law, and both the first and second participate symmetrically. A characteristic feature of such additional conservation laws is that not only are the functions of gas-dynamic parameters and their derivatives taken along streamlines but so are the integrals, i.e., the functionals, and their derivatives participate in them. All these facts reveal the physical sense of the topical conservation laws constructed. A comparison with the asymmetric conservation laws constructed previously by the author (Doklady Physics, 2002) is performed. As an example, the relation that connects four additional laws comparable by dimensionality is constructed as an example. This is the conservation law of the momentum and its three analogs. Two laws are asymmetric (from Doklady Physics, 2002), while two others are constructed in this study.

Boundary value problems for quasi-linear elliptic second order equations in unbounded cone-like domains

Abstract We study the behaviour of weak solutions (as well as their gradients) of boundary value problems for quasi-linear elliptic divergence equations in domains extending to infinity along a cone.

A limiting case for the divergence equation

We consider the equation \(\text{ div}\,\mathbb{Y }=f\), with \(f\) a zero average function on the torus \(\mathbb{T }^d\). In their seminal paper, Bourgain and Brezis [J Am Math Soc 16(2):393–426, 2003 (electronic)] proved the existence of a solution \(\mathbb{Y }\in W^{1,d}\cap L^\infty \) for a datum \(f\in L^d\). We extend their result to the critical Sobolev spaces \(W^{s,p}\) with \((s+1)p=d\) and \(p\ge 2\). More generally, we prove a similar result in the scale of Triebel–Lizorkin spaces. We also consider the equation \(\text{ div} \,\mathbb{Y }=f\) in a bounded domain \(\varOmega \) subject to zero Dirichlet boundary condition.

Regularization Methods for the Numerical Solution of the Divergence Equation $∇· u = f$

Regularization Methods for the Numerical Solution of the Divergence Equation $∇· u = f$

Potential vorticity and inertia–gravity waves

The dynamics of balanced and wave flow components is investigated using the theoretical structure of potential vorticity (PV) and a vector potential ϕ = (ϕ, ψ, φ) under the isochoric (volume-preserving), Boussinesq, and f-plane approximations. The vector potential ϕ = (ϕ, ψ, φ) defines both the three-dimensional velocity (proportional to the curl of ϕ ) and the density anomaly (proportional to the divergence of ϕ ). The vertical potential φ is obtained through PV inversion from the three-dimensional (3D) PV field and the horizontal potential ϕ h = (ϕ, ψ). As a first example of this approach, it is shown that the dipole wave packet spontaneously generated in rotating and stably stratified dipolar flows has a clear signal in the vertical potential φ diagnosed from this kind of PV inversion. It is also shown that the wave packet, despite departing from a plane inertia–gravity wave solution, does not significantly contribute to the PV field (i.e. it has zero PV). A new interpretation of φ is introduced. It is proved that the Laplacian of φ may be dynamically interpreted, using the divergence equation, as the buoyancy force and Coriolis acceleration contributions to the difference between the amounts of 3D stretching and vorticity. It is also found that the use of zero PV inversion in high-order dynamics (inertia–gravity wave permitting dynamics) is useful to set up inertia–gravity wave fields satisfying required boundary conditions. The combined use of PV inversion and vector potential ϕ enables a simple way by which a vortical flow can be put into correspondence with a wave field which has initially some similarities with the vortical flow. Finally, it is suggested that a good candidate for a wave quantity could be the vertical component of the isopycnal vector ω a × ∇ρ, where ω a is the absolute vorticity and ρ is mass density. The pattern of this component is very consistent with that of the dipole wave packet at different vertical levels.

Convergence Analysis of a Residual Local Projection Finite Element Method for the Navier–Stokes Equations

This work presents and analyzes a new residual local projection stabilized finite element method (RELP) for the nonlinear incompressible Navier-Stokes equations. Stokes problems defined elementwise drive the construction of the residual-based terms which make the present method stable for the finite element pairs $\mathbb{P}_1/\mathbb{P}_l$, $l=0,1$. Numerical upwinding is incorporated through an extra control on the advective derivative and on the residual of the divergence equation. Well-posedness of the discrete problem as well as optimal error estimates in natural norms are proved under standard assumptions. Next, a divergence-free velocity field is provided by a simple postprocessing of the computed velocity and pressure using the lowest order Raviart-Thomas basis functions. This updated velocity is proved to converge optimally to the exact solution. Numerics assess the theoretical results and validate the RELP method.

Dual regularization and Pontryagin’s maximum principle in a problem of optimal boundary control for a parabolic equation with nondifferentiable functionals

A problem of optimal boundary control is considered for a divergent linear parabolic equation. Equality constraints of the problem are given by nondifferentiable functionals. A dual regularization algorithm stable to errors in initial data is constructed for solving the problem. Pontryagin’s maximum principle plays the key role in this algorithm.

The divergence equation in rough spaces

We aim at extending the existence theory for the equation div v = f in a bounded or exterior domain with homogeneous Dirichlet boundary conditions, to a class of solutions which need not have a trace at the boundary. Typically, the weak solutions that we shall consider will belong to some Besov space B p , q s ( Ω ) with s ∈ ( − 1 + 1 / p , 1 / p ) . After generalizing the notion of a solution for this equation, we propose an explicit construction by means of the classical Bogovskiĭ formula. This construction enables us to keep track of a “marginal” information about the trace of solutions. In particular, it ensures that the trace is zero if f is smooth enough. We expect our approach to be of interest for the study of rough solutions to systems of fluid mechanics.

Ensemble prediction for nowcasting with a convection-permitting model – II: forecast error statistics

A 24-member ensemble of 1-h high-resolution forecasts over the Southern United Kingdom is used to study shortrange forecast error statistics. The initial conditions are found from perturbations from an ensemble transform Kalman filter. Forecasts from this system are assumed to lie within the bounds of forecast error of an operational forecast system. Although noisy, this system is capable of producing physically reasonable statistics which are analysed and compared to statistics implied from a variational assimilation system. The variances for temperature errors for instance show structures that reflect convective activity. Some variables, notably potential temperature and specific humidity perturbations, have autocorrelation functions that deviate from 3-D isotropy at the convective-scale (horizontal scales less than 10 km). Other variables, notably the velocity potential for horizontal divergence perturbations, maintain 3-D isotropy at all scales. Geostrophic and hydrostatic balances are studied by examining correlations between terms in the divergence and vertical momentum equations respectively. Both balances are found to decay as the horizontal scale decreases. It is estimated that geostrophic balance becomes less important at scales smaller than 75 km, and hydrostatic balance becomes less important at scales smaller than 35 km, although more work is required to validate these findings. The implications of these results for high-resolution data assimilation are discussed.

Evidence for critical-like behavior in ultraslowing glass-forming systems

Glass transition constitutes one of main problems of condensed matter physics and material engineering that remains unsolved. The common acceptance of the Vogel-Fulcher-Tammann (VFT) equation for portraying the primary relaxation time or shear viscosity indicated a possible phase transition, hidden below the glass transition temperature (T(g)). Recently Hecksher [Nat. Phys. 4, 737 (2008)] delivered strong empirical arguments that VFT description lacks a direct experimental basis and thus theories not predicting a dynamic divergence should be focused on. We present clear evidence for a superiority of critical-like divergent equation τ(T)=τ(0)(T-T(C))(-ϕ) and T(C)<T(g) for liquid crystalline (LC) glass formers and orientationally disordered crystals (ODIC). Such dependence was already known for spin-glasslike systems and the dynamical scaling model, although the latter was hardly explored so far. The pressure-related behavior is also discussed. Results obtained support arguments for the suggested direct link between critical phenomena and vitrification [Tanaka, Nat. Mater. 9, 324 (2010)]. LCs and ODICs may be considered as simple experimental model systems for the structural glass formers group.

A pseudo-spectral algorithm and test cases for the numerical solution of the two-dimensional rotating Green–Naghdi shallow water equations

A pseudo-spectral algorithm is presented for the solution of the rotating Green–Naghdi shallow water equations in two spatial dimensions. The equations are first written in vorticity–divergence form, in order to exploit the fact that time-derivatives then appear implicitly in the divergence equation only. A nonlinear equation must then be solved at each time-step in order to determine the divergence tendency. The nonlinear equation is solved by means of a simultaneous iteration in spectral space to determine each Fourier component. The key to the rapid convergence of the iteration is the use of a good initial guess for the divergence tendency, which is obtained from polynomial extrapolation of the solution obtained at previous time-levels. The algorithm is therefore best suited to be used with a standard multi-step time-stepping scheme (e.g. leap-frog). Two test cases are presented to validate the algorithm for initial value problems on a square periodic domain. The first test is to verify cnoidal wave speeds in one-dimension against analytical results. The second test is to ensure that the Miles–Salmon potential vorticity is advected as a parcel-wise conserved tracer throughout the nonlinear evolution of a perturbed jet subject to shear instability. The algorithm is demonstrated to perform well in each test. The resulting numerical model is expected to be of use in identifying paradigmatic behavior in mesoscale flows in the atmosphere and ocean in which both vortical, nonlinear and dispersive effects are important.

Source of Gravity Waves within a Vortex-Dipole Jet Revealed by a Linear Model

Abstract This study develops a linear numerical model to address the source mechanism of the gravity waves generated within a vortex dipole simulated in a fully nonlinear nonhydrostatic mesoscale model. The background flow for this linear model is obtained from potential vorticity inversion constrained by the nonlinear balance equation. The forcing imposed in the linear model is derived from an imbalance in the large-scale flow—that is, the forcing or imbalance in the vorticity, divergence, and thermodynamic equations, respectively. The response from the sum of these imbalanced forcings obtained from the linear dynamics shows well-defined gravity wave signals, which compare reasonably well in terms of location, phase, and amplitude with the gravity waves simulated in a fully nonlinear nonhydrostatic mesoscale model. It is found that the vorticity forcing, largely due to the advection of balanced relative vorticity, is the leading contributor to the gravity waves in the exit region of the vortex-dipole jet.

A decomposition technique for John domains

We develop a method to decompose functions with mean value zero that are defined on a (possibly unbounded) John domain into a countable sum of functions with mean value zero and support in cubes or balls. This method enables us to generalize results known for simple domains to the class of John domains and domains satisfying a certain chain condition. As applications we present the solvability of the divergence equation divu = f, the negative norm theorem, Korn's inequality, Poincare's inequality and a localized version of the Feerman-Stein inequality. We present the results for weighted Lebesgue spaces and Orlicz spaces.

Hamilton’s principle and associated variational principles in polar thermopiezoelectricity

We express Hamilton’s principle for a regular region of thermopiezoelectric polar materials. First, we obtain a four-field variational principle which leads, as its Euler–Lagrange equations, to the divergence equations and the associated natural boundary conditions only. Next, we adjoin the rest of the fundamental equations into the variational principle through an involutory transformation. Thus, we formulate a differential type of unified variational principles operating on all the field variables. The unified variational principle is extended for the region with a fixed internal surface of discontinuity and for a curvilinear laminated region as well. The principles derived in invariant form are expressible in a system of particular coordinate system most appropriate to the geometry of the regions. They are indicated to recover some of earlier principles as special cases.