Divergence Form Research Articles

Intense d-p Hybridization in Nb3O15 Tripolymer Induced the Largest Second Harmonic Generation Response and Birefringence in Germanates.

We herein report two asymmetric germanate crystals, KNbGe3O9 and K3Nb3Ge2O13, with different structures and optical properties derived from divergent polymerized forms of GeO4 and NbO6 groups. Remarkably, K3Nb3Ge2O13 achieved a rare combination of the strongest second harmonic generation (SHG) response of 17.5×KDP@1064nm and the largest birefringence of 0.196@546nm in germanates. It features unique [Nb3O12]∞ tubular chains constructed by circular Nb3O15 tripolymers. Theoretical calculations reveal that the d-p interactions in the Nb3O15 group are responsible for outstanding optical properties. This work emphasizes the significance of the polymerizable functional units in obtaining high-performance nonlinear optical (NLO) crystals.

Intensed‐pHybridization in Nb3O15Tripolymer Induced the Largest Second Harmonic Generation Response and Birefringence in Germanates

AbstractWe herein report two asymmetric germanate crystals, KNbGe3O9and K3Nb3Ge2O13, with different structures and optical properties derived from divergent polymerized forms of GeO4and NbO6groups. Remarkably, K3Nb3Ge2O13achieved a rare combination of the strongest second harmonic generation (SHG) response of 17.5×KDP @ 1064 nm and the largest birefringence of 0.196 @ 546 nm in germanates. It features unique [Nb3O12]∞tubular chains constructed by circular Nb3O15tripolymers. Theoretical calculations reveal that thed‐pinteractions in the Nb3O15group are responsible for outstanding optical properties. This work emphasizes the significance of the polymerizable functional units in obtaining high‐performance nonlinear optical (NLO) crystals.

The rescaling method for some quasilinear wave equations with the divergence forms of the nonlinearity

In this paper, we will study the general kind of quasilinear wave equation □u=u2+∑i,j=132bij(uuxi)xj+∑i=132ciuuxi,inR3×[0,+∞). It is a special term of the divergence forms, but this is the first attempt to this direction without any non-local term which comes from the derivative loss. We obtain that for the compactly supported smooth initial values, the solution must blow up in finite time if the initial data are nonnegative and positive somewhere no matter how small the initial data are. The originality of the paper is the choice of “rescaled” test function (2.3)(One can refer to Section 2 for details). And also we give the sharp lifespan estimate of solution for the problem. This solves a part of the famous Strauss conjecture with regard to the general kind of quasilinear wave equations with subcritical exponent p=2 in three space dimensions.

Boundedness of solutions to Dirichlet, Neumann and Robin problems for elliptic equations in Orlicz spaces

Boundary value problems for second-order elliptic equations in divergence form, whose nonlinearity is governed by a convex function of non-necessarily power type, are considered. The global boundedness of their solutions is established under boundary conditions of Dirichlet, or Neumann, or Robin type. A decisive role in the results is played by optimal forms of Orlicz-Sobolev embeddings and boundary trace embeddings, which allow for critical growths of the coefficients.

Wolff potentials and measure data vectorial problems with Orlicz growth

We study solutions to measure data elliptic systems with Uhlenbeck-type structure that involve operator of divergence form, depending continuously on the spacial variable, and exposing doubling Orlicz growth with respect to the second variable. Pointwise estimates for the solutions that we provide are expressed in terms of a nonlinear potential of generalized Wolff type. Not only we retrieve the recent sharp results proven for p-Laplace systems, but additionally our study covers the natural scope of operators with similar structure and natural class of Orlicz growth.

Upper heat kernel estimates for nonlocal operators via Aronson’s method

In his celebrated article, Aronson established Gaussian bounds for the fundamental solution to the Cauchy problem governed by a second order divergence form operator with uniformly elliptic coefficients. We extend Aronson’s proof of upper heat kernel estimates to nonlocal operators whose jumping kernel satisfies a pointwise upper bound and whose energy form is coercive. A detailed proof is given in the Euclidean space and extensions to doubling metric measure spaces are discussed.

A single–region Finite Volume framework for modeling discontinuous magnetic field distributions

In this paper we present a novel method to model discontinuous distributions of magnetic fields. The method is based on a finite volume discretization of a divergence form of the magnetostatic equations. The theoretical framework is written in terms of the vector potential, therefore, the computational implementation is capable of predicting the magnetic field distribution in permeable, permanently magnetized, and current-carrying media. The use of a single-region meshing scheme simplifies the simulation set-up due to the fact that it avoids the use of interface boundary conditions. This renders it computationally more effective than the multi-region counterpart. In order to test the performance of the present approach we execute several numerical experiments and compare the results against a multi-region discontinuous method and a single-region boundary layerless method. The experiments demonstrate that provided the boundary layer complies with certain topological rules, the method gives accurate results of the magnitude and direction of the magnetic field, both near and far from the interfaces. We show that, for orthogonal meshes, the results obtained with a single-region approach using boundary layers are comparable to the results obtained with a multi-region framework that imposes the magnetic field gradient discontinuity as a boundary condition.

Infinitely many solutions for an elliptic equation in divergent form with critical Sobolev exponent and concave-convex nonlinearity

Infinitely many solutions for an elliptic equation in divergent form with critical Sobolev exponent and concave-convex nonlinearity

Stabilization in degenerate parabolic equations in divergence form and application to chemotaxis systems

This paper presents a stabilization result for weak solutions of degenerate parabolic equations in divergence form. More precisely, the result asserts that the global-in-time weak solution converges to the average of the initial data in some topology as time goes to infinity. It is also shown that the result can be applied to a degenerate parabolic-elliptic Keller-Segel system.

Stochastic processes associated to multidimensional parabolic transmission problems in divergence form

In this note we define and study the stochastic process $X$ in link with a parabolic transmission operator $(A,D(A))$ in divergence form. The transmission operator involves a diffraction condition along a transmission boundary. To that aim we gather and clarify some results coming from the theory of Dirichlet forms as exposed in [6] and [14] for general divergence form operators. We show that $X$ is a semimartingale and that it is solution of a stochastic differential equation involving partial reflections in the co-normal directions along the transmission boundary.

Two-dimensional-one-dimensional splitting scheme for numerical solution of problems of suspended matter transport in coastal systems

This article discusses the problems of numerical solution of non-stationary convection-diffusion-reaction problems using the model problem of suspended matter transport as an example. In the difference scheme proposed by the authors, at each time layer, the original spatial-three-dimensional problem is split along horizontal directions into a chain of two-dimensional and one-dimensional problems. In order to ensure the unconditional skew-symmetry of the convective transfer operator and its energy neutrality, the convective terms are written in symmetric form (half the sum of the non-divergent and divergent forms). The approximation of the initial boundary value problem, to which the suspended matter transport model is reduced, is considered in the Hilbert space of grid functions, which in subsequent discussions will allow us to focus on the use of general results of the theory of stability (correctness) of operator-difference schemes.

Exponential convergence of hp-time-stepping in space-time discretizations of parabolic PDES

For linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and for incompatible, time-analytic forcing term in polygonal/polyhedral domains D, we prove time-analyticity of solutions. Temporal analyticity is quantified in terms of weighted, analytic function classes, for data with finite, low spatial regularity and without boundary compatibility. Leveraging this result, we prove exponential convergence of a conforming, semi-discrete hp-time-stepping approach. We combine this semi-discretization in time with first-order, so-called “h-version’’ Lagrangian Finite Elements with corner-refinements in space into a tensor-product, conforming discretization of a space-time formulation. We prove that, under appropriate corner- and corner-edge mesh-refinement of D, error vs. number of degrees of freedom in space-time behaves essentially (up to logarithmic terms), to what standard FEM provide for one elliptic boundary value problem solve in D. We focus on two-dimensional spatial domains and comment on the one- and the three-dimensional case.

Parabolic homogenization with an interface

This paper considers a family of second-order parabolic equations in divergence form with rapidly oscillating and time-dependent periodic coefficients and an interface between two periodic structures. Following a framework initiated by Blanc, Le Bris and Lions and a generalized two-scale expansion in divergence form of elliptic homogenization with an interface by Josien, we can determine the effective (or homogenized) equation with the coefficient matrix being piecewise constant and discontinuous across the interface. Moreover, we obtain the $O(\varepsilon)$ convergence rates in $L^{2(d+2)/{d}}_{x,t}$ with $\varepsilon$-smoothing method and the uniform interior Lipschitz estimates via compactness argument.

Weak solvability of the first boundary value problem for nonuniformly and strongly degenerate second-order elliptic-parabolic equations in divergent form

Weak solvability of the first boundary value problem for nonuniformly and strongly degenerate second-order elliptic-parabolic equations in divergent form

Nonlinear perturbations of BLMP and YTSF equations

Some nonlinear evolutive equations of Mathematical Physics present infinitely many solutions described in many paper by different methods. For example Körteweg De Vries equation and Kadomtsev - Petviashvili equation are completely integrable despite the presence of quasilinear terms. In the present paper we perturb these kind of equations by positive nonlinear terms having polynomial growth. Assuming that the quasilinear term in the original equation has divergence form, we may apply test function method and establish a range of exponents for the perturbation so that a non-existence result of global weak solutions holds. Concerning initial data condition, an important difference with other equations studied by similar methods (wave, Tricomi and so on) appears. Indeed, we present a class of quasilinear equations for which the sign assumption on the initial data can be omitted and non-existence results still hold. Our basic examples are the perturbation of Boiti Leon Manna Pempinelli equation and Yu Toda Sasa Fukuyama equation. Finally we suggest open problems for other equations and other kind of perturbations.

Effect of external magnetic field on helix plasma plume

<sec>Electric phenomena and magnetic phenomena are inseparable. The magnetic field affects the ionization balance and spatial distribution of the plasma. A new type of plasma discharges has been found in nitrogen gas under sub atmospheric pressure condition without external magnetic field. Because of its regular helical propagation pattern, it is called helical plasma (HP) plume. Although a great deal of research has been carried out on the key characteristics of the HP plume, the formation mechanism of it remains unclear, which affects its applications in materials and nanotechnology.</sec><sec>By applying magnetic field to HP with different chirality, the regulation behavior of the external magnetic field on HP is studied. It is found that the external magnetic field will make the HP shrink or stretch. With the magnetic field intensity increasing from 0 mT to 200 mT, the left-handed HP plume stretches under the magnetic field of S-pole. Conversely, the left-handed chiral HP plume contracts when the magnetic pole changes to N-pole. However, when the chirality of HP plume is changed to being right-handed by adjusting voltage, phenomenon opposite to the previous one is observed. Moreover, the applied magnetic field also affects the divergence of HP. With the increase of S-pole external magnetic field, the HP plume stretches until the external magnetic field reaches to 80 mT. When the magnetic field intensity is 80 mT, the HP disappears. The plasma in the quartz tube appears in the divergent form. With the magnetic field intensity increasing to 160 mT, the plasma in the form of a helix appears again. Finally, in the process of changing the intensity of the magnetic field, the clarity of the HP plume will also change.</sec><sec>The radial electric field of HP is calculated by electromagnetic wave theory, and the mechanism of the influence of external magnetic field on the behavior of HP is clarified. It is found that the magnetic field force component of Lorentz force is responsible for the external magnetic field regulating the HP behavior. This study lays a theoretical foundation for understanding the motion behavior of HP, which is conducive to the practical applications in the field of materials and nanotechnology.</sec>

Einstein's model of "the movement of small particles in a stationary liquid" revisited: finite propagation speed

The aforementioned celebrated model, though a breakthrough in Stochastic processes and a great step toward the construction of the Brownian motion, leads to a paradox: infinite propagation speed and violation of the 2nd law of thermodynamics. We adapt the model by assuming the diffusion matrix is dependent on the concentration of particles, rather than constant it was up to Einstein, and prove a finite propagation speed under the assumption of a qualified decrease of the diffusion for small concentrations. The method involves a nonlinear degenerated parabolic PDE in divergent form, a parabolic Sobolev-type inequality, and the Ladyzhenskaya-Ural'tseva iteration lemma.

G-convergence of elliptic and parabolic operators depending on vector fields

We consider sequences of elliptic and parabolic operators in divergence form and depending on a family of vector fields. We show compactness results with respect to G-convergence, or H-convergence, by means of the compensated compactness theory, in a setting in which the existence of affine functions is not always guaranteed, due to the nature of the family of vector fields.

Inverse conductivity Problem with Internal Data

This paper concerns the reconstruction of a scalar coefficient of a second-order elliptic equation in divergence form posed on a bounded domain from internal data. This theory finds applications in multi-wave imaging, greedy methods to approximate parameter-dependent elliptic problems, and image treatment with partial differential equations. We first show that the inverse problem for smooth coefficients can be rewritten as a linear transport equation. Assuming that the coefficient is known near the boundary, we study the well-posedness of associated transport equation as well as its numerical resolution using discontinuous Galerkin method. We propose a regularized transport equation that allow us to derive rigorous convergence rates of the numerical method in terms of the order of the polynomial approximation as well as the regularization parameter. We finally provide numerical examples for the inversion assuming a lower regularity of the coefficient, and using synthetic data.

Local boundedness of weak solutions to elliptic equations with $ p, q- $growth

<abstract><p>This article is dedicated to Giuseppe Mingione for his $ 50^{th} $ birthday, a leading expert in the regularity theory and in particular in the subject of this manuscript. In this paper we give conditions for the <italic>local boundedness</italic> of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form of the type considered below in (1.1), under $ p, q- $growth assumptions. The novelties with respect to the mathematical literature on this topic are the general growth conditions and the explicit dependence of the differential equation on $ u $, other than on its gradient $ Du $ and on the $ x $ variable.</p></abstract>