On the Stability of Linear Elliptic Equations with $$L^2$$-Drifts of Negative Divergence and Singular Zero-Order Terms

We show the existence and uniqueness as well as boundedness of weak solutions to linear elliptic equations with L 2 L^2 -drifts of negative divergence and singular zero-order terms which are positive. Our main target is to show the L r L^r -contraction properties of the unique weak solutions. Indeed, using the Dirichlet form theory, we construct a sub-Markovian C 0 C_0 -resolvent of contractions and identify it to the weak solutions. Furthermore, we derive an L 1 L^1 -stability result through an extended version of the L 1 L^1 -contraction property.

The influence of non-singular terms on the precision of stress description near a sharp material inclusion tip

Problems related to the study of the properties of solutions of partial differential equations have attracted the attention of many authors in recent decades. The main qualitative properties of solutions of homogeneous linear elliptic equations of the second-order divergent type with measurable coefficients without lower-order terms are already known from the results of De Giorgi, Nash, and Moser. These results are generalized by Serrin, Ladyzhenska and Uraltseva, Aronson and Serrin, and Trudinger for wide classes of elliptic and parabolic equations with lower-order terms from the corresponding $ L^{q} $-classes. Analogous results for evolution equations with $ p \,-$Laplacian appeared much later. The first significant transition to the $ p \,-$Laplace equation with the measure $~\mu~$ in the right-hand side was achieved by Kilpelainen and Maly. They established point estimates of the solutions in terms of the nonlinear Wolff potential. These results were later extended by \linebreak Trudinger and Wang and Laboutin to nonlinear and subelliptic quasilinear equations. Irregularly elliptic and inhomogeneous parabolic equations without/or with singular lower terms have been studied for a long time. The first results in this direction were obtained by Fabes, Kenig and Separioni and Gutierrez for a weighted linear elliptic equation with weight representing $ A_{2} $ of the Mackenhaupt class. In recent decades, there has been a growing interest in parabolic and elliptic equations due to their application in modeling nonlinear physical processes occurring in heterogeneous media. Also, these equations are interesting because a general qualitative theory has not been constructed for them. Among the researchers who obtained the first significant results, we note Di Benedetto E., Bogelein V., Ivanov A. V., Duzaar F., Gianazza U., Vespri V..

A probabilistic approach to Neumann problems for elliptic PDEs with nonlinear divergence terms

Ocean surface winds observed by the Quick Scatterometer (QuikSCAT) satellite prior to the geneses of 36 tropical cyclones (TCs) in the South China Sea (SCS) are investigated in this paper. The results show that there are areas with negative mean horizontal divergence around the TC genesis locations three days prior to TC formation. The divergence term [−(f+ζ)(∂u/∂x+∂v/∂y)] in the vorticity equation is calculated based upon the QuikSCAT ocean surface wind data. The calculated mean divergence term is about 10.3 times the mean relative vorticity increase rate around the TC genesis position one day prior to TC genesis, which shows the important contributions of the divergence term to the vorticity increase prior to TC formation. It is suggested that criteria related with the divergence and divergence term be applied in early detections of tropical cyclogenesis using the QuikSCAT satellite data.

Medieval logic can often ‘seem to consist of a variety of unsystematic and disparate remarks, and it is not at all obvious whether or how they fit together’ (p. 1). In this ambitious book, Terence Parsons seeks to demonstrate how ‘medieval logic can […] be seen as a group of theories and practices clustered around a core theory which is a paradigm of logic; this theory consists of a number of widely known principles, all of which can be derived from a very simple core of rules and axioms’ (p. 1). Starting from the beginning—that is, Aristotle—Parsons takes the reader from the semantics of the simplest categorical (subject-predicate) statements through a semi-formal notation called ‘Linguish’ (a mix of Latin and English, plus some symbols), to the modes of personal supposition, and eventually to complex statements involving tenses, relatives, anaphora, and other phenomena; along the way, comparisons with contemporary logic and lingusitic theory are regularly made.
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\nBecause the Organon provided the foundation for developments in the Middle Ages, Parsons begins with Aristotelian logic in ch. 1, presenting not so much Aristotle's views per se but rather what the theory of categorical sentences and syllogisms looked like to medieval logicians (p. 6). This approach focuses on the structure of categorical sentences and arguments, and mostly glosses over questions about what makes these sentences true (though see pp. 9 and 10). In ch. 2, the basic building blocks are extended to categorical sentences which have quantified predicates, predicates which are singular terms, and negative terms, all of which require special analyses, and were explicitly dealt with by medieval authors.
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\nThe next two chapters are primarily modern in orientation, introducing a new notation, called ‘Linguish’ for representing explicitly the sentence types (‘logical forms’) and proof rules discussed in chs 1 …

Symmetry classification and joint invariants for the scalar linear (1 + 1) elliptic equation

In this paper, we give a probabilistic interpretation for solutions to the Neumann boundary problems for a class of semi-linear parabolic partial differential equations (PDEs for short) with singular non-linear divergence terms. This probabilistic approach leads to the study on a new class of backward stochastic differential equations (BSDEs for short). A connection between this class of BSDEs and semi-linear PDEs is established.

Reconstruction of a 2D stress field around the tip of a sharp material inclusion

The work objective is to develop a collocation method for numerical solving nonlinear boundary value problems of mathematical physics. A feature of the being testing numerical method, the direct collocation method, is the irregular arrangement of the collocation nodes in the solution domain. That can drastically increase the accuracy of the numerical solution by improving the quality of the linear algebraic equations system that the boundary value problem leads to. Using various systems of basis functions, we present a numerical solution in the form of a polynomial, trigonometric series, and a series of local basis functions. The proposed method allows us to obtain an approximate solution of boundary value problems for a wide range of elliptic, parabolic, wave linear and nonlinear equations in an analytical form. To confirm the effectiveness of the studied numerical methods, two-dimensional and three-dimensional boundary value problems were solved for linear and nonlinear elliptic and parabolic equations with known solutions. The dependences of the numerical solution error on the number of linear equations in the resulting system are obtained. It is shown that even with a small number of equations in the system, the achieved solution accuracy is higher than the accuracy of alternative numerical methods. The investigated numerical method allows us to significantly expand the application area of traditional numerical methods in solving applied problems of modeling various physical fields, described by linear and nonlinear elliptic and parabolic equations. The results obtained in this paper show the high potential capabilities of the direct collocation method, which are based on the universality of the method and the high accuracy of numerical solutions. These qualities of the method indicate the prospects of its use in solving a wide range of applied problems.

Solution of Partial Differential Equations (PDEs) in some region R of the space of independent variables is a function, which has all the derivatives that appear on the equation, and satisfies the equation everywhere in the region R. Some linear and most nonlinear differential equations are virtually impossible to solve using exact solutions, so it is often possible to find numerical or approximate solutions for such type of problems. Therefore, numerical methods are used to approximate the solution of such type of partial differential equation to the exact solution of partial differential equation. The finite-volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages.

We analyse the vorticity production of lake-scale circulation in wind-induced shallow flows using a linear elliptic partial differential equation. The linear equation is derived from the vorticity form of the shallow-water equation using a linear bed friction formula. The features of the wind-induced steady-state flow are analysed in a circular basin with topography as a concave paraboloid, having a quadratic pile in the middle of the basin. In our study, the size of the pile varies by a size parameter. The vorticity production due to the gradient in the topography (and the distance of the boundary) makes the streamlines parallel to topographical contours, and beyond a critical size parameter, it results in a secondary vortex pair. We compare qualitatively and quantitatively the steady-state circulation patterns and vortex evolution of the flow fields calculated by our linear vorticity model and the full, nonlinear shallow-water equations. From these results, we hypothesize that the steady-state topographical vorticity production in lake-scale wind-induced circulations can be described by the equilibrium of the wind friction field and the bed friction field. Moreover, the latter can also be considered as a linear function of the velocity vector field, and hence the problem can be described by a linear equation.

We analyse the vorticity production of lake-scale circulation in wind-induced shallow flows using a linear elliptic partial differential equation. The linear equation is derived from the vorticity form of the shallow-water equation using a linear bed friction formula. The features of the wind-induced steady-state flow are analysed in a circular basin with topography as a concave paraboloid, having a quadratic pile in the middle of the basin. In our study, the size of the pile varies by a size parameter. The vorticity production due to the gradient in the topography (and the distance of the boundary) makes the streamlines parallel to topographical contours, and beyond a critical size parameter, it results in a secondary vortex pair. We compare qualitatively and quantitatively the steady-state circulation patterns and vortex evolution of the flow fields calculated by our linear vorticity model and the full, nonlinear shallow-water equations. From these results, we hypothesize that the steady-state topographical vorticity production in lake-scale wind-induced circulations can be described by the equilibrium of the wind friction field and the bed friction field. Moreover, the latter can also be considered as a linear function of the velocity vector field, and hence the problem can be described by a linear equation. doi:10.1017/S1446181119000051

Cotton-type invariants for a subclass of a system of two linear elliptic equations, obtainable from a complex base linear elliptic equation, are derived both by spliting of the corresponding complex Cotton invariants of the base complex equation and from the Laplace-type invariants of the system of linear hyperbolic equations equivalent to the system of linear elliptic equations via linear complex transformations of the independent variables. It is shownthat Cotton-type invariants derived from these two approaches are identical. Furthermore, Cotton-type and joint invariants for a general system of two linear elliptic equations are also obtained from the Laplace-type and joint invariants for a system of two linear hyperbolic equations equivalent to the system of linear elliptic equations by complex changes of the independent variables. Examples are presented to illustrate the results.

In this paper, we consider the existence and multiplicity of solutions to the elliptic equation with resonance. We classify the linear elliptic equation and obtain some new conditions on the existence and multiplicity for asymptotically linear elliptic equation by using critical point theory.