Linear Potentials in Nonlinear Potential Theory

Structures in circumstellar matter reflect both fast processes and quasi-equilibrium states. A geometrical diversity of emitting circumstellar matter is observed around evolved massive stars, in particular around B[e] supergiants. We recapitulate classical analytical tools of linear and nonlinear potential theory, such as Cole–Hopf transformation and Grad–Shafranov theory, and develop them further to explain the occurrence of the circumstellar matter structures and their dynamics. We use potential theory to formulate the nonlinear hydrodynamical equations and test dilatations of the quasi-equilibrium initial conditions. We find that a wide range of flow patterns can basically be generated and the timescales can switch, based on initial conditions, and lead to eruptive processes, reinforcing that the nonlinear fluid environment includes both quasi-stationary structures and fast processes like finite-time singularities. Some constraints and imposed symmetries can lead to Keplerian orbits, while other constraints can deliver quasi-Keplerian ones. The threshold is given by a characteristic density at the stellar surface.

The classification theory of Riemann surfaces is generalized to Riemanniann-manifolds in the conformally invariant case. This leads to the study of the existence ofA-harmonic functions of typen with various properties and to an extension of the definition of the classical notions with inclusionsOG⊂OHP⊂OHB⊂OHD. In the classical case the properness of the inclusions were proved rather late, in the 50's by Ahlfors and Toki. Our main objective is to show that such inclusions are proper also in the generalized case.

Originally, harmonic morphisms were defined as continuous mappings φ:X → X' between harmonic spaces such that h'∘φ remains harmonic whenever h' is harmonic, see [1], p. 20. In general linear axiomatic potential theory, one has to replace harmonic function

In this paper, we obtain gradient continuity estimates for viscosity solutions of [Formula: see text] in terms of the scaling critical [Formula: see text]-norm of [Formula: see text], where [Formula: see text] is the normalized [Formula: see text]-Laplacian operator. Our main result corresponds to the borderline gradient continuity estimate in terms of the modified Riesz potential [Formula: see text]. Moreover, for [Formula: see text] with [Formula: see text], we also obtain [Formula: see text] estimates. This improves one of the regularity results in [A. Attouchi, M. Parviainen and E. Ruosteenoja, [Formula: see text] regularity for the normalized [Formula: see text]-Poisson problem, J. Math. Pures Appl. (9) 108(4) (2017) 553–591], where a [Formula: see text] estimate was established depending on the [Formula: see text]-norm of [Formula: see text] under the additional restriction that [Formula: see text] and [Formula: see text]. We also mention that differently from the approach in the above paper, which uses methods from divergence form theory and nonlinear potential theory, the method in this paper is more non-variational in nature, and it is based on separation of phases inspired by the ideas in [L. Wang, Compactness methods for certain degenerate elliptic equations, J. Differential Equations 107(2) (1994) 341–350]. Moreover, for [Formula: see text] continuous, our approach also gives a somewhat different proof of the [Formula: see text] regularity result.

Approximation in Sobolev spaces and nonlinear potential theory

<div class="htmlview paragraph">This paper describes a software system called FREPS (Forced REsponse Prediction System) that integrates structural dynamic, steady and unsteady aerodynamic analyses to efficiently predict the forced response dynamic stresses in axial flow turbomachinery blades due to aerodynamic and mechanical excitations. A flutter analysis capability is also incorporated into the system. The FREPS system performs aeroelastic analysis by modeling the motion of the blade in terms of its normal modes. The structural dynamic analysis is performed by a finite element code such as MSC/NASTRAN. The steady aerodynamic analysis is based on nonlinear potential theory and the unsteady aerodynamic analysis is based on a linearization of the non-uniform potential mean flow. The program description and presentation of the capabilities are reported herein. The effectiveness of the FREPS package is demonstrated on the High Pressure Oxygen Turbopump turbine of the Space Shuttle Main Engine. Both flutter and forced response analyses are performed and typical results are illustrated.</div>

Nonlinear potential theory in metric spaces

The existence problem is solved, and global pointwise estimates of solu- tions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems: or viscosity sense has been an open problem even for good data fi E Z>s(fi), s > 1. Such results are deduced from our existence criteria with the sharp exponents s = n(q~p+1) for the first equation, and s = ^ for the second one. Furthermore, a complete characterization of removable singularities is given. Our methods are based on systematic use of Wolff's potentials, dyadic models, and nonlinear trace inequalities. We make use of recent advances in potential theory and PDE due to Kilpelainen and Maly, Trudinger and Wang, and Labutin. This enables us to treat singular solutions, nonlocal operators, and distributed singularities, and develop the theory simultaneously for quasi- linear equations and equations of Monge- Ampere type.

The linearized Vlasov and Poisson equations governing electron plasma oscillations are solved numerically for the case of a Maxwellian electron gas confined by a linear potential. An iterative method of solving the wave equation is developed using a local Fourier transform to demonstrate convergence of the method. The numerical solution verifies and extends the results obtained analytically with the WKB theory by Berk, Horton, Rosenbluth, and Sudan. The reflection coefficient obtained from the numerical solution is compared with the first-order WKB reflection coefficient. The numerical calculation demonstrates that the higher-order reflection processes are smaller than the first-order process.

This paper numerically investigates the energy evolution of the focused wave and its correlation with input parameters for the wave generation. The focused wave is numerically generated based on the fully nonlinear potential wave theory which is solved by the Quasi Arbitrary Lagrangian Eulerian-Finite Element Method. The investigation is performed by dividing the generated wave energy into three categories according to their frequency intervals, i.e., the initially assigned frequency interval [f1fN], the frequencies lower (f < f1) and higher (f > fN) than the initial interval. The amount of the generated energy falling into three energy categories and the energy distribution in the initial frequency interval are analysed against amplitude parameters and frequency bands under three amplitude spectra. Four indicators are proposed to indicate the actual energy variations comparing to the initial design. It is found that for all the three frequency intervals the energies increase when the amplitude parameters increases or the given frequency shifts towards the lower frequency domain. The choice of wave amplitude spectrum plays a significant role to maintain the distribution of the energy in the design frequency range and to minimise the energies out of the design frequency range. The results from the second-order wave group theory are used to assist analyses of the mechanism of the energy evolution alongside the focused wave generation. It is found that the lower-frequency energy in the vicinity of the focusing point is dominated by the second-order difference waves, whereas the higher-frequency energy is mainly produced by the wave flap. The waves which are higher than the second-order and the complex interactions not considered by the second-order wave theory are found significantly disturbing the energy distribution in the originally assigned wave-frequency interval.

Hydroelasticity is an important problem in the field of ocean engineering. It can be noted from most of the works published as well as theories proposed earlier that this particular problem was addressed based on the time independent/ frequency domain approach. In this paper, we propose a novel numerical method to address the fluid-structure interaction problem in time domain simulations. The hybrid numerical model proposed earlier for hydro-elasticity (Sriram and Ma, 2012) as well as for breaking waves (Sriram et al 2014) has been extended to study the problem of breaking wave-elastic structure interaction. The method involves strong coupling of Fully Nonlinear Potential Flow Theory (FNPT) and Navier Stokes (NS) equation using a moving overlapping zone in space and Runge kutta 2nd order with a predictor corrector scheme in time. The fluid structure interaction is achieved by a near strongly coupled partitioned procedure. The simulation was performed using Finite Element method (FEM) in the FNPT domain, Particle based method (Improved Meshless Local Petrov Galerkin based on Rankine source, IMPLG_R) in the NS domain and FEM for the structural dynamics part. The advantage of using this approach is due to high computational efficiency. The method has been applied to study the interaction between breaking waves and elastic wall.

An initial boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann–Dirichlet type is investigated in this work. From a physical point of view, the initial boundary value problem considered here is a mathematical model of quasistationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. On the basis of a priori estimates, we prove a local existence theorem and uniqueness for a weak generalized solution of the initial boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow‐up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow‐up of a solution in a finite time in a limited area with a nonlinear Neumann–Dirichlet boundary condition are obtained.

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] The study of Strichartz s inequality is very important in the theory of nonlinear Shrodinger equations. The inequalities do not hold for all choice of constants. Many cases were settled. for instance, some cases were settled by Montegomery-Smith. In this paper we give a generalization to the usual Strichartz inequality using the Riesz potential. We then prove the inequality to hold in some cases and give counterexamples on other cases.

A summary of water entry problem of a wedge based on the fully nonlinear velocity potential theory

Many problems of science and technique lead to partial differential equations. The mathematical theories of such equations, especially of the nonlinear ones, are very intricate, such that their numerical study becomes inevitable. To classify the partial differential equations, this chapter uses various criteria, that is, considering the order of the derivatives (first order, second order, or nth order), the linearity character (linear, quasilinear, or nonlinear equations), the influence of the integration domain at a point (equations of elliptic, parabolic, or hyperbolic type), and the types of limit conditions (Dirichlet, Neumann, or mixed problems). It presents the numerical methods for the integration of partial differential equations and of systems of partial differential equations. The numerical methods are point matching method, Ritz's method, Galerkin's method, and method of the least squares. Finally, the chapter gives numerical examples and some applications.