Second-order Quasilinear Equations Research Articles

Two Dimensional Subsonic and Subsonic-Sonic Spiral Flows Outside a Porous Body

In this paper, we investigate two dimensional subsonic and subsonic-sonic spiral flows outside a porous body. The existence and uniqueness of the subsonic spiral flow are obtained via variational formulation, which tends to a given radially symmetric subsonic spiral flow at far field. The optimal decay rate at far field is also derived by Kelvin’s transformation and some elliptic estimates. By extracting spiral subsonic solutions as the approximate sequences, we obtain the spiral subsonic-sonic limit solution by utilizing the compensated compactness. The main ingredients of our analysis are methods of calculus of variations, the theory of second-order quasilinear equations and the compensated compactness framework.

Compact Difference Schemes on a Three-Point Stencil for Second-Order Hyperbolic Equations

We consider compact difference schemes of approximation order 4+2 on a three-point spatial stencil for theKlein–Gordon equations with constant and variable coefficients. New compact schemes areproposed for one type of second-order quasilinear hyperbolic equations. In the case of constantcoefficients, we prove the strong stability of the difference solution under small perturbations ofthe initial conditions, the right-hand side, and the coefficients of the equation. A priori estimatesare obtained for the stability and convergence of the difference solution in strong mesh norms.

Blow up dynamics for the hyperbolic vanishing mean curvature flow of surfaces asymptotic to the Simons cone

In this article, we establish the existence of a family of hypersurfaces (\Gamma(t))_{0< t \leq T} that evolve by the vanishing mean curvature flow in Minkowski space and that, as t tends to 0 , blow up towards a hypersurface that behaves like the Simons cone both near the origin and at infinity. This issue amounts to singularity formation for a second-order quasilinear wave equation. Our constructive approach consists in proving the existence of finite-time blow up solutions of this hyperbolic equation of the form u(t,x) \sim t^ {\nu+1} Q( {x} / {t^ {\nu+1}}) , where Q is a stationary solution and \nu an arbitrarily large positive irrational number. Our approach roughly follows that of Krieger, Schlag and Tataru [22–24]. However, in contrast to these works, the equation to be handled in this article is quasilinear. This brings about a number of difficulties to overcome.

Boundary-value problems for singular p– and p(x)– Laplacian equations in a domain with conical point on the boundary

This paper is a survey of our last results about solutions to the Dirichlet and Robin boundary problems, the Robin transmission problem for an elliptic quasilinear second-order equation with the constant p– and variable p(x)-Laplacians, as well as to the degenerate oblique derivative problem for elliptic linear and quasilinear second-order equations in a conical bounded n-dimensional domain.

Boundary value problems for singular p- and p(x)- Laplacian equations in a domain with conical point on the boundary

This paper is a survey of our last results about solutions to the Dirichlet and Robin boundary problems, the Robin transmission problem for an elliptic quasilinear second-order equation with the constant p- and variable p(x)-Laplacians, as well as to the degenerate oblique derivative problem for elliptic linear and quasilinear second-order equations in a conical bounded n-dimensional domain.

An analytical mechanics approach to the first law of thermodynamics and construction of a variational hierarchy

A simple procedure is presented for the study of the conservation of energy equation with dissipation in continuum mechanics in 1D. This procedure is used to transform this nonlinear evolution-diffusion equation into a hyperbolic PDE; specifically, a second-order quasi-linear wave equation. An immediate implication of this procedure is the formation of a least action principle for the balance of energy with dissipation. The corresponding action functional enables us to establish a complete analytic mechanics for thermomechanical systems: a Lagrangian?Hamiltonian theory, integrals of motion, bracket formalism, and Noether?s theorem. Furthermore, we apply our procedure iteratively and produce an infinite sequence of interlocked variational principles, a variational hierarchy, where at each level or iteration the full implication of the least action principle can be shown again.

Blow-up dynamics for the hyperbolic vanishing mean curvature flow of surfaces asymptotic to a Simons cone

In this paper, we establish the existence of a family of surfaces ( Γ ( t ) ) 0 < t ≤ T that evolve by the vanishing mean curvature flow in Minkowski space and, as t tends to 0, blow up towards a surface that behaves like the Simons cone at infinity. This issue amounts to investigate the singularity formation for a second-order quasilinear wave equation. Our constructive approach consists in proving the existence of a finite-time blow-up solution to this hyperbolic equation under the form u ( t , x ) ∼ t ν + 1 Q ( x t ν + 1 ) , where Q is a stationary solution and ν is an irrational number strictly larger than 1/2. Our strategy roughly follows that of Krieger, Schlag and Tataru in [7] , [8] , [9] . However, contrary to these articles, the equation to be handled in this work is quasilinear. This induces a number of difficulties to face. Dans cette note, on étabit l'existence d'une famille de surfaces ( Γ ( t ) ) 0 < t ≤ T qui évoluent sous le flot de courbure moyenne nulle dans l'espace de Minkowski et qui explosent lorsque t tend vers 0 vers une surface asymptotique au cône de Simons à l'infini. Ce problème revient à étudier la formation de singularités pour une équation d'ondes quasi-linéaire du second ordre. Notre approche constructive consiste à démontrer l'existence de solutions à cette équation hyperbolique explosant en temps fini sous la forme u ( t , x ) ∼ t ν + 1 Q ( x t ν + 1 ) , où Q est une solution stationnaire et ν > 1 / 2 est un nombre irrationnel. Notre démarche s'inspire de celle de Krieger, Schlag et Tataru dans [7] , [8] , [9] . Cependant contrairement à ces travaux, l'équation en question dans cette note est quasi-linéaire, ce qui génère des difficultés que l'on doit surmonter.

Global large smooth solutions for 3-D Hall-magnetohydrodynamics

In this paper, the global smooth solution of Cauchy's problem of incompressible, resistive, viscous Hall-magnetohydrodynamics (Hall-MHD) is studied. By exploring the nonlinear structure of Hall-MHD equations, a class of large initial data is constructed, which can be arbitrarily large in $H^3(\mathbb{R}^3)$. Our result may also be considered as the extension of work of Lei-Lin-Zhou from the second-order semi-linear equations to the second-order quasilinear equations, because the Hall term elevates the Hall-MHD system to the quasilinear level.

Oscillation of Generalized Second-Order Quasi Linear Difference Equations

Authorsipresent sufficienticonditions for theioscillation of the generalizediperturbed quasilinearidifferenceiequation where , and . Examplesiillustrates the importanceiof our results are alsoiincluded.

On the Solvability of a Boundary-Value Problem for a Second-Order Singular Quasilinear Equation

We obtain solvability conditions for a two-point boundary-value problem for a second-order quasilinear equation. The equation is singular with respect to the independent variable. The result is based on the properties of the Green operator of the corresponding linear problem. In particular, we prove its boundedness and obtain an upper estimate of its norm. Conditions of existence of a solution of the original problem are obtained from the solvability condition of an auxiliary operator equation.

Analytical Modeling of an 15N Isotope Separation Column by Second-Order Quasi-Linear Equations with Two Independent Variable

Analytical Modeling of an 15N Isotope Separation Column by Second-Order Quasi-Linear Equations with Two Independent Variable

The Robin problem for singular p-Laplacian equation in a cone

ABSTRACTWe study the behaviour near the boundary angular or conical point of weak solutions to the Robin problem for elliptic quasi-linear second-order equation with the p-Laplacian.

Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function

For a system that is governed by the isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a second-order quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, we consider non-stationary solutions locally around a stationary state on a finite time interval and discuss the well-posedness of this kind of problem. We introduce a strict $H^2$-Lyapunov function and show that the boundary feedback constant can be chosen such that the $H^2$-Lyapunov function and hence also the $H^2$-norm of the difference between the non-stationary and the stationary state decays exponentially with time.

The Dirichlet problem in a cone for second order elliptic quasi-linear equation with the p-Laplacian

We study the behavior near the boundary conical point of weak solutions to the Dirichlet problem for elliptic quasi-linear second-order equation with the p -Laplacian and the strong nonlinearity on the right side.

Singular reduction modules of differential equations

The notion of singular reduction modules, i.e., of singular modules of nonclassical (conditional) symmetry, of differential equations is introduced. It is shown that the derivation of nonclassical symmetries for differential equations can be improved by an in-depth prior study of the associated singular modules of vector fields. The form of differential functions and differential equations possessing parameterized families of singular modules is described up to point transformations. Singular cases of finding reduction modules are related to lowering the order of the corresponding reduced equations. As examples, singular reduction modules of evolution equations and second-order quasi-linear equations are studied. Reductions of differential equations to algebraic equations and to first-order ordinary differential equations are considered in detail within the framework proposed and are related to previous no-go results on nonclassical symmetries.

On a Variant of a Nonlocal Problem for a Quasilinear Equation with Rectilinear Characteristics

A nonlocal problem of Goursat type for a second-order quasilinear equation with rectilinear characteristics is investigated.

The Nonlinear Cauchy Problem with Solutions Defined in Domains with Gaps

In this paper, we consider the Cauchy problem on the closed data support for a secondorder quasilinear equation with an admissible parabolic degeneration. Cases with solutions defined in domains with gaps are examined.

On One Nonlinear Variant of the Nonlocal Characteristic Problem

A nonlocal problem of Darboux type is examined for a second-order quasilinear equation with admissible parabolic degeneration.

Multiplicity of solutions for Dirichlet boundary conditions of second-order quasilinear equations with impulsive effects

Multiplicity of solutions for Dirichlet boundary conditions of second-order quasilinear equations with impulsive effects

A fourth-order finite difference method based on spline in tension approximation for the solution of one-space dimensional second-order quasi-linear hyperbolic equations

In this paper, we propose a new three-level implicit nine-point compact finite difference formulation of order two in time and four in space directions, based on spline in tension approximation in x-direction and central finite difference approximation in t-direction for the numerical solution of one-space dimensional second-order quasi-linear hyperbolic equations with first-order space derivative term. We describe the mathematical formulation procedure in detail and also discuss how our formulation is able to handle a wave equation in polar coordinates. The proposed method, when applied to a general form of the telegrapher equation, is also shown to be unconditionally stable. Numerical examples are used to illustrate the usefulness of the proposed method. MSC: 65M06, 65M12.