Boundary blow-up solutions of second order quasilinear equation on infinite cylinders

On similarity solutions of a third order quasilinear equation

In this article it is proved that bounded sequences of measure-valued solutions of a non-degenerate first order quasilinear equation are precompact in the topology of strong convergence. The general case of flow functions which are merely continuous is considered.

Solution of the first order quasi-linear equation that defines the evolution of plasma turbulence: PMM vol. 40, n≗ 5, 1976, pp. 823–833

Existence of eventually positive solutions of fourth order quasilinear differential equations

This paper presents a new method for constructing entropy solutions of first order quasilinear equations of conservation type, which is illustrated in terms of the kinetic theory of gases. Regarding a quasilinear equation as a model of macroscopic conservation laws in gas dynamics, we introduce as the corresponding microscopic model an auxiliary linear equation involving a real parameter $\xi$ which plays the role of the velocity argument. Approximate solutions for the quasilinear equation are then obtained by integrating solutions of the linear equation with respect to the parameter $\xi$. All of these equations are treated in the Fréchet space $L_{{\text {loc}}}^1({R^n})$, and a convergence theorem for such approximate solutions to the entropy solutions is established with the aid of nonlinear semigroup theory.

This paper presents a new method for constructing entropy solutions of first order quasilinear equations of conservation type, which is illustrated in terms of the kinetic theory of gases. Regarding a quasilinear equation as a model of macroscopic conservation laws in gas dynamics, we introduce as the corresponding microscopic model an auxiliary linear equation involving a real parameter $\xi$ which plays the role of the velocity argument. Approximate solutions for the quasilinear equation are then obtained by integrating solutions of the linear equation with respect to the parameter $\xi$. All of these equations are treated in the Fréchet space $L_{{\text {loc}}}^1({R^n})$, and a convergence theorem for such approximate solutions to the entropy solutions is established with the aid of nonlinear semigroup theory.

A class of two- and three-level implicit methods of order two in time and four in space based on half-step discretization for two-dimensional fourth order quasi-linear parabolic equations

We implemented a two-dimensional finite-difference time-domain (FDTD) method for the calculation of the scattering by turbid slabs containing cylindrical scatterers. We present validation results of the FDTD method used for the calculation of the scattering by an infinite dielectric cylinder. In particular the error caused by numerical dispersion due to an expansion of the simulation grid is discussed. Finally, an analytical solution of the scattering by an infinite cylinder has been used to analyze the error caused by the discrete near- to far-field transformation.

First order quasilinear equations with boundary conditions in the [formula omitted] framework

We study the Dirichlet problem for a first order quasilinear equation on a smooth manifold with boundary. The existence and uniqueness of a generalized entropy solution are established. The uniqueness is proved under some additional requirement on the field of coefficients. It is shown that generally the uniqueness fails. The nonuniqueness occurs because of the presence of the characteristics not outgoing from the boundary (including closed ones). The existence is proved in a general case. Moreover, we establish that among generalized entropy solutions laying in the ball‖u‖∞≤R\|u\|_\infty \le Rthere exist unique maximal and minimal solutions. To prove our results, we use the kinetic formulation similar to the one by C. Imbert and J. Vovelle.

We solve the initial and boundary condition problem for a general first order quasilinear equation in several space variables by using a vanishing viscosity method and give a definition which chara...

A Kinetic Approximation of Entropy Solutions of First Order Quasilinear Equations

We propose a level set method for systems of PDEs which is consistent with the previous research pursued by Evans (1996) for the heat equation and by Giga and Sato (2001) for Hamilton–Jacobi equations. Our approach follows a geometric construction related to the notion of barriers introduced by De Giorgi. The main idea is to force a comparison principle between manifolds of different codimension and require each nonzero sub-level of a solution of the level set equation to be a barrier for the graph of a solution of the corresponding system. We apply the method to a class of systems of first order quasi-linear equations. We compute the level set equation associated with suitable first order systems of conservation laws, with the mean curvature flow of a manifold of arbitrary codimension and with systems of reaction–diffusion equations.

We consider the prescribed Levi curvature equation, a second order quasilinear equation whose associated operator can be represented as a sum of squares of nonlinear vector fields. For this equation we introduce a notion of derivatives modeled on the geometry of the associated operator and prove an a priori $L^2$ estimate for these second order intrinsic derivatives of a viscosity solution. We then show that viscosity solutions are strong solutions in a natural sense and satisfy the equation almost everywhere.

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