High-resolution numerical simulation of 2D nonlinear wave structures in electromagnetic fluids with absorbing boundary conditions

Here we show how the full set of governing equations for the dynamics of charged-particle fluids in an electromagnetic field may be solved numerically in order to model nonlinear wave structures propagating in two dimensions. We employ a source-term adaptation and two-fluid extension of the second-order high-resolution central scheme of Balbas et al. (2004) [1]. The model employed is a 2D extension of that used by Baboolal and Bharuthram (2007) [5] in studies of 1D shocks and solitons in a two-fluid plasma under 3D electromagnetic fields. Further, we outline the use of free-flow boundary conditions to obtain stable wave structures over sufficiently long modelling times. As illustrative results, we examine the formation and evolution of shock-like and soliton structures of the magnetosonic mode.     

On the well-posedness of a linearized plasma-vacuum interface problem in ideal compressible MHD

We study the initial-boundary value problem resulting from the linearization of the plasma-vacuum interface problem in ideal compressible magnetohydrodynamics (MHD). We suppose that the plasma and the vacuum regions are unbounded domains and the plasma density does not go to zero continuously, but jumps. For the basic state upon which we perform linearization we find two cases of well-posedness of the “frozen” coefficient problem: the “gas dynamical” case and the “purely MHD” case. In the “gas dynamical” case we assume that the jump of the normal derivative of the total pressure is always negative. In the “purely MHD” case this condition can be violated but the plasma and the vacuum magnetic fields are assumed to be non-zero and non-parallel to each other everywhere on the interface. For this case we prove a basic a priori estimate in the anisotropic weighted Sobolev space for the variable coefficient problem.     

Pointwise Green's function estimates toward stability for multidimensional fourth-order viscous shock fronts

For the case of multidimensional viscous conservation laws with fourth-order smoothing only, we develop detailed pointwise estimates on the Green's function for the linear fourth-order convection equation that arises upon linearization of the conservation law about a viscous planar wave solution. As in previous analyses in the case of second-order smoothing, our estimates are sufficient to establish that spectral stability implies nonlinear stability, though the full development of this result will be considered in a companion paper.     

Solution to the initial boundary-value problem for the regularized Buckley-Leverett system

We solve the initial boundary-value problem for the regularized Buckley-Leverett system, which describes the flow of two immiscible incompressible fluids through a porous medium. This is the case of the flow of water and oil in an oil reservoir. The system is formed by a hyperbolic equation and an elliptic equation coupled by a vector field which represents the total velocity of the mixture. The regularization is done by means of a filter acting on the velocity field. We consider the critical situation in which we inject pure water into the reservoir. At this critical value for the water saturation, the spatial components of the characteristics of the hyperbolic equation vanish and this motivates the use of a new technique to prove the achievement of the boundary condition for the hyperbolic equation. We treat the case of a horizontal plane reservoir. We also prove that the time averages of the saturation component of the solution converge to one, as the time interval increases indefinitely, for almost all points of the reservoir, with a rate of convergence which depends only on the flux function.     

A finite volume scheme for a model coupling free surface and pressurised flows in pipes

A model is derived for the coupling of transient free surface and pressurized flows. The resulting system of equations is written under a conservative form with discontinuous gradient of pressure. We treat the transition point between the two types of flows as a free boundary associated to a discontinuity of the gradient of pressure. The numerical simulation is performed by making use of a Roe-like finite volume scheme that we adapted to such discontinuities in the flux. The validation is performed by comparison with experimental results.     

Lie-group theoretic method for analyzing interaction of discontinuous waves in a relaxing gas

A Lie group of transformations method is used to establish self-similar solutions to the problem of shock wave propagation through a relaxing gas and its interaction with the weak discontinuity wave. The forms of the equilibrium value of the vibrational energy and the relaxation time, varying with the density and pressure are determined for which the system admits self-similar solutions. A particular solution to the problem has been found out and used to study the effects of specific heat ratio and ambient density exponent on the flow parameters. The coefficients of amplitudes of reflected and transmitted waves after the interaction are determined.     

Stability of contact discontinuities for the nonisentropic Euler equations

We study the stability of contact discontinuities for the nonisentropic Euler equations in two or three space dimensions. A simple criterion predicting neutral stability or violent instability is given.

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Sunto Si studia la stabilità delle discontinuità di contatto per le equazioni di Eulero non isentropiche in dimensione di spazio 2 e 3. Viene presentato un criterio semplice per la stabilità neutrale e l’instabilità violenta.

     

A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners

Fourth order hinged plate type problems are usually solved via a system of two second order equations. For smooth domains such an approach can be justified. However, when the domain has a concave corner the bi-Laplace problem with Navier boundary conditions may have two different types of solutions, namely u₁ with and . We will compare these two solutions. A striking difference is that in general only the first solution, obtained by decoupling into a system, preserves positivity, that is, a positive source implies that the solution is positive. The other type of solution is more relevant in the context of the hinged plate. We will also address the higher-dimensional case. Our main analytical tools will be the weighted Sobolev spaces that originate from Kondratiev. In two dimensions we will show an alternative that uses conformal transformation. Next to rigorous proofs the results are illustrated by some numerical experiments for planar domains.     

A variational framework for nonlinear viscoelastic models in finite deformation regime

This work presents a general framework for constitutive viscoelastic models in the finite deformation regime. The approach is qualified as variational since the constitutive updates consist of a minimization problem within each load increment. The set of internal variables is strain-based and uses a multiplicative decomposition of strain in elastic and viscous components. Spectral decomposition is explored in order to accommodate, into analytically tractable expressions, a wide set of specific models. Moreover, it is shown that, through appropriate choices of the constitutive potentials, the proposed formulation is able to reproduce results obtained elsewhere in the literature. Finally, numerical examples are included to illustrate the characteristics of the present formulation.     

Solutions to the seepage face model for dual porosity flows with hysteresis

This paper is concerned with theoretical analysis of the initial-boundary value problem for the dual porosity model of Darcian flows with the Preisach hysteresis operator. Due to the presence of the so-called unilateral boundary condition, the problem comes as an evolution variational inequality. We apply the penalty method to prove the global-in-time existence of the variational solution by constructing approximate solutions satisfying suitable energy type estimates.     

Ultra-analytic effect of Cauchy problem for a class of kinetic equations

The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous nonlinear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker-Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat equation. In the second part, we study a model of spatially inhomogeneous linear Landau equation with Maxwellian molecules, and show the analytic effect of the Cauchy problem.     

Weak solutions to stochastic porous media equations

A stochastic version of the porous medium equation is studied. The corresponding Kolmogorov equation is solved in a space where is an invariant measure. Then a weak solution, that is a solution in the sense of the corresponding martingale problem, is constructed.     

On a nonlinear hyperbolic problem arising in two-phase flows in naturally fractured reservoirs

A system of two first-order quasilinear equations consisting of one nonhomogenous hyperbolic conservation law and an ordinary differential equation is investigated in two spatial dimensions. The initial boundary-value problem is solved for the system and existence, uniqueness, and stability theorems are proved. We also obtain a result on the behavior of the solution when time goes to infinity which agrees with practical experience. These results offer mathematical validation to computer models in current usage for the numerical simulation of multiphase flow in naturally fractured reservoirs.     

Global classical solutions of mixed initial-boundary value problem for quasilinear hyperbolic systems

In this paper, we consider the mixed initial-boundary value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant . Under the assumptions that the system is strictly hyperbolic and linearly degenerate or weakly linearly degenerate, the global existence and uniqueness of C¹ solutions are obtained for small initial and boundary data. We also present two applications for physical models.     

Global stability of the Armstrong-Frederick model with periodic biaxial inputs

The paper is concerned with the study of plasticity models described by differential equations with stop and play operators. We suggest sufficient conditions for the global stability of a unique periodic solution for the scalar models and for the vector models with biaxial inputs of a particular form, namely the sum of a uniaxial function and a constant term. For another class of simple biaxial inputs, we present an example of the existence of unstable periodic solutions. The paper was written during the research stay of D. Rachinskii at the Technical University Munich supported by the research fellowship from the Alexander von Humboldt Foundation. His work was partially supported by the Russian Science Support Foundation, Russian Foundation for Basic Research (Grant No. 01-01-00146, 03-01-00258), and the Grants of the President of Russia (Grant No. MD-87.2003.01, NS-1532.2003.1). The support is gratefully acknowledged.     

Existence and continuous dependence of large solutions for the magnetohydrodynamic equations

Global solutions of the nonlinear magnetohydrodynamic (MHD) equations with general large initial data are investigated. First the existence and uniqueness of global solutions are established with large initial data in H ¹. It is shown that neither shock waves nor vacuum and concentration are developed in a finite time, although there is a complex interaction between the hydrodynamic and magnetodynamic effects. Then the continuous dependence of solutions upon the initial data is proved. The equivalence between the well-posedness problems of the system in Euler and Lagrangian coordinates is also showed.     

The ignition problem for a scalar nonconvex combustion model

The ignition problem for the scalar Chapman-Jouguet combustion model without convexity is considered. Under the pointwise and global entropy conditions, we constructively obtain the existence and uniqueness of the solution and show that the unburnt state is stable (unstable) when the binding energy is small (large), which is the desired property for a combustion model. The transitions between deflagration and detonation are shown, which do not appear in the convex case.     

Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow

We establish global solutions of nonconcave hyperbolic equations with relaxation arising from traffic flow. One of the characteristic fields of the system is neither linearly degenerate nor genuinely nonlinear. Furthermore, there is no dissipative mechanism in the relaxation system. Characteristics travel no faster than traffic. The global existence and uniqueness of the solution to the Cauchy problem are established by means of a finite difference approximation. To deal with the nonconcavity, we use a modified argument of Oleinik (Amer. Math. Soc. Translations 26 (1963) 95). It is also shown that the zero relaxation limit of the solutions exists and is the unique entropy solution of the equilibrium equation.     

Global existence of shock front solutions in 1-dimensional piston problem in the relativistic Euler equations

The paper studies the 1-D piston problem of the relativistic Euler equations when the speed of the piston is a perturbation of a constant. A sequence of approximate solutions constructed by a modified Glimm scheme is proved to be convergent to the weak solution (which includes a strong leading shock) to the piston problem. In particular, we give the precise estimates on the reflection of the perturbed waves on the piston and the leading shock. The paper is supported by the National Natural Science Foundation of China (Grant 10626034) and the Special Research Fund for Selecting Excellent Young Teachers of the Universities in Shanghai.     

On the wave equation with a large rough potential

We prove an optimal dispersive L^∞ decay estimate for a three-dimensional wave equation perturbed with a large nonsmooth potential belonging to a particular Kato class. The proof is based on a spectral representation of the solution and suitable resolvent estimates for the perturbed operator.