Addendum to: “Dissipative and Entropy Solutions to Non-Isotropic Degenerate Parabolic Balance Laws”

This note generalizes the notion of dissipative solutions to non-isotropic degenerate parabolic balance laws introduced in [3]. The new definition allows us to use a larger class of test functions than the one used in [3] to study the equivalence between dissipative and entropy solutions. As a result, it is possible to study general relaxation-type approximations (limits).Arch. Rational Mech. Anal. 170 (2003) 359–370 Digital Object Identifier (DOI) 10.1007/s00205-003-0282-5 Published online September 30, 2003Acknowledgement This work was partially supported by HYKE programme HPRN-CT-2002-00282 (http://www.hyke.org), and PANAGIOTIS E. SOUGANIDIS by the National Science Foundation.     

Asymptotic Linear Stability of Solitary Water Waves

In this paper, we study the well-posedness of Cahn–Hilliard equations with degenerate phase-dependent diffusion mobility. We consider a popular form of the equations which has been used in phase field simulations of phase separation and microstructure evolution in binary systems. We define a notion of weak solutions for the nonlinear equation. The existence of such solutions is obtained by considering the limits of Cahn–Hilliard equations with non-degenerate mobilities.     

Weakly Nonlinear-Dissipative Approximations of Hyperbolic–Parabolic Systems with Entropy

Hyperbolic–parabolic systems have spatially homogenous stationary states. When the dissipation is weak, one can derive weakly nonlinear-dissipative approximations that govern perturbations of these constant states. These approximations are quadratically nonlinear. When the original system has an entropy, the approximation is formally dissipative in a natural Hilbert space. We show that when the approximation is strictly dissipative it has global weak solutions for all initial data in that Hilbert space. We also prove a weak-strong uniqueness theorem for it. In addition, we give a Kawashima type criterion for this approximation to be strictly dissipative. We apply the theory to the compressible Navier–Stokes system.     

Well-Posedness in Smooth Function Spaces for the Moving-Boundary Three-Dimensional Compressible Euler Equations in Physical Vacuum

We prove well-posedness for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) =  C _γ ρ ^γ for γ > 1. The physical vacuum singularity requires the sound speed c to go to zero as the square-root of the distance to the moving boundary, and thus creates a degenerate and characteristic hyperbolic free-boundary system wherein the density vanishes on the free-boundary, the uniform Kreiss–Lopatinskii condition is violated, and manifest derivative loss ensues. Nevertheless, we are able to establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary, and our estimates have no derivative loss with respect to initial data. Our proof is founded on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term, chosen to preserve as much of the geometric structure of the Euler equations as possible. We first construct solutions to this degenerate parabolic regularization using a higher-order version of Hardy’s inequality; we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter. Solutions to the compressible Euler equations are found in the limit as the artificial viscosity tends to zero. Our regular solutions can be viewed as degenerate viscosity solutions. Our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws.     

Existence of multidimensional traveling waves in the transport of reactive solutes through periodic porous media

We prove the existence of multidimensional traveling-wave solutions to the scalar equation for the transport of solutes (contaminants) with nonlinear adsorption and spatially periodic convection-diffusion-adsorption coefficients under the assumption that the nonlinear adsorption function satisfies the Lax and Oleinik entropy conditions. In the nondegenerate case, we also prove the uniqueness of the traveling waves. These traveling waves are analogues of viscous shock profiles. They propagate with effective speeds that depend on the periodic porous media only up to their mean states, and are given by an averaged Rankine-Hugoniot relation. This is a direct consequence of the fact that the transport equation is in conservation form. We use the sliding domain method, the continuation method, spectral theory, maximum principles, and a priori estimates. In the degenerate case, the traveling waves are weak solutions of a degenerate parabolic equation and are only Holder continuous. We obtain them by taking suitable limits on the non-degenerate traveling waves. The uniqueness of the degenerate traveling waves is open.     

Asymptotic behaviors of solutions for dissipative quantum Zakharov equations

The dissipative quantum Zakharov equations are mainly studied. The existence and uniqueness of the solutions for the dissipative quantum Zakharov equations are proved by the standard Galerkin approximation method on the basis of a priori estimate. Meanwhile, the asymptotic behavior of solutions and the global attractor which is constructed in the energy space equipped with the weak topology are also investigated.     

Evolving Graphs by Singular Weighted Curvature

A new notion of solutions is introduced to study degenerate nonlinear parabolic equations in one space dimension whose diffusion effect is so strong at particular slopes of the unknowns that the equation is no longer a partial differential equation. By extending the theory of viscosity solutions, a comparison principle is established. For periodic continuous initial data a unique global continuous solution (periodic in space) is constructed. The theory applies to motion of interfacial curves by crystalline energy or more generally by anisotropic interfacial energy with corners when the curves are the graphs of functions. Even if the driving force term (homogeneous in space) exists, the initial-value problem is solvable for general nonadmissible continuous (periodic) initial data. (Accepted July 5, 1996)     

On some diffusive waves in non-linear heat conduction

We address the non-linear heat conduction in the presence of absorption for the case of spherical symmetry geometry. The non-linear model is based on both a temperature-dependent thermal conductivity and a non-linear generalization of the Fourier law. The governing equation belongs to a class of degenerate parabolic equations. We obtain similarity solutions in closed form for the Cauchy problem corresponding to an instantaneous point source problem. We investigate the non-linear effects on the propagation of the temperature distrubances. We find that in certain cases the temperature distribution displays travelling wave characteristics. The solution for the Cauchy problem is recovered by considering a suitable first boundary value problem.     

非线性Mohr－Coulomb体边坡的极限荷载

本文根据拟线性偏微分方程组的特征线理论，求得无重双曲、抛物Ｍｏｈｒ－Ｃｏｕｌｏｍｂ体边坡极限荷载的解析解．用两参数的双曲线和抛物线可以较好拟合由实验提供的破坏曲线．由其解发现屈服准则和坡角对极限荷载影响很大，理想塑性体的极限荷载也非最小．     

On gravity and centrifugal settling of polydisperse suspensions forming compressible sediments

A mathematical model describing both the hindered settling and the consolidation of suspensions with particles of different sizes and densities forming compressible sediments is presented. The specific new element is a centrifugal configuration, which gives rise to a non-constant body force. Within a range of angular velocities, the model can be reduced to one (radial) space dimension. The result is a system of second-order strongly degenerate parabolic–hyperbolic convection–diffusion equations. For the special subcase of suspensions of rigid spheres, which do not form compressible sediments and for which the effective solid stress can be assumed to vanish, these equations form a first-order system of conservation laws. A type analysis shows that these equations include hyperbolic, hyperbolic–parabolic, and hyperbolic–elliptic systems, depending on the sizes and densities of the solid particles. A numerical high-resolution central difference scheme for the hyperbolic and hyperbolic–parabolic models is applied to solve the model numerically, and thereby to simulate centrifugation of two polydisperse suspensions.     

Preventing Blow up by Convective Terms in Dissipative PDE’s

We study the impact of the convective terms on the global solvability or finite time blow up of solutions of dissipative PDEs. We consider the model examples of 1D Burger’s type equations, convective Cahn–Hilliard equation, generalized Kuramoto–Sivashinsky equation and KdV type equations. The following common scenario is established: adding sufficiently strong (in comparison with the destabilizing nonlinearity) convective terms to equation prevents the solutions from blowing up in a finite time and makes the considered system globally well-posed and dissipative and for weak enough convective terms the finite time blow up may occur similar to the case, when the equation does not involve convective term. This kind of result has been previously known for the case of Burger’s type equations and has been strongly based on maximum principle. In contrast to this, our results are based on the weighted energy estimates which do not require the maximum principle for the considered problem.     

Existence and Uniqueness of Solutions¶of an Asymptotic Equation¶Arising from a Variational Wave Equation¶with General Data

We establish here the global existence and uniqueness of admissible (both dissipative and conservative) weak solutions to a canonical asymptotic equation () for weakly nonlinear solutions of a class of nonlinear variational wave equations with any L ²(ℝ) initial datum. We use the method of Young measures and mollification techniques. Accepted April 25, 2000?Published online November 16, 2000     

Riesz Potentials and Nonlinear Parabolic Equations

The spatial gradient of solutions to nonlinear degenerate parabolic equations can be pointwise estimated by the caloric Riesz potential of the right hand side datum, exactly as in the case of the heat equation. Heat kernels type estimates persist in the nonlinear case.     

Degenerate Conformally Invariant Fully Nonlinear Elliptic Equations

We establish Liouville theorems for , entire solutions and locally Lipschitz entire weak solutions to general degenerate conformally invariant fully nonlinear elliptic equations of second order. For applications to local gradient estimates of solutions of general conformally invariant fully nonlinear elliptic equations of second order, see [20].     

On group properties and conservation laws for second-order quasi-linear differential equations

A sufficient condition for the absence of tangent transformations admitted by second-order quasi-linear differential equations and a sufficient condition for linear autonomy of operators of the Lie group of transformations admitted by second-order weakly nonlinear differential equations are found. A theorem on the structure of the first-order conservation laws for second-order weakly nonlinear differential equations is proved. A classification of second-order linear differential equations with two independent variables in terms of first-order conservation laws is proposed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 64–70, May–June, 2009.     

Nonlinear Fluctuations of Weakly Asymmetric Interacting Particle Systems

We introduce what we call the second-order Boltzmann–Gibbs principle, which allows one to replace local functionals of a conservative, one-dimensional stochastic process by a possibly nonlinear function of the conserved quantity. This replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states. As an application of this second-order Boltzmann–Gibbs principle, we introduce the notion of energy solutions of the KPZ and stochastic Burgers equations. Under minimal assumptions, we prove that the density fluctuations of one-dimensional, stationary, weakly asymmetric, conservative particle systems are sequentially compact and that any limit point is given by energy solutions of the stochastic Burgers equation. We also show that the fluctuations of the height function associated to these models are given by energy solutions of the KPZ equation in this sense. Unfortunately, we lack a uniqueness result for these energy solutions. We conjecture these solutions to be unique, and we show some regularity results for energy solutions of the KPZ/Burgers equation, supporting this conjecture.     

Propagation of weak waves in elastic-plastic solids

Wave fronts admitting discontinuities only in the derivatives of the dependent variables are by convention called ‘weak’ waves. For the special case of discontinuous first-order derivatives, the fronts are customarily called ‘acceleration’ waves. If the governing equations are quasi-linear, then the weak waves are necessarily characteristic surfaces. Sometimes, these surfaces are also referred to as ‘singular surfaces’ of order r ? 1, where r stands for the order of the first discontinuous derivatives. This latter approach is adopted in this paper and applied to governing equations which form a set of first-order quasi-linear hyperbolic equations. When these equations are written in terms of singular surface coordinates, simplification occurs upon differencing equations written on the front and rear sides of the surface: a set of algebraic (‘connection’) equations is generated for the discontinuities in the normal derivatives of the dependent variables across the surface. When a similar operation is performed on the governing equations written for second-order derivatives, a set of first-order differential (‘transport’) equations is generated.     

Weak–Strong Uniqueness of Dissipative Measure-Valued Solutions for Polyconvex Elastodynamics

For the equations of elastodynamics with polyconvex stored energy, and some related simpler systems, we define a notion of a dissipative measure-valued solution and show that such a solution agrees with a classical solution with the same initial data, when such a classical solution exists. As an application of the method we give a short proof of strong convergence in the continuum limit of a lattice approximation of one dimensional elastodynamics in the presence of a classical solution. Also, for a system of conservation laws endowed with a positive and convex entropy, we show that dissipative measure-valued solutions attain their initial data in a strong sense after time averaging.     

Periodic solutions of linear degenerate systems of ordinary differential equations of the second order

We propose sufficient conditions for the existence of a periodic solution of a system of linear ordinary differential equations of the second order with a degenerate symmetric matrix in the coefficient of the second-order derivative in the case of an arbitrary periodic inhomogeneity.     

Application of Topological Methods to Quasilinear Parabolic Boundary-Value Problems

We apply a topological approach to the investigation of quasilinear parabolic boundary-value problems. The class of problems under investigation is reduced to an operator equation with an operator satisfying condition (S)₊. We establish theorems on solvability and give an example of the application of the approach indicated to the case of second-order parabolic equations.