We propose and analyze a finite element method for a semi-stationary Stokes system modeling compressible fluid flow subject to a Navier-slip boundary condition. The velocity (momentum) equation is approximated by a mixed ...

We propose a nonconforming finite element method for isentropic viscous gas flow in situations where convective effects may be neglected. We approximate the continuity equation by a piecewise constant discontinuous Galerkin ...

We develop a viscosity solution theory for a system of nonlinear degenerate parabolic integro-partial differential equations (IPDEs) related to stochastic optimal switching and control problems or stochastic games. In the ...

We study the pricing of American put and call options in a market with jumps. We extend and make rigorous previous work that characterizes the price as a solution of an integro-differential equation set on the whole domain. ...

We derive error estimates for finite difference-quadrature schemes approximating viscosity solutions of nonlinear degenerate parabolic integro-PDEs with variable diffusion coefficients. The relevant equations can be viewed ...

We consider higher-order Camassa--Holm equations describing exponential curves of the manifold of smooth orientation preserving diffeomorphisms of the unit circle in the plane. We establish the existence of a strongly ...

We present a well-balanced, large time stepping method for conservation laws with source terms. The numerical method is based on a local reformulation of the balance law as a conservation law with a discontinuous flux ...

We consider general triangular systems of conservation laws which arise in applications like multi-phase flows in porous media and are non-strictly hyperbolic. We device simple and efficient finite volume schemes of the ...

We propose and analyse a finite volume scheme of the Godunov type that preserves discrete steady states. The scheme works in resonance regime as well as for problems with discontinuous flux. Moreover, no additional ...

Recent work [4] has shown that the Degasperis-Procesi equation is well-posed in the class of (discontinuous) entropy solutions. In the present paper we construct numerical schemes and prove that they converge to entropy ...

We consider a fully parabolic model for chemotaxis with volume-filling effect and a nonlinear diffusion that degenerates in a two-sided fashion. We address the questions of existence of weak solutions and of their regularity ...

We derive error estimates for certain approximate solutions of Bellman equations associated to a class of controlled jump-diffusion (Lévy) processes. These Bellman equations are fully nonlinear degenerate integro-PDEs ...

We consider non-strictly hyperbolic systems of conservation laws in triangular form, which arise in applications like three-phase flows in porous media. We device simple and efficient finite volume schemes of Godunov type ...

We suggest a finite dfference scheme for the Camassa-Holm equation that can handle general H1 initial data. The form of the difference scheme is judiciously chosen to ensure that it satisfies a total energy inequality. We ...

We prove well-posedness (existence and uniqueness) results for aÊ class of degenerate reaction-diffusion systems. A prototype system belongingÊ to this class is provided by the bidomain model, which is frequently used toÊ ...

We consider a shallow water equation of Camassa-Holm type, containing nonlinear dispersive effects as well as fourth order dissipative effects. We prove that as the diffusion and dispersion parameters tend to zero, with a ...

We derive error estimates for approximate (viscosity) solutions of Bellman equations associated to controlled jump-diffusion processes, which are fully nonlinear integro-partial differential equations. Two main results are ...

A well-studied one-dimensional model for the operation of clarifier-thickener units in engineering applications can be expressed as a conservation law with a flux that is discontinuous with respect to the spatial variable. ...

We consider a weakly dissipative hyperelastic-rod wave equation (or weakly dissipative Camassa-Holm equation) describing nonlinear dispersive dissipative waves in compressible hyperelastic rods. By fixed a smooth solution, ...

We prove existence of a renormalized soluion to a system of nonlinear partial differential equations with anisotropic diffusivities and transport effects, supplemented with initial and Dirichlet boundary conditions. The ...

We show existence of a unique, regular global solution of the parabolic-elliptic system $u_t +f(t,x,u)_x+g(t,x,u)+P_x=(a(t,x) u_x)_x$ and $-P_{xx}+P=h(t,x,u,u_x)+k(t,x,u)$ with initial data $u|_{t=0} = u_0$. Here ...

We consider doubly nonlinear anisotropic degenerate parabolic equations, supplemented with an initial condition and a homogeneous Dirichlet boundary condition. We introduce a notion of entropy solution and prove that the ...

We consider a generalized hyperelastic-rod wave equation (or generalized Camassa--Holm equation) describing nonlinear dispersive waves in compressible hyperelastic rods. We establish existence of a strongly continuous ...

We give the first convergence proof for the Lax-Friedrichs finite difference scheme for non-convex genuinely nonlinear scalar conservation laws of the form $$ u_t+f(k(x,t),u)_x=0, $$ where the coefficient $k(x,t)$ is ...

We prove existence results for distributional solutions of anisotropic nonlinear elliptic systems with a measure valued right-hand side. The functional setting involves anisotropic Sobolev spaces as well as weak Lebesgue ...

We prove a comparison principle for unbounded semicontinuous viscosity sub- and supersolutions of nonlinear degenerate parabolic integro-partial differential equations coming from applications in mathematical finance in ...

We analyze the classical Merton's portfolio optimization problem when the risky asset follows an exponential Ornstein-Uhlenbeck process, also known as the Schwartz mean-reversion dynamics. The corresponding Hamilton-Jacobi-Bellman ...

The wear of steel balls in continuously operated grinding mills, used in mineral processing to comminute metalliferous rocks, can be described by a simple population balance model. This model gives rise to a scalar transport ...

We present a general framework for deriving continuous dependence estimates for, possibly polynomially growing, viscosity solutions of fully nonlinear degenerate parabolic integro-PDEs. We use this framework to provide ...

The chief purpose of this paper is to formulate and partly analyze a new mathematical model for continuous sedimentation-consolidation processes of flocculated suspensions in clarifier-thickener units. This model appears ...

We study the well-posedness of discontinuous entropy solutions to quasilinear aniso-tropic degenerate parabolic equations with explicit $(t,x)$--dependence: $$ \pt u + \si \pxi f_i(u,t,x)=\sij \pxj\left(\aij(u,t,x)\pxi ...

We study a semi-discrete splitting method for computing approximate viscosity solutions of the initial value problem for a class of nonlinear degenerate parabolic equations with source terms. It is fairly standard to prove ...

We consider a scalar conservation law modeling the settling of particles in an ideal clarifier-thickener unit. The conservation law has a nonconvex flux which is spatially dependent on two discontinuous parameters. We ...

Under general conditions stated in Rheinländer 30], we prove that in a stochastic volatility market the Radon-Nikodym density of the minimal entropy martingale measure can be expressed in terms of the solution of a semilinear ...

We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2x2 quasilinear system without a source term, and ...

We continue the work that was initiated in [7] on a fast marching-like method for simulating two-phase incompressible immiscible flow of water and oil in a porous medium. The purpose ot the present paper is threefold: (1) ...

We formulate and prove a non-local ``maximum principle for semicontinuous functions'' in the setting of fully nonlinear degenerate elliptic integro-partial differential equations with integro operators of second order. ...

The well-known Lighthill-Whitham-Richards kinematic traffic flow model for unidirectional flow on a single-lane highway is extended to include both abruptly changing road surface conditions and drivers' reaction time and ...

We develop a general L1-framework for deriving continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the Chen-Perthame kinetic approach [9]. We apply L1-framework ...

We propose a Kruzkov-type entropy condition for nonlinear degenerate parabolic equations with discointinuous coefficients. We establish L1 stability, and thus uniqueness, for weak solutions satisfying the entropy condition, ...

We prove well posedness (existence and uniqueness) of renormalized entropy solutions to the Cauchy problem for quasilinear anisotropic degenerate parabolic equations with L1 data. This paperÊ complements the work by Chen ...

In this paper we investigate the recently introduced Malliavin approach compared to more classical approaches to find sensitivities of options in commodity and energy markets. The Malliavin approach has been developed in ...