34 research outputs found
The discontinuous Galerkin method for fractional degenerate convection-diffusion equations
We propose and study discontinuous Galerkin methods for strongly degenerate
convection-diffusion equations perturbed by a fractional diffusion (L\'evy)
operator. We prove various stability estimates along with convergence results
toward properly defined (entropy) solutions of linear and nonlinear equations.
Finally, the qualitative behavior of solutions of such equations are
illustrated through numerical experiments
On Nonlinear Stochastic Balance Laws
We are concerned with multidimensional stochastic balance laws. We identify a
class of nonlinear balance laws for which uniform spatial bounds for
vanishing viscosity approximations can be achieved. Moreover, we establish
temporal equicontinuity in of the approximations, uniformly in the
viscosity coefficient. Using these estimates, we supply a multidimensional
existence theory of stochastic entropy solutions. In addition, we establish an
error estimate for the stochastic viscosity method, as well as an explicit
estimate for the continuous dependence of stochastic entropy solutions on the
flux and random source functions. Various further generalizations of the
results are discussed
A theory of -dissipative solvers for scalar conservation laws with discontinuous flux
We propose a general framework for the study of contractive semigroups
of solutions to conservation laws with discontinuous flux. Developing the ideas
of a number of preceding works we claim that the whole admissibility issue is
reduced to the selection of a family of "elementary solutions", which are
certain piecewise constant stationary weak solutions. We refer to such a family
as a "germ". It is well known that (CL) admits many different contractive
semigroups, some of which reflects different physical applications. We revisit
a number of the existing admissibility (or entropy) conditions and identify the
germs that underly these conditions. We devote specific attention to the
anishing viscosity" germ, which is a way to express the "-condition" of
Diehl. For any given germ, we formulate "germ-based" admissibility conditions
in the form of a trace condition on the flux discontinuity line (in the
spirit of Vol'pert) and in the form of a family of global entropy inequalities
(following Kruzhkov and Carrillo). We characterize those germs that lead to the
-contraction property for the associated admissible solutions. Our
approach offers a streamlined and unifying perspective on many of the known
entropy conditions, making it possible to recover earlier uniqueness results
under weaker conditions than before, and to provide new results for other less
studied problems. Several strategies for proving the existence of admissible
solutions are discussed, and existence results are given for fluxes satisfying
some additional conditions. These are based on convergence results either for
the vanishing viscosity method (with standard viscosity or with specific
viscosities "adapted" to the choice of a germ), or for specific germ-adapted
finite volume schemes
Renormalized solutions of semilinear equations involving measure data and operator corresponding to Dirichlet form
Does the influence of peers and parents on adolescents' drunkenness differ between Roma and non-Roma adolescents in Slovakia?
An Engquist–Osher-Type Scheme for Conservation Laws with Discontinuous Flux Adapted to Flux Connections
A singular limit problem for conservation laws related to the Camassa-Holm shallow water equation
We consider a shallow water equation
of Camassa-Holm type, which contains nonlinear dispersive effects
as well as fourth order dissipative effects.
We prove that as the diffusion and dispersion
parameters tend to zero, with a condition on
the relative balance between these two parameters,
smooth solutions of the shallow water equation
converge to discontinuous weak solutions
of a scalar conservation law. The
proof relies on deriving
suitable a priori estimates together
with an application of the compensated
compactness method in the setting
Bounded solutions for the Degasperis-Procesi equation
This paper deals with the well-posedness in of the Cauchy problem for the Degasperis-Procesi equation.
This is a third order nonlinear dispersive equation in one spatial variable and describes the dynamics of shallow water waves
Wellposedness for a parabolic-elliptic system
We show existence of a unique, regular global solution of the parabolic-elliptic system
and
with initial data
. Here . Furthermore, we show that the solution is
stable with respect to variation in the initial data and the functions , etc.
Explicit stability estimates are provided.
The regularized generalized Camassa--Holm equation is a special case of the model we discuss
A hyperbolic-elliptic model of two-phase flow in porous media-existence of entropy solutions
We consider the flow of two-phases in a porous medium and propose a modified
version of the fractional flow model for incompressible, two-phase flow based on a Helmholtz
regularization of the Darcy phase velocities. We show the existence of global-in-time entropy
solutions for this model with suitable assumptions on the boundary conditions. Numerical
experiments demonstrating the approximation of the classical two-phase flow equations with
the new model are presented