Computational study of the Willmore flow on graphs
Connections between spatial decay of initial data and time asymptotics in a supercritical parabolic equation
Consistent stable difference schemes for nonlinear Black-Scholes equations modelling option pricing with transaction costs
This paper deals with the numerical solution of nonlinear Black-Scholes equation modeling European vanilla call option pricing under transaction costs. Using an explicit finite difference scheme consistent with the partial differential equation valuation problem, a sufficient condition for the stability of the solution is given in terms of the stepsize discretization variables and the parameter measuring the transaction costs. This stability condition is linked to some properties of the numerical...
Construction of entire solutions for semilinear parabolic equations.
Continuity of solutions of the porous media equation
Continuous dependence estimates for the ergodic problem of Bellman-Isaacs operators via the parabolic Cauchy problem
This paper concerns continuous dependence estimates for Hamilton-Jacobi-Bellman-Isaacs operators. We establish such an estimate for the parabolic Cauchy problem in the whole space [0, +∞) × ℝn and, under some periodicity and either ellipticity or controllability assumptions, we deduce a similar estimate for the ergodic constant associated to the operator. An interesting byproduct of the latter result will be the local uniform convergence for some classes of singular perturbation problems.
Continuous dependence for BV-entropy solutions to strongly degenerate parabolic equations with variable coefficients
We consider the Cauchy problem for degenerate parabolic equations with variable coefficients. The equation has nonlinear convective term and degenerate diffusion term which depends on the spatial and time variables. In this paper, we prove the continuous dependence for entropy solutions in the space BV to the problem not only initial function but also all coefficients.
Continuous dependence for the Brinkman equations of flow in double-diffusive convection.
Control and estimation of the boundary heat transfer function in Stefan problems
Control of the flux substrate entering an enzymatic membrane.
Convergence analysis of finite difference schemes for semi-linear initial-value problems
Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation
We consider a degenerate parabolic system which models the evolution of nematic liquid crystal with variable degree of orientation. The system is a slight modification to that proposed in [Calderer et al., SIAM J. Math. Anal.33 (2002) 1033–1047], which is a special case of Ericksen's general continuum model in [Ericksen, Arch. Ration. Mech. Anal.113 (1991) 97–120]. We prove the global existence of weak solutions by passing to the limit in a regularized system. Moreover, we propose a practical...
Convergence of a lattice numerical method for a boundary-value problem with free boundary and nonlinear Neumann boundary conditions.
Convergence of a variational lagrangian scheme for a nonlinear drift diffusion equation
We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusion equation on an interval. The discretization is based on the equation’s gradient flow structure with respect to the Wasserstein distance. The scheme inherits various properties from the continuous flow, like entropy monotonicity, mass preservation, metric contraction and minimum/ maximum principles. As the main result, we give a proof of convergence in the limit of vanishing mesh size under a CFL-type condition. We...
Convergence of the 'relativistic' heat equation to the heat equation as c → ∞.
We prove that the entropy solutions of the so-called relativistic heat equation converge to solutions of the heat equation as the speed of light c tends to ∞ for any initial condition u0 ≥ 0 in L1(RN) ∩ L∞(RN).
Convergent semidiscretization of a nonlinear fourth order parabolic system
A semidiscretization in time of a fourth order nonlinear parabolic system in several space dimensions arising in quantum semiconductor modelling is studied. The system is numerically treated by introducing an additional nonlinear potential. Exploiting the stability of the discretization, convergence is shown in the multi-dimensional case. Under some assumptions on the regularity of the solution, the rate of convergence proves to be optimal.
Convergent semidiscretization of a nonlinear fourth order parabolic system
A semidiscretization in time of a fourth order nonlinear parabolic system in several space dimensions arising in quantum semiconductor modelling is studied. The system is numerically treated by introducing an additional nonlinear potential. Exploiting the stability of the discretization, convergence is shown in the multi-dimensional case. Under some assumptions on the regularity of the solution, the rate of convergence proves to be optimal.
Corner defects in almost planar interface propagation
Counterexample to the regularity of weak solution of the quasilinear parabolic system
Counterexamples for Parabolic Differential Equations.