Structure preserving schemes and kinetic models for approximating measure valued solutions of hyperbolic equations

In this thesis we consider approximate schemes and models for hyperbolic conservation laws. Systems of conservation laws are fundamental mathematical models and have received a lot of attention from the point of view of analysis, modelling and computations. They include the wave equations in elastic media and fundamental equations in fluid mechanics. We consider structure preserving schemes and kinetic models for approximating measure valued solutions of hyperbolic equations. Such solutions are of interest given their application to problems in uncertainty quantification and in statistical inference. This thesis contains new results on (i) the design of new schemes for the computation of entropy consistent approximations, with particular emphasis on the consistency of the computational algorithms to entropic measure valued solutions for HCL, (ii) the introduction of discrete and generalised kinetic models designed to directly approximate measure valued solutions by using a combination of approximate Young measures and the kinetic formulation of the conservation law and (iii) stability analysis of generalised viscus kinetic models. We obtain uniqueness within a particular class of vanishing viscosity limits of these models and of their corresponding measure valued solutions

Stochastic phase-field modeling of brittle fracture: computing multiple crack patterns and their probabilities

In variational phase-field modeling of brittle fracture, the functional to be minimized is not convex, so that the necessary stationarity conditions of the functional may admit multiple solutions. The solution obtained in an actual computation is typically one out of several local minimizers. Evidence of multiple solutions induced by small perturbations of numerical or physical parameters was occasionally recorded but not explicitly investigated in the literature. In this work, we focus on this issue and advocate a paradigm shift, away from the search for one particular solution towards the simultaneous description of all possible solutions (local minimizers), along with the probabilities of their occurrence. Inspired by recent approaches advocating measure-valued solutions (Young measures as well as their generalization to statistical solutions) and their numerical approximations in fluid mechanics, we propose the stochastic relaxation of the variational brittle fracture problem through random perturbations of the functional. We introduce the concept of stochastic solution, with the main advantage that point-to-point correlations of the crack phase fields in the underlying domain can be captured. These stochastic solutions are represented by random fields or random variables with values in the classical deterministic solution spaces. In the numerical experiments, we use a simple Monte Carlo approach to compute approximations to such stochastic solutions. The final result of the computation is not a single crack pattern, but rather several possible crack patterns and their probabilities. The stochastic solution framework using evolving random fields allows additionally the interesting possibility of conditioning the probabilities of further crack paths on intermediate crack patterns

An Integral Spectral Representation of the Massive Dirac Propagator in the Kerr Geometry in Eddington-Finkelstein-type Coordinates

We consider the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating advanced Eddington-Finkelstein-type coordinates and derive a functional analytic integral representation of the associated propagator using the spectral theorem for unbounded self-adjoint operators, Stone's formula, and quantities arising in the analysis of Chandrasekhar's separation of variables. This integral representation describes the dynamics of Dirac particles outside and across the event horizon, up to the Cauchy horizon. In the derivation, we first write the Dirac equation in Hamiltonian form and show the essential self-adjointness of the Hamiltonian. For the latter purpose, as the Dirac Hamiltonian fails to be elliptic at the event and the Cauchy horizon, we cannot use standard elliptic methods of proof. Instead, we employ a new, general method for mixed initial-boundary value problems that combines results from the theory of symmetric hyperbolic systems with near-boundary elliptic methods. In this regard and since the time evolution may not be unitary because of Dirac particles impinging on the ring singularity, we also impose a suitable Dirichlet-type boundary condition on a time-like inner hypersurface placed inside the Cauchy horizon, which has no effect on the dynamics outside the Cauchy horizon. We then compute the resolvent of the Dirac Hamiltonian via the projector onto a finite-dimensional, invariant spectral eigenspace of the angular operator and the radial Green's matrix stemming from Chandrasekhar's separation of variables. Applying Stone's formula to the spectral measure of the Hamiltonian in the spectral decomposition of the Dirac propagator, that is, by expressing the spectral measure in terms of this resolvent, we obtain an explicit integral representation of the propagator.Comment: 31 pages, 1 figure, details added, references added, minor correction

The aggregation equation with power-law kernels: ill-posedness, mass concentration and similarity solutions

We study the multidimensional aggregation equation u_t+\Div(uv)=0, $v=-\nabla K*u$ with initial data in \cP_2(\bR^d)\cap L_{p}(\bR^d). We prove that with biological relevant potential $K(x)=|x|$, the equation is ill-posed in the critical Lebesgue space L_{d/(d-1)}(\bR^d) in the sense that there exists initial data in \cP_2(\bR^d)\cap L_{d/(d-1)}(\bR^d) such that the unique measure-valued solution leaves L_{d/(d-1)}(\bR^d) immediately. We also extend this result to more general power-law kernels $K(x)=|x|^\alpha$, $0<\alpha<2$ for $p=p_s:=d/(d+\alpha-2)$, and prove a conjecture in Bertozzi, Laurent and Rosado [5] about instantaneous mass concentration for initial data in \cP_2(\bR^d)\cap L_{p}(\bR^d) with $p<p_s$. Finally, we classify all the "first kind" radially symmetric similarity solutions in dimension greater than two.Comment: typos corrected, 18 pages, to appear in Comm. Math. Phy

Relative entropy in diffusive relaxation

We establish convergence in the diffusive limit from entropy weak solutions of the equations of compressible gas dynamics with friction to the porous media equation away from vacuum. The result is based on a Lyapunov type of functional provided by a calculation of the relative entropy. The relative entropy method is also employed to establish convergence from entropic weak solutions of viscoelasticity with memory to the system of viscoelasticity of the rate-type

On the convergence of a shock capturing discontinuous Galerkin method for nonlinear hyperbolic systems of conservation laws

In this paper, we present a shock capturing discontinuous Galerkin (SC-DG) method for nonlinear systems of conservation laws in several space dimensions and analyze its stability and convergence. The scheme is realized as a space-time formulation in terms of entropy variables using an entropy stable numerical flux. While being similar to the method proposed in [14], our approach is new in that we do not use streamline diffusion (SD) stabilization. It is proved that an artificial-viscosity-based nonlinear shock capturing mechanism is sufficient to ensure both entropy stability and entropy consistency, and consequently we establish convergence to an entropy measure-valued (emv) solution. The result is valid for general systems and arbitrary order discontinuous Galerkin method.Comment: Comments: Affiliations added Comments: Numerical results added, shortened proo