Related papers: Extensions for Systems of Conservation Laws
The notions of generating sets of conservation laws of systems of differential equations with respect to symmetry groups and equivalence groups are introduced and applied. This allows us to generalize essentially the procedure of finding…
A novel principle is presented which allows for the proof of bounded weak solutions to a class of physically relevant, strongly coupled parabolic systems exhibiting a formal gradient-flow structure. The main feature of these systems is that…
Hamiltonian perturbations of the simplest hyperbolic equation $u_t + a(u) u_x=0$ are studied. We argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be…
In this paper, we study diagonal hyperbolic systems in one space dimension. Based on a new gradient entropy estimate, we prove the global existence of a continuous solution, for large and non-decreasing initial data. We remark that these…
In this paper we consider generalization of procedure of construction of potential systems for systems of partial differential equations with multidimensional spaces of conservation laws. More precisely, for construction of potential…
We consider hyperbolic systems of conservation laws in one spatial dimension. For any limit of front tracking solutions $v$, and for a general weak solution $u\in L^\infty$ with no BV assumption, we prove the following H\"older-type…
We present a generalization of Kruzkov's theory to manifolds. Nonlinear hyperbolic conservation laws are posed on a differential (n+1)-manifold, called a spacetime, and the flux field is defined as a field of n-forms depending on a…
We provide rigorous and computable a-posteriori error estimates for first order finite-volume approximations of nonlinear systems of hyperbolic conservation laws in one spatial dimension. Our estimators rely on recent stability results by…
We have developed a new embedding method for solving scalar hyperbolic conservation laws on surfaces. The approach represents the interface implicitly by a signed distance function following the typical level set method and some embedding…
In this work, we modify a continuous Galerkin discretization of a scalar hyperbolic conservation law using new algebraic correction procedures. Discrete entropy conditions are used to determine the minimal amount of entropy stabilization…
Alternative finite difference Weighted Essentially Non-Oscillatory (AFD-WENO) schemes allow us to very efficiently update hyperbolic systems even in complex geometries. Recent innovations in AFD-WENO methods allow us to treat hyperbolic…
In this work we consider companion conservation laws to general systems of conservation laws. We investigate sufficient regularity for weak solutions to satisfy companion laws, assuming the fluxes to be $C^{1,\gamma}$, $0<\gamma<1$,…
We address numerical challenges in solving hyperbolic free boundary problems described by spherically symmetric conservation laws that arise in the modeling of tumor growth due to immune cell infiltrations. In this work, we normalize the…
Entropy, its production, and its change in a dynamical system can be understood from either a fully stochastic dynamic description or from a deterministic dynamics exhibiting chaotic behavior. By taking the former approach based on the…
We carry out an extensive investigation of conservation laws and potential symmetries for the class of linear (1+1)-dimensional second-order parabolic equations. The group classification of this class is revised by employing admissible…
Reduced order models of nonlinear conservation laws in fluid dynamics do not typically inherit stability properties of the full order model. We introduce projection-based hyper-reduced models of nonlinear conservation laws which are…
We seek to define statistical solutions of hyperbolic systems of conservation laws as time-parametrized probability measures on $p$-integrable functions. To do so, we prove the equivalence between probability measures on $L^p$ spaces and…
In this paper we study a system of partial differential equations which models lithium-ion batteries. The system describes the conservation of Lithium and conservation of charges in the solid and electrolyte phases, together with the…
The relations between two construction methods (called multiplier and embedding methods) for conserved currents of general systems of ordinary or partial differential equations (DEs) are investigated. Recent studies indicate that the…
Linear and nonlinear Hodge-like systems for 1-forms are studied, with an assumption equivalent to complete integrability substituted for the requirement of closure under exterior differentiation. The systems are placed in a variational…