相关论文: Hyperbolic conservation laws and spacetimes with l…
We obtain necessary and sufficient conditions for the existence of "conservation laws" on null hypersurfaces for the wave equation on general four-dimensional Lorentzian manifolds. Examples of null hypersurfaces exhibiting such conservation…
The theory of weak solutions for nonlinear conservation laws is now well developed in the case of scalar equations [3] and for one-dimensional hyperbolic systems [1, 2]. For systems in several space dimensions, however, even the global…
Well-posedness of the initial (boundary) value problem is an essential property, both of meaningful physical models and of numerical applications. To prove well-posedness of wave-type equations their level of hyperbolicity is an essential…
Following Ben-Artzi and LeFloch, we consider nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L1-error estimate for a class of finite volume schemes allowing for the approximation of entropy…
A system of hyperbolic conservation laws $$ \partial_t u + \partial_x \partial_u Q = 0, \quad Q = u_1^3 / 3 + u_1 u_2^2, \qquad u = u(x,t) \in\mR^2, $$ as well as its viscous regularization $$ \partial_t u + \partial_x \partial_u Q = \calM…
Noether's theorem on the equivalence of symmetry and conservation laws has applications to geometric problems on symmetric spaces. We remind the reader of the theorem and give an application to a variational problem on hyperbolic surfaces.
We study the problem of constructing systems of hyperbolic conservation laws in one space dimension with prescribed eigencurves, i.e. the eigenvector fields of the Jacobian of the flux are given. We formulate this as a typically…
Nonlinear scalar conservation laws are traditionally viewed as transport equations. We take instead the viewpoint of these PDEs as continuity equations with an implicitly defined velocity field. We show that a weak solution is the entropy…
Aim of these notes is provide a brief review of the current well-posedness theory for hyperbolic systems of conservation laws in one space dimension, also pointing out open problems and possible research directions. They supplement the…
We investigate the numerical approximation of (discontinuous) entropy solutions to nonlinear hyperbolic conservation laws posed on a Lorentzian manifold. Our main result establishes the convergence of monotone and first-order finite volume…
In this paper we introduce a new geometric flow --- the hyperbolic gradient flow for graphs in the $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$. This kind of flow is new and very natural to understand the geometry of manifolds. We…
Consider a strictly hyperbolic $n\times n$ system of conservation laws, where each characteristic field is either genuinely nonlinear or linearly degenerate. In this standard setting, it is well known that there exists a Lipschitz semigroup…
We prove logarithm laws for unipotent flows on non-compact finite-volume hyperbolic manifolds. Our method depends on the estimate of norms of certain incomplete Eisenstein series.
We provide an informal overview on the theory of transport equations with non smooth velocity fields, and on some applications of this theory to the well-posedness of hyperbolic systems of conservation laws.
In this paper, we study the intrinsic mean curvature flow on certain closed spacelike manifolds, and prove the existence of hyperbolic structures on them.
In this paper we establish the short-time existence and uniqueness theorem for hyperbolic geometric flow, and prove the nonlinear stability of hyperbolic geometric flow defined on the Euclidean space with dimension larger than 4. Wave…
We present some recent developments on shock capturing methods for nonlinear hyperbolic systems of balance laws, whose prototype is the Euler system of compressible fluid flows, and especially discuss {structure-preserving} techniques. The…
An explicit determination of all local conservation laws of kinematic type on moving domains and moving surfaces is presented for the Euler equations of inviscid compressible fluid flow on curved Riemannian manifolds in n>1 dimensions. All…
Onsager's conjecture, which relates the conservation of energy to the regularity of weak solutions of the Euler equations, was completely resolved in recent years. In this work, we pursue an analogue of Onsager's conjecture in the context…
For the hyperbolic system of quasilinear first-order partial differential equations, linearizable by hodograph transformation, the conservation laws are used to solve the Cauchy problem. The equivalence of the initial problem for…