We develop a level set method for the computation of multivalued solutions to quasi-linear hyperbolic partial differential equations and Hamilton-Jacobi equations in any number of space dimensions. We use the classic idea of Courant and Hilbert to define the solution of the quasi-linear hyperbolic PDEs or the gradient of the solution to the Hamilton-Jacobi equations as zero level sets of level ...

We introduce a generalization of Glimm’s random choice method, which provides us with an approximation of entropy solutions to quasilinear hyperbolic system of balance laws. The flux-function and the source term of the equations may depend on the unknown as well as on the time and space variables. The method is based on local approximate solutions of the generalized Riemann problem, which form ...

We discuss the local existence results of the solutions for the nonlocal hyperbolic problem 2 ( ) || ( ) || ( ) 0, tt t u x u t u u        , N x R  0, t  with initial conditions 0 ( ,0) ( ) u x u x  and 1 ( ,0) ( ), t u x u x  in the case where 3, 0 N    and 1 ( ( )) ( ) x g x    is a positive function lying in / ( ) ( ). N N N L R L R  When the initial energy 0 1 ( , ) E u u...

Kato's theory on the construction of strongly continuous evolution systems associated with hyperbolic equations is applied to linear equation describing an age-structured population that subject time-dependent diffusion. The system used provide conditions for well-posedness corresponding quasilinear equation.

In this paper the author considers the motion of a relativistic perfect fluid with self-interaction mediated by Nordström’s scalar theory of gravity. The evolution of the fluid is determined by a quasilinear hyperbolic system of PDEs, and a cosmological constant is introduced in order to ensure the existence of non-zero constant solutions. Accordingly, the initial value problem for a compact pe...

We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full L1 setting requiring no growth assumptions on the nonline...