Weighted Sobolev descent for singular first order partial differential equations.
Well posed Cauchy problems for complex nonlinear equations must be semilinear.
Well posedness and control of semilinear wave equations with iterated logarithms
Well posedness and control of semilinear wave equations with iterated logarithms
Motivated by a classical work of Erdős we give rather precise necessary and sufficient growth conditions on the nonlinearity in a semilinear wave equation in order to have global existence for all initial data. Then we improve some former exact controllability theorems of Imanuvilov and Zuazua.
Well posedness in for a weakly hyperbolic second order equation
Well posedness under Levi conditions for a degenerate second order Cauchy problem
Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system
We introduce a new second-order central-upwind scheme for the Saint-Venant system of shallow water equations on triangular grids. We prove that the scheme both preserves “lake at rest” steady states and guarantees the positivity of the computed fluid depth. Moreover, it can be applied to models with discontinuous bottom topography and irregular channel widths. We demonstrate these features of the new scheme, as well as its high resolution and robustness in a number of numerical examples.
Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system
We introduce a new second-order central-upwind scheme for the Saint-Venant system of shallow water equations on triangular grids. We prove that the scheme both preserves “lake at rest” steady states and guarantees the positivity of the computed fluid depth. Moreover, it can be applied to models with discontinuous bottom topography and irregular channel widths. We demonstrate these features of the new scheme, as well as its high resolution and robustness in a number of numerical examples.
Well-posedness and asynchronous exponential growth of solutions of a two-phase cell division model.
Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain
We study a class of hyperbolic partial differential equations on a one dimensional spatial domain with control and observation at the boundary. Using the idea of feedback we show these systems are well-posed in the sense of Weiss and Salamon if and only if the state operator generates a C0-semigroup. Furthermore, we show that the corresponding transfer function is regular, i.e., has a limit for s going to infinity.
Well-posedness and regularity results for a dynamic von Kármán plate.
Well-posedness in the Gevrey classes of the Cauchy problem for a non-strictly hyperbolic equation with coefficients depending on time
Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients
We prove that the Cauchy problem for a class of hyperbolic equations with non-Lipschitz coefficients is well-posed in and in Gevrey spaces. Some counter examples are given showing the sharpness of these results.
Well-posedness of the Cauchy problem for hyperbolic equations with non-Lipschitz coefficients.
Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds