arXiv subjects

Amru Hussein

Publications and source records attributed to Amru Hussein.

24 records · Page 2Linked to original sources

arXiv2016

Global Strong $L^p$ Well-Posedness of the 3D Primitive Equations with Heat and Salinity Diffusion

Consider the full primitive equations, i.e. the three dimensional primitive equations coupled to the equation for temperature and salinity, and subject to outer forces. It is shown that this set of equations is globally strongly well-posed for arbitrary large initial data lying in certain interpolation spaces, which are explicitly characterized as subspaces of $H^{2/p,p}$, $1<p<\infty$, satisfying certain boundary conditions. In particular, global well-posedeness of the full primitive equations is obtained for initial data having less differentiability properties than $H^1$, hereby generalizing by result by Cao and Titi (Ann. of Math. (2) 166 (2007), no. 1, 245-267) to the case of non-smooth data. In addition, it is shown that the solutions are exponentially decaying provided the outer forces possess this property.

math.AP↗

arXiv2013

Non-self-adjoint graphs

On finite metric graphs we consider Laplace operators, subject to various classes of non-self-adjoint boundary conditions imposed at graph vertices. We investigate spectral properties, existence of a Riesz basis of projectors and similarity transforms to self-adjoint Laplacians. Among other things, we describe a simple way how to relate the similarity transforms between Laplacians on certain graphs with elementary similarity transforms between matrices defining the boundary conditions.

math.SP↗

arXiv2013

Quantum graphs with mixed dynamics: the transport/diffusion case

We introduce a class of partial differential equations on metric graphs associated with mixed evolution: on some edges we consider diffusion processes, on other ones transport phenomena. This yields a system of equations with possibly nonlocal couplings at the boundary. We provide sufficient conditions for these to be governed by a contractive semigroup on a Hilbert space naturally associated with the system. We show that our setting is also adequate to discuss specific systems of diffusion equations with boundary delays.

math.AP↗

arXiv2013

Bounds on the negative eigenvalues of Laplacians on finite metric graphs

For a self--adjoint Laplace operator on a finite, not necessarily compact, metric graph lower and upper bounds on each of the negative eigenvalues are derived. For compact finite metric graphs Poincaré type inequalities are given.

math.SP↗

arXiv2013

Maximal-quasi-accretive Laplacians on finite metric graphs

For a finite not necessarily compact metric graph, one considers the differential expression $-\frac{d^2}{d x^2}$ on each edge. The boundary conditions at the vertices of the graph yielding quasi-m-accretive as well as m-accretive operators are completely characterized.

math.AP↗

arXiv2013

Sign-indefinite second order differential operators on finite metric graphs

The question of self-adjoint realizations of sign-indefinite second order differential operators is discussed in terms of a model problem. Operators of the type $-\frac{d}{dx} \sgn (x) \frac{d}{dx}$ are generalized to finite, not necessarily compact, metric graphs. All self-adjoint realizations are parametrized using methods from extension theory. The spectral and scattering theory of the self-adjoint realizations are studied in detail.

math-ph↗