Mixed problems for degenerate abstract parabolic equations and applications

Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed $L_{p}$ spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering

Nonlinear dynamical analysis for globally modified incompressible non-Newtonian fluids

We present the global modification of the Ladyzhenskaya equations, for incompressible non-Newtonian fluids. This modification is through a cut-off function that multiplies the convective term of the equation and an additional artificial smoothing dissipation term as part of the viscous term of the equation. The goal of this work is the comparative analysis between the modified system and the non-modified system. Therefore, we show the existence and regularity of weak solutions, the existence of global attractors, the estimation of the fractal dimension of the global attractors, and finally, the relationship of the autonomous dynamics between the modified system and the non-modified system

Construction of Radial and Non-radial Solutions for Local and Non-local Equations of Liouville Type

This paper deals with radial and non-radial solutions for local and nonlocal Liouville type equations. At first non-degenerate and degenerate mean field equations are studied and radially symmetric solutions to the Dirichlet problem for them are written into explicit form. Non-radial solution is constructed in the case of Blaschke type nonlinearity. The Cauchy boundary value problem for nonlinear Laplace equation with several exponential nonlinearities is considered and C^2 smooth monotonically decreasing radial solution u ( r ) is found. Moreover, u ( r ) has logarithmic growth at ∞. Our results are applied to the differential geometry, more precisely, minimal non-superconformal degenerate two dimensional surfaces are constructed in R^4 and their Gaussian, respectively normal curvatures are written into explicit form. At the end of the paper several examples of local Liouville type PDE with radial coefficients which do not have radial solutions are given

Scaling and Balancing for High-Performance Computation of Optimal Controls

The article of record may be found at ttps://doi.org/10.2514/1.G003382It is well known that proper scaling can increase the efficiency of computational problems. In this paper, we define and show that a balancing technique can substantially improve the computational efficiency of optimal-control algorithms. We also show that noncanonical scaling and balancing procedures may be used quite effectively to reduce the computational difficulty of some hard problems. These results have been used successfully for several flight and field operations at NASA and the U.S. Department of Defense. A surprising aspect of our analysis shows that it may be inadvisable to use autoscaling procedures employed in some software packages. The new results are agnostic to the specifics of the computational method; hence, they can be used to enhance the utility of any existing algorithm or software

Exact Characterization of the Convex Hulls of Reachable Sets

We study the convex hulls of reachable sets of nonlinear systems with bounded disturbances. Reachable sets play a critical role in control, but remain notoriously challenging to compute, and existing over-approximation tools tend to be conservative or computationally expensive. In this work, we exactly characterize the convex hulls of reachable sets as the convex hulls of solutions of an ordinary differential equation from all possible initial values of the disturbances. This finite-dimensional characterization unlocks a tight estimation algorithm to over-approximate reachable sets that is significantly faster and more accurate than existing methods. We present applications to neural feedback loop analysis and robust model predictive control