Gradient bounds for nonlinear degenerate parabolic equations and application to large time behavior of systems

We obtain new oscillation and gradient bounds for the viscosity solutions of fully nonlinear degenerate elliptic equations where the Hamiltonian is a sum of a sublinear and a superlinear part in the sense of Barles and Souganidis (2001). We use these bounds to study the asymptotic behavior of weakly coupled systems of fully nonlinear parabolic equations. Our results apply to some "asymmetric systems" where some equations contain a sublinear Hamiltonian whereas the others contain a superlinear one. Moreover, we can deal with some particular case of systems containing some degenerate equations using a generalization of the strong maximum principle for systems

Lipschitz regularity results for nonlinear strictly elliptic equations and applications

Most of lipschitz regularity results for nonlinear strictly elliptic equations are obtained for a suitable growth power of the nonlinearity with respect to the gradient variable (subquadratic for instance). For equations with superquadratic growth power in gradient, one usually uses weak Bernstein-type arguments which require regularity and/or convex-type assumptions on the gradient nonlinearity. In this article, we obtain new Lipschitz regularity results for a large class of nonlinear strictly elliptic equations with possibly arbitrary growth power of the Hamiltonian with respect to the gradient variable using some ideas coming from Ishii-Lions' method. We use these bounds to solve an ergodic problem and to study the regularity and the large time behavior of the solution of the evolution equation

Large time behavior for some nonlinear degenerate parabolic equations

We study the asymptotic behavior of Lipschitz continuous solutions of nonlinear degenerate parabolic equations in the periodic setting. Our results apply to a large class of Hamilton-Jacobi-Bellman equations. Defining S as the set where the diffusion vanishes, i.e., where the equation is totally degenerate, we obtain the convergence when the equation is uniformly parabolic outside S and, on S, the Hamiltonian is either strictly convex or satisfies an assumption similar of the one introduced by Barles-Souganidis (2000) for first-order Hamilton-Jacobi equations. This latter assumption allows to deal with equations with nonconvex Hamiltonians. We can also release the uniform parabolic requirement outside S. As a consequence, we prove the convergence of some everywhere degenerate second-order equations

Some new results on Lipschitz regularization for parabolic equations

It is well known that the bounded solution u(t, x) of the heat equation posed in RN×(0,T) for any continuous initial condition becomes Lipschitz continuous as soon as t>0, even if the initial datum is not Lipschitz continuous. We investigate this Lipschitz regularization for both strictly and degenerate parabolic equations of Hamilton–Jacobi type. We give proofs avoiding Bernstein’s method which leads to new, less restrictive conditions on the Hamiltonian, i.e., the first-order term. We discuss also whether the Lipschitz constant depends on the oscillation for the initial datum or not. Finally, some important applications of this Lipschitz regularization are presented

Affections of Turbine Nozzle Cross-Sectional Area to the Marine Diesel Engine Working

After a long period of use, some important technical parameters of the main marine diesel engines (MDE) gradually become worse, such as the turbine speed, intake pressure, exhaust temperature, engine power, and specific fuel oil consumption (SFOC). This paper studies the affections of the turbine nozzle cross-sectional area (AT) to MDE and presents a method of AT adjustment to improve the performances of MDE. A mathematical model of an engine was built based on the existent engine construction and the theory of the diesel engine working cycle and the simulation was programmed by Matlab/Simulink. This simulation model accuracy was evaluated through the comparison of simulation results and experimental data of the MDE. The accuracy testing results were acceptable (within 5%). The influences of AT on the engine working parameters and the finding optimization point were conducted by using the simulation program to study. The predicted optimization point of the nozzle was used to improve the engine’s performances on board. The integration of the simulation and experiment studies showed its effectiveness in the practical application of the marine diesel engine field

Aspects of private sector development in Vietnam

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Large time behavior of weakly coupled systems of first-order Hamilton-Jacobi equations

We show a large time behavior result for class of weakly coupled systems of first-order Hamilton-Jacobi equations in the periodic setting. We use a PDE approach to extend the convergence result proved by Namah and Roquejoffre (1999) in the scalar case. Our proof is based on new comparison, existence and regularity results for systems. An interpretation of the solution of the system in terms of an optimal control problem with switching is given

Towards Designing Spatial Robots that are Architecturally Motivated

While robots are increasingly integrated into the built environment, little is known how their qualities can meaningfully influence our spaces to facilitate enjoyable and agreeable interaction, rather than robotic settings that are driven by functional goals. Motivated by the premise that future robots should be aware of architectural sensitivities, we developed a set of exploratory studies that combine methods from both architectural and interaction design. While we empirically discovered that dynamically moving spatial elements, which we coin as spatial robots, can indeed create unique life-sized affordances that encourage or resist human activities, we also encountered many unforeseen design challenges originated from how ordinary users and experts perceived spatial robots. This discussion thus could inform similar design studies in the areas of human-building architecture (HBI) or responsive and interactive architecture