Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the control issues for higher-order nonlinear dispersive equations on the circle
195
0
0.0
(
0
)
Publication date
2021
fields
and research's language is
English
Authors
Roberto de A. Capistrano-Filho
Ask ChatGPT about the research
No Arabic abstract
The local and global control results for a general higher-order KdV-type operator posed on the unit circle are presented. Using spectral analysis, we are able to prove local results, that is, the equation is locally controllable and exponentially stable. To extend the local results to the global one we captured the smoothing properties of the Bourgain spaces, the so-called propagation of singularities, which are proved with a new perspective. These propagation, together with the Strichartz estimates, are the key to extending the local control properties to the global one, precisely, higher-order KdV-type equations are globally controllable and exponentially stabilizable in the Sobolev space $H^{s}(mathbb{T})$ for any $s geq 0$. Our results recover previous results in the literature for the KdV and Kawahara equations and extend, for a general higher-order operator of KdV-type, the Strichartz estimates as well as the propagation results, which are the main novelties of this work.
rate research
Read More
We develop the notion of higher Cheeger constants for a measurable set $Omega subset mathbb{R}^N$. By the $k$-th Cheeger constant we mean the value [h_k(Omega) = inf max {h_1(E_1), dots, h_1(E_k)},] where the infimum is taken over all $k$-tuples of mutually disjoint subsets of $Omega$, and $h_1(E_i)$ is the classical Cheeger constant of $E_i$. We prove the existence of minimizers satisfying additional adjustment conditions and study their properties. A relation between $h_k(Omega)$ and spectral minimal $k$-partitions of $Omega$ associated with the first eigenvalues of the $p$-Laplacian under homogeneous Dirichlet boundary conditions is stated. The results are applied to determine the second Cheeger constant of some planar domains.
In this article we study ergodic problems in the whole space $mathbb{R}^N$ for a weakly coupled systems of viscous Hamilton-Jacobi equations with coercive right-hand sides. The Hamiltonians are assumed to have a fairly general structure and the switching rates need not be constant. We prove the existence of a critical value $lambda^*$ such that the ergodic eigenvalue problem has a solution for every $lambdaleqlambda^*$ and no solution for $lambda>lambda^*$. Moreover, the existence and uniqueness of non-negative solutions corresponding to the value $lambda^*$ are also established. We also exhibit the implication of these results to the ergodic optimal control problems of controlled switching diffusions.
For initial data in Sobolev spaces $H^s(mathbb T)$, $frac 12 < s leqslant 1$, the solution to the Cauchy problem for the Benjamin-Ono equation on the circle is shown to grow at most polynomially in time at a rate $(1+t)^{3(s-frac 12) + epsilon}$, $0<epsilon ll 1$. Key to establishing this result is the discovery of a nonlinear smoothing effect for the Benjamin-Ono equation, according to which the solution to the equation satisfied by a certain gauge transform, which is widely used in the well-posedness theory of the Cauchy problem, becomes smoother once its free solution is removed.
The well-known Greens function method has been recently generalized to nonlinear second order differential equations. In this paper we study possibilities of exact Greens function solutions of nonlinear differential equations of higher order. We show that, if the nonlinear term satisfies a generalized homogeneity property, then the nonlinear Greens function can be represented in terms of the homogeneous solution. Specific examples and a numerical error analysis support the advantage of the method. We show how, for the Bousinesq and Kortweg-de Vries equations, we are forced to introduce higher order Green functions to obtain the solution to the inhomogeneous equation. The method proves to work also in this case supporting our generalization that yields a closed form solution to a large class of nonlinear differential equations, providing also a formula easily amenable to numerical evaluation.
Corrector estimates constitute a key ingredient in the derivation of optimal convergence rates via two-scale expansion techniques in homogenization theory of random uniformly elliptic equations. The present work follows up - in terms of corrector estimates - on the recent work of Fischer and Neukamm (arXiv:1908.02273) which provides a quantitative stochastic homogenization theory of nonlinear uniformly elliptic equations under a spectral gap assumption. We establish optimal-order estimates (with respect to the scaling in the ratio between the microscopic and the macroscopic scale) for higher-order linearized correctors. A rather straightforward consequence of the corrector estimates is the higher-order regularity of the associated homogenized monotone operator.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
