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Steven I. Marcus - One of the best experts on this subject based on the ideXlab platform.

Yoshihiro Shibata - One of the best experts on this subject based on the ideXlab platform.

  • on some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface
    Discrete and Continuous Dynamical Systems, 2016
    Co-Authors: Takayuki Kubo, Yoshihiro Shibata, Kohei Soga
    Abstract:

    In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic viscous fluid flow without surface tension in the $L_p$ in time and the $L_q$ in space framework with $2< p <\infty$ and $N< q <\infty$ under the assumption that the initial domain is a uniform $W^{2-1/q}_q$ domain in $\mathbb{R}^N (N\ge 2)$. After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal $L_p$-$L_q$ regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key step of our method is to prove the existence of $\mathcal{R}$-Bounded Solution operator to resolvent problem corresponding to linearized problem. The $\mathcal{R}$-Boundedness combined with Weis's operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal $L_p$-$L_q$ regularity theorem.

  • on the mathscr Bounded Solution operator and the maximal l_p lp l_q lq egularity of the stokes equations with free boundary condition
    8th CREST-SBM nternational Conference on Mathematical Fluid Dynamics Present and Future 2014, 2016
    Co-Authors: Yoshihiro Shibata
    Abstract:

    In this paper, we consider the boundary value problem of Stokes operator arising in the study of free boundary problem for the Navier-Stokes equations with surface tension in a uniform \(W^{3-1/r}_r\) domain of N-dimensional Euclidean space \({{\mathbb {R}}}^N\) (\(N\geqslant 2\), \(N< r < \infty \)). We prove the existence of \({\mathscr {R}}\)-Bounded Solution operator with spectral parameter \(\lambda \) varying in a sector \(\varSigma _{\varepsilon , \lambda _0} = \{\lambda \in {{\mathbb {C}}}\mid |\arg \lambda | \leqslant \pi -\varepsilon , \ |\lambda | \geqslant \lambda _0\}\) (\(0< \varepsilon < \pi /2\)), and the maximal \(L_p\)-\(L_q\) regularity with the help of the \({\mathscr {R}}\)-Bounded Solution operator and the Weis operator valued Fourier multiplier theorem. The essential assumption of this paper is the unique solvability of the weak Dirichlet-Neumann problem, namely it is assumed the unique existence of Solution \({\mathfrak {p}}\in {\mathscr {W}}^1_q(\varOmega )\) to the variational problem: \((\nabla {\mathfrak {p}}, \nabla \varphi )_\varOmega = (f, \nabla \varphi )_\varOmega \) for any \(\varphi \in {\mathscr {W}}^1_{q'}(\varOmega )\) with \(1< q < \infty \) and \(q' = q/(q-1)\), where \({\mathscr {W}}^1_q(\varOmega )\) is a closed subspace of \(\hat{W}^1_{q,\varGamma }(\varOmega ) = \{{\mathfrak {p}}\in L_{q, \mathrm{loc}} (\varOmega ) \mid \nabla {\mathfrak {p}}\in L_q(\varOmega )^N, \ {\mathfrak {p}}|_\varGamma = 0\}\) with respect to gradient norm \(\Vert \nabla \cdot \Vert _{L_q(\varOmega )}\) that contains a space \(W^1_{q,\varGamma }(\varOmega ) = \{{\mathfrak {p}}\in W^1_q(\varOmega ) \mid {\mathfrak {p}}|_\varGamma = 0\}\), and \(\varGamma \) is one part of boundary on which free boundary condition is imposed. The unique solvability of such weak Dirichlet-Neumann problem is necessary for the unique existence of a Solution to the resolvent problem with uniform estimate with respect to spectral parameter varying in \((\lambda _0, \infty )\), which was proved in Shibata [13]. Our assumption is satisfied for any \(q \in (1, \infty )\) by the following domains: half space, perturbed half space, Bounded domains, layer, perturbed layer, straight cube, and exterior domains with \({\mathscr {W}}^1_q(\varOmega ) = \hat{W}^1_{q, \varGamma }(\varOmega )\).

  • on the mathcal r Boundedness for the two phase problem with phase transition compressible incompressible model problem
    arXiv: Analysis of PDEs, 2015
    Co-Authors: Yoshihiro Shibata
    Abstract:

    In this paper, we prove the maximal $L_p$-$L_q$ regularity of the compressible and incompressible two phase flow with phase transition in the model problem case with the help of ${\mathcal R}$-Bounded Solution operators corresponding to generalized resolvent problem. The problem arises from the mathematical study of the motion of two-phase flows having gaseous phase and liquid phase separated by a sharp interface with phase transition. Using the result obtained in this paper, in \cite{S0} we proved the local well-posedness of free boundary problem for the compressible and incompressible two phase flow separated by sharp interface with phase transition.

  • on the ℛ Boundedness for the two phase problem compressible incompressible model problem
    Boundary Value Problems, 2014
    Co-Authors: Takayuki Kubo, Yoshihiro Shibata, Kohei Soga
    Abstract:

    The situation of this paper is that the Stokes equation for the compressible viscous fluid flow in the upper half-space is coupled via inhomogeneous interface conditions with the Stokes equations for the incompressible one in the lower half-space, which is the model problem for the evolution of compressible and incompressible viscous fluid flows with a sharp interface. We show the existence of ℛ-Bounded Solution operators to the corresponding generalized resolvent problem, which implies the generation of analytic semigroup and maximal - regularity for the corresponding time dependent problem with the help of the Weis’ operator valued Fourier multiplier theorem. The problem was studied by Denisova (Interfaces Free Bound. 2(3):283-312, 2000) under some restriction on the viscosity coefficients and one of our purposes is to eliminate the assumption in (Denisova in Interfaces Free Bound. 2(3):283-312, 2000). MSC: 35Q35, 76T10.

  • on some free boundary problem for a compressible barotropic viscous fluid flow
    Annali Dell'universita' Di Ferrara, 2013
    Co-Authors: Yuko Enomoto, Lorenz Von Below, Yoshihiro Shibata
    Abstract:

    In this paper, we prove a local in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the \(L_p\) in time and \(L_q\) in space framework with \(2 < p < \infty \) and \(N < q < \infty \) under the assumption that the initial domain is a uniform \(W^{2-1/q}_q\) one in \({\mathbb {R}}^{N}\, (N \ge 2\)). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve problem by the Banach contraction mapping principle based on the maximal \(L_p\)–\(L_q\) regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key issue for the linear theorem is the existence of \({\mathcal {R}}\)-Bounded Solution operator in a sector, which combined with Weis’s operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal \(L_p\)–\(L_q\) regularity theorem. The nonlinear problem we studied here was already investigated by several authors (Denisova and Solonnikov, St. Petersburg Math J 14:1–22, 2003; J Math Sci 115:2753–2765, 2003; Secchi, Commun PDE 1:185–204, 1990; Math Method Appl Sci 13:391–404, 1990; Secchi and Valli, J Reine Angew Math 341:1–31, 1983; Solonnikov and Tani, Constantin caratheodory: an international tribute, vols 1, 2, pp 1270–1303, World Scientific Publishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin, 1992; Tani, J Math Kyoto Univ 21:839–859, 1981; Zajaczkowski, SIAM J Math Anal 25:1–84, 1994) in the \(L_2\) framework and Holder spaces, but our approach is different from them.

Daniel Hernandezhernandez - One of the best experts on this subject based on the ideXlab platform.

  • risk sensitive control of markov processes in countable state space
    Systems & Control Letters, 1996
    Co-Authors: Daniel Hernandezhernandez, Steven I. Marcus
    Abstract:

    In this paper we consider infinite horizon risk-sensitive control of Markov processes with discrete time and denumerable state space. This problem is solved by proving, under suitable conditions, that there exists a Bounded Solution to the dynamic programming equation. The dynamic programming equation is transformed into an Isaacs equation for a stochastic game, and the vanishing discount method is used to study its Solution. In addition, we prove that the existence conditions are also necessary.

E Fernandezgaucherand - One of the best experts on this subject based on the ideXlab platform.

  • controlled markov chains with risk sensitive criteria average cost optimality equations and optimal Solutions
    Mathematical Methods of Operations Research, 1999
    Co-Authors: Rolando Cavazoscadena, E Fernandezgaucherand
    Abstract:

    We study controlled Markov chains with denumerable state space and Bounded costs per stage. A (long-run) risk-sensitive average cost criterion, associated to an exponential utility function with a constant risk sensitivity coefficient, is used as a performance measure. The main assumption on the probabilistic structure of the model is that the transition law satisfies a simultaneous Doeblin condition. Working within this framework, the main results obtained can be summarized as follows: If the constant risk-sensitivity coefficient is small enough, then an associated optimality equation has a Bounded Solution with a constant value for the optimal risk-sensitive average cost; in addition, under further standard continuity-compactness assumptions, optimal stationary policies are obtained. However, it is also shown that the above conclusions fail to hold, in general, for large enough values of the risk-sensitivity coefficient. Our results therefore disprove previous claims on this topic. Also of importance is the fact that our developments are very much self-contained and employ only basic probabilistic and analysis principles. Copyright Springer-Verlag Berlin Heidelberg 1999

  • remarks on the existence of Solutions to the average cost optimality equation in markov decision processes
    Systems & Control Letters, 1991
    Co-Authors: E Fernandezgaucherand, Aristotle Arapostathis, Steven I. Marcus
    Abstract:

    Necessary conditions are given for the existence of a Bounded Solution to the optimality equation arising in Markov decision processes, under a long-run, expected average cost criterion. The relationships of some of our results to known sufficient conditions are also shown.

Bin Zhou - One of the best experts on this subject based on the ideXlab platform.

  • construction of strict lyapunov krasovskii functionals for time varying time delay systems
    Automatica, 2019
    Co-Authors: Bin Zhou
    Abstract:

    For stability analysis of time-varying time-delay systems, it is known that time-derivatives (time-shifts in the discrete-time setting) of the Lyapunov–Krasovskii functionals (LKFs) for both internal stability and input-to-state stability (ISS) can be indefinite, which can improve the resulting stability conditions in some cases. However, the non-strict LKFs may be insufficient and inconvenient to use in practice. In this paper, based on the non-strict ISS LKFs for time-varying time-delay systems and with the help of uniformly asymptotically stable (UAS) and uniformly exponentially Bounded (UEB) scalar functions, three classes of strict ISS LKFs are constructed such that their time-derivatives are strictly negative definite along the trajectories of the considered system. A general construction of strict ISS LKFs by using the positive definite and uniformly Bounded Solution to a scalar Lyapunov differential equation is established, which includes the proposed three classes of strict ISS LKFS as special cases. The approaches are also extended to deal with a Lyapunov inequality whose right hand side is indefinite and nonlinear. Both continuous-time and discrete-time systems are considered. The effectiveness of the proposed methods is illustrated by some examples borrowed from the literature.