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Yoshihiro Shibata - One of the best experts on this subject based on the ideXlab platform.
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global well posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface
Evolution Equations and Control Theory, 2018Co-Authors: Yoshihiro ShibataAbstract:In this paper, we prove the global well-posedness of free boundary problems of the Navier-Stokes equations in a bounded Domain with surface tension. The velocity field is obtained in the \begin{document}$L_p$\end{document} in time \begin{document}$L_q$\end{document} in space maximal regularity class, ( \begin{document}$2 , \begin{document}$N , and \begin{document}$2/p + N/q ), under the assumption that the Initial Domain is close to a ball and Initial data are sufficiently small. The essential point of our approach is to drive the exponential decay theorem in the \begin{document}$L_p$\end{document} - \begin{document}$L_q$\end{document} framework for the linearized equations with the help of maximal \begin{document}$L_p$\end{document} - \begin{document}$L_q$\end{document} regularity theory for the Stokes equations with free boundary conditions and spectral analysis of the Stokes operator and the Laplace-Beltrami operator.
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on some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface
Discrete and Continuous Dynamical Systems, 2016Co-Authors: Takayuki Kubo, Yoshihiro Shibata, Kohei SogaAbstract: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.
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on some free boundary problem for a compressible barotropic viscous fluid flow
Annali Dell'universita' Di Ferrara, 2013Co-Authors: Yuko Enomoto, Lorenz Von Below, Yoshihiro ShibataAbstract: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.
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on some free boundary problem for a compressible barotropic viscous fluid flow
Annali Dell'universita' Di Ferrara, 2013Co-Authors: Yuko Enomoto, Lorenz Von Below, Yoshihiro ShibataAbstract: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.
Fabio Toninelli - One of the best experts on this subject based on the ideXlab platform.
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Zero'' temperature stochastic 3D Ising model and dimer covering fluctuations: a first step towards interface mean curvature motion
Communications on Pure and Applied Mathematics, 2011Co-Authors: Pietro Caputo, Fabio Martinelli, François Simenhaus, Fabio ToninelliAbstract:We consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at a temperature that goes to zero with the system size (hence the quotation marks in the title). In dimension d = 3 we prove that an Initial Domain of linear size L of "−" spins disappears within a time τ+, which is at most L2(log L)c and at least L2/(c log L) for some c > 0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe, on large time scales, the evolution of the interface between "+" and "−" Domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimmer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factors, is the first rigorous confirmation of the Lifshitz law τ+ ≃ const × L2, conjectured on heuristic grounds [8, 13]. In dimension d = 2, τ+ can be shown to be of order L2 without logarithmic corrections: the upper bound was proven in [6], and here we provide the lower bound. For d = 2, we also prove that the spectral gap of the generator behaves like equation image for L large, as conjectured in [2].
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zero temperature stochastic 3d ising model and dimer covering fluctuations a first step towards interface mean curvature motion
arXiv: Mathematical Physics, 2010Co-Authors: Pietro Caputo, Fabio Martinelli, François Simenhaus, Fabio ToninelliAbstract:We consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at temperature which goes to zero with the system size (hence the quotation marks in the title). In dimension d=3 we prove that an Initial Domain of linear size L of "-" spins disappears within a time \tau_+ which is at most L^2(\log L)^c and at least L^2/(c\log L), for some c>0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe, on large time-scales, the evolution of the interface between "+" and "-" Domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factor, is the first rigorous confirmation of the expected behavior \tau_+\simeq const\times L^2, conjectured on heuristic grounds. In dimension d=2, \tau_+ can be shown to be of order L^2 without logarithmic corrections: the upper bound was proven in [Fontes, Schonmann, Sidoravicius, 2002] and here we provide the lower bound. For d=2, we also prove that the spectral gap of the generator behaves like c/L for L large, as conjectured in [Bodineau-Martinelli, 2002].
Pietro Caputo - One of the best experts on this subject based on the ideXlab platform.
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Zero'' temperature stochastic 3D Ising model and dimer covering fluctuations: a first step towards interface mean curvature motion
Communications on Pure and Applied Mathematics, 2011Co-Authors: Pietro Caputo, Fabio Martinelli, François Simenhaus, Fabio ToninelliAbstract:We consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at a temperature that goes to zero with the system size (hence the quotation marks in the title). In dimension d = 3 we prove that an Initial Domain of linear size L of "−" spins disappears within a time τ+, which is at most L2(log L)c and at least L2/(c log L) for some c > 0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe, on large time scales, the evolution of the interface between "+" and "−" Domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimmer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factors, is the first rigorous confirmation of the Lifshitz law τ+ ≃ const × L2, conjectured on heuristic grounds [8, 13]. In dimension d = 2, τ+ can be shown to be of order L2 without logarithmic corrections: the upper bound was proven in [6], and here we provide the lower bound. For d = 2, we also prove that the spectral gap of the generator behaves like equation image for L large, as conjectured in [2].
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zero temperature stochastic 3d ising model and dimer covering fluctuations a first step towards interface mean curvature motion
arXiv: Mathematical Physics, 2010Co-Authors: Pietro Caputo, Fabio Martinelli, François Simenhaus, Fabio ToninelliAbstract:We consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at temperature which goes to zero with the system size (hence the quotation marks in the title). In dimension d=3 we prove that an Initial Domain of linear size L of "-" spins disappears within a time \tau_+ which is at most L^2(\log L)^c and at least L^2/(c\log L), for some c>0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe, on large time-scales, the evolution of the interface between "+" and "-" Domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factor, is the first rigorous confirmation of the expected behavior \tau_+\simeq const\times L^2, conjectured on heuristic grounds. In dimension d=2, \tau_+ can be shown to be of order L^2 without logarithmic corrections: the upper bound was proven in [Fontes, Schonmann, Sidoravicius, 2002] and here we provide the lower bound. For d=2, we also prove that the spectral gap of the generator behaves like c/L for L large, as conjectured in [Bodineau-Martinelli, 2002].
Doru C Lupascu - One of the best experts on this subject based on the ideXlab platform.
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fatigue induced evolution of Domain structure in ferroelectric lead zirconate titanate ceramics investigated by piezoresponse force microscopy
Journal of Applied Physics, 2005Co-Authors: V V Shvartsman, A L Kholkin, Cyril Verdier, Doru C LupascuAbstract:The evolution of the Domain structure in lead zirconate titanate ceramics in the course of polarization fatigue is investigated by piezoresponse force microscopy. It is found that fatigue results in a strong modification of the Domain structure. The Domain patterns Initially consisting of mainly 180° Domains split into fine ferroelastic Domains to relieve the mechanical stresses arising due to the continuous polarization switching. The observed distorted Domain walls (or “wavy” Domain patterns) are attributed to clamping by defect agglomerates. The biggest concentration of clamped Domains is found in grains close to the electrode interface signifying that these are most damaged by fatigue. Furthermore, a preferred polarization orientation is observed near the electrodes. Postannealing of fatigued samples promotes the partial recovery of the Initial Domain structure. The results indicate the importance of nearby electrode grains in the polarization switching in polycrystalline materials.
Fabio Martinelli - One of the best experts on this subject based on the ideXlab platform.
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Zero'' temperature stochastic 3D Ising model and dimer covering fluctuations: a first step towards interface mean curvature motion
Communications on Pure and Applied Mathematics, 2011Co-Authors: Pietro Caputo, Fabio Martinelli, François Simenhaus, Fabio ToninelliAbstract:We consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at a temperature that goes to zero with the system size (hence the quotation marks in the title). In dimension d = 3 we prove that an Initial Domain of linear size L of "−" spins disappears within a time τ+, which is at most L2(log L)c and at least L2/(c log L) for some c > 0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe, on large time scales, the evolution of the interface between "+" and "−" Domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimmer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factors, is the first rigorous confirmation of the Lifshitz law τ+ ≃ const × L2, conjectured on heuristic grounds [8, 13]. In dimension d = 2, τ+ can be shown to be of order L2 without logarithmic corrections: the upper bound was proven in [6], and here we provide the lower bound. For d = 2, we also prove that the spectral gap of the generator behaves like equation image for L large, as conjectured in [2].
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zero temperature stochastic 3d ising model and dimer covering fluctuations a first step towards interface mean curvature motion
arXiv: Mathematical Physics, 2010Co-Authors: Pietro Caputo, Fabio Martinelli, François Simenhaus, Fabio ToninelliAbstract:We consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at temperature which goes to zero with the system size (hence the quotation marks in the title). In dimension d=3 we prove that an Initial Domain of linear size L of "-" spins disappears within a time \tau_+ which is at most L^2(\log L)^c and at least L^2/(c\log L), for some c>0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe, on large time-scales, the evolution of the interface between "+" and "-" Domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factor, is the first rigorous confirmation of the expected behavior \tau_+\simeq const\times L^2, conjectured on heuristic grounds. In dimension d=2, \tau_+ can be shown to be of order L^2 without logarithmic corrections: the upper bound was proven in [Fontes, Schonmann, Sidoravicius, 2002] and here we provide the lower bound. For d=2, we also prove that the spectral gap of the generator behaves like c/L for L large, as conjectured in [Bodineau-Martinelli, 2002].