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Toshiaki Hishida - One of the best experts on this subject based on the ideXlab platform.
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Decay Estimates of Gradient of a Generalized Oseen Evolution Operator Arising from Time-Dependent Rigid Motions in Exterior Domains
Archive for Rational Mechanics and Analysis, 2020Co-Authors: Toshiaki HishidaAbstract:Let us consider the motion of a viscous incompressible fluid past a rotating rigid body in three dimensions, where the translational and angular velocities of the body are prescribed but time-dependent. In a reference frame attached to the body, we have the Navier–Stokes system with the drift and (one half of the) Coriolis terms in a fixed exterior domain. The existence of the evolution operator T ( t , s ) in the space $$L^q$$ L q generated by the linearized non-Autonomous system was proved by Hansel and Rhandi (J Reine Angew Math 694:1–26, 2014) and the large time behavior of T ( t , s ) f in $$L^r$$ L r for $$(t-s)\rightarrow \infty $$ ( t - s ) → ∞ was then developed by Hishida (Math Ann 372:915–949, 2018) when f is taken from $$L^q$$ L q with $$q\leqq r$$ q ≦ r . The contribution of the present paper concerns such $$L^q$$ L q - $$L^r$$ L r decay estimates of $$\nabla T(t,s)$$ ∇ T ( t , s ) with optimal rates, which must be useful for the study of stability/attainability of the Navier–Stokes flow in several physically relevant situations. Our main theorem completely recovers the $$L^q$$ L q - $$L^r$$ L r estimates for the Autonomous Case (Stokes and Oseen semigroups, those semigroups with rotating effect) in three dimensional exterior domains, which were established by Hishida and Shibata (Arch Ration Mech Anal 193:339–421, 2009), Iwashita (Math Ann 285, 265–288, 1989), Kobayashi and Shibata (Math Ann 310:1–45, 1998), Maremonti and Solonnikov (Ann Sc Norm Super Pisa 24:395–449, 1997) and Shibata (in: Amann, Arendt, Hieber, Neubrander, Nicaise, von Below (eds) Functional analysis and evolution equations, the Günter Lumer volume. Birkhäuser, Basel, pp 595–611, 2008).
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decay estimates of gradient of a generalized oseen evolution operator arising from time dependent rigid motions in exterior domains
arXiv: Analysis of PDEs, 2019Co-Authors: Toshiaki HishidaAbstract:Let us consider the motion of a viscous incompressible fluid past a rotating rigid body in 3D, where the translational and angular velocities of the body are prescribed but time-dependent. In a reference frame attached to the body, we have the Navier-Stokes system with the drift and (one half of the) Coriolis terms in a fixed exterior domain. The existence of the evolution operator $T(t,s)$ in the space $L^q$ generated by the linearized non-Autonomous system was proved by Hansel and Rhandi [26] and the large time behavior of $T(t,s)f$ in $L^r$ for $(t-s)\to\infty$ was then developed by the present author [33] when $f$ is taken from $L^q$ with $q\leq r$. The contribution of the present paper concerns such $L^q$-$L^r$ decay estimates of $\nabla T(t,s)$ with optimal rates, which must be useful for the study of stability/attainability of the Navier-Stokes flow in several physically relevant situations. Our main theorem completely recovers the $L^q$-$L^r$ estimates for the Autonomous Case (Stokes and Oseen semigroups, those semigroups with rotating effect) in 3D exterior domains, which were established by [37], [42], [39], [36] and [44].
Carlota Rebelo - One of the best experts on this subject based on the ideXlab platform.
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persistence in seasonally varying predator prey systems via the basic reproduction number
Nonlinear Analysis-real World Applications, 2016Co-Authors: Maurizio Garrione, Carlota RebeloAbstract:Abstract We study persistence in general seasonally varying predator–prey models. Using the notion of basic reproduction number R 0 and the theoretical results proved in Rebelo et al. (2012) in the framework of epidemiological models, we show that uniform persistence is obtained as long as R 0 > 1 . In this way, we extend previous results obtained in the Autonomous Case for models including competition among predators, prey–mesopredator–superpredator models and Leslie–Gower systems.
Razvan Gabriel Iagar - One of the best experts on this subject based on the ideXlab platform.
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blow up profiles for a quasilinear reaction diffusion equation with weighted reaction with linear growth
Journal of Dynamics and Differential Equations, 2019Co-Authors: Razvan Gabriel Iagar, Ariel SanchezAbstract:We study the blow up profiles associated to the following second order reaction–diffusion equation with non-homogeneous reaction: $$\begin{aligned} \partial _tu=\partial _{xx}(u^m) + |x|^{\sigma }u, \end{aligned}$$ with \(\sigma >0\). Through this study, we show that the non-homogeneous coefficient \(|x|^{\sigma }\) has a strong influence on the blow up behavior of the solutions. First of all, it follows that finite time blow up occurs for self-similar solutions u, a feature that does not appear in the well known Autonomous Case \(\sigma =0\). Moreover, we show that there are three different types of blow up self-similar profiles, depending on whether the exponent \(\sigma \) is closer to zero or not. We also find an explicit blow up profile. The results show in particular that global blow up occurs when \(\sigma >0\) is sufficiently small, while for \(\sigma >0\) sufficiently large blow up occurs only at infinity, and we give prototypes of these phenomena in form of self-similar solutions with precise behavior. This work is a part of a larger program of understanding the influence of non-homogeneous weights on the blow up sets and rates.
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blow up profiles for a quasilinear reaction diffusion equation with weighted reaction with linear growth
arXiv: Analysis of PDEs, 2018Co-Authors: Razvan Gabriel Iagar, Ariel SanchezAbstract:We study the blow up profiles associated to the following second order reaction-diffusion equation with non-homogeneous reaction: $$ \partial_tu=\partial_{xx}(u^m) + |x|^{\sigma}u, $$ with $\sigma>0$. Through this study, we show that the non-homogeneous coefficient $|x|^{\sigma}$ has a strong influence on the blow up behavior of the solutions. First of all, it follows that finite time blow up occurs for self-similar solutions $u$, a feature that does not appear in the well known Autonomous Case $\sigma=0$. Moreover, we show that there are three different types of blow up self-similar profiles, depending on whether the exponent $\sigma$ is closer to zero or not. We also find an explicit blow up profile. The results show in particular that \emph{global blow up} occurs when $\sigma>0$ is sufficiently small, while for $\sigma>0$ sufficiently large blow up \emph{occurs only at infinity}, and we give prototypes of these phenomena in form of self-similar solutions with precise behavior. This work is a part of a larger program of understanding the influence of non-homogeneous weights on the blow up sets and rates.
Lu Yang - One of the best experts on this subject based on the ideXlab platform.
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asymptotic regularity and attractors of the reaction diffusion equation with nonlinear boundary condition
Nonlinear Analysis-real World Applications, 2012Co-Authors: Lu YangAbstract:Abstract We consider the dynamical behavior of the reaction–diffusion equation with nonlinear boundary condition for both Autonomous and non-Autonomous Cases. For the Autonomous Case, under the assumption that the internal nonlinear term f is dissipative and the boundary nonlinear term g is non-dissipative, the asymptotic regularity of solutions is proved. For the non-Autonomous Case, we obtain the existence of a compact uniform attractor in H 1 ( Ω ) with dissipative internal and boundary nonlinearities.
Maurizio Garrione - One of the best experts on this subject based on the ideXlab platform.
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persistence in seasonally varying predator prey systems via the basic reproduction number
Nonlinear Analysis-real World Applications, 2016Co-Authors: Maurizio Garrione, Carlota RebeloAbstract:Abstract We study persistence in general seasonally varying predator–prey models. Using the notion of basic reproduction number R 0 and the theoretical results proved in Rebelo et al. (2012) in the framework of epidemiological models, we show that uniform persistence is obtained as long as R 0 > 1 . In this way, we extend previous results obtained in the Autonomous Case for models including competition among predators, prey–mesopredator–superpredator models and Leslie–Gower systems.
