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Hideo Kozono - One of the best experts on this subject based on the ideXlab platform.

  • necessary and sufficient condition on initial data in the Besov Space for solutions in the serrin class of the navier stokes equations
    Journal of Evolution Equations, 2020
    Co-Authors: Hideo Kozono, Akira Okada, Senjo Shimizu
    Abstract:

    The Cauchy problem of the Navier–Stokes equations in $$\mathbb {R}^n$$ with the initial data a in the Besov Space $${B}^{-1+\frac{n}{p}}_{{p},{q}}(\mathbb {R}^n)$$ for $$nSpace. Conversely, if the solution belongs to $$L^{\alpha ,q}(0,T;L^{r}(\mathbb {R}^n))$$ with $$\frac{2}{\alpha }+\frac{n}{r}=1$$ , then the initial data a necessarily belong to $${B}^{-1+\frac{n}{r}}_{{r},{q}}(\mathbb {R}^n)$$ . It implies that the initial data in the Besov Space $$B^{-1+\frac{n}{p}}_{p,q} (\mathbb {R}^n)$$ are a necessary and sufficient condition for the existence of solutions in the Serrin class.

  • characterization of initial data in the homogeneous Besov Space for solutions in the serrin class of the navier stokes equations
    Journal of Functional Analysis, 2020
    Co-Authors: Hideo Kozono, Akira Okada, Senjo Shimizu
    Abstract:

    Abstract Consider the Cauchy problem of the Navier-Stokes equations in R n with initial data a in the homogeneous Besov Space B ˙ p , q − 1 + n p ( R n ) for n p ∞ and 1 ≦ q ≦ ∞ . We show that the Stokes flow e t Δ a can be controlled in L α , q ( 0 , ∞ ; B ˙ r , 1 0 ( R n ) ) for 2 α + n r = 1 with p ≦ r ∞ , where L α , q denotes the Lorentz Space. As an application, the global existence theorem of mild solutions for the small initial data is established in the above class which is slightly stronger than Serrin's. Conversely, if the global solution belongs to the usual Serrin class L α , q ( 0 , ∞ ; L r ( R n ) ) for r and α as above with 1 q ≦ ∞ , then the initial data a necessarily belongs to B ˙ r , q − 1 + n r ( R n ) . Moreover, we prove that such solutions are analytic in the Space variables. Our method for the proof of analyticity is based on a priori estimates of higher derivatives of solutions in L p ( R n ) with Holder continuity in time.

  • navier stokes equations in the Besov Space near l and bmo
    Kyushu Journal of Mathematics, 2003
    Co-Authors: Hideo Kozono, Takayoshi Ogawa, Yasushi Taniuchi
    Abstract:

    We prove a local existence theorem for the Navier-Stokes equations with the initial data in B0∞,∞ containing functions which do not decay at infinity. Then we establish an extension criterion on our local solutions in terms of the vorticity in the homogeneous Besov Space B·0∞,∞.

B W Silverman - One of the best experts on this subject based on the ideXlab platform.

  • wavelet thresholding via a bayesian approach
    Journal of The Royal Statistical Society Series B-statistical Methodology, 1998
    Co-Authors: Felix Abramovich, Theofanis Sapatinas, B W Silverman
    Abstract:

    We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion that is common to most applications. For the prior specified, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any specific Besov Space. We establish a relationship between the hyperparameters of the prior model and the parameters of those Besov Spaces within which realizations from the prior will fall. Such a relationship gives insight into the meaning of the Besov Space parameters. Moreover, the relationship established makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coefficients. However, prior knowledge about a function's regularity properties might be difficult to elicit; with this in mind, we propose a standard choice of prior hyperparameters that works well in our examples. Several simulated examples are used to illustrate our method, and comparisons are made with other thresholding methods. We also present an application to a data set that was collected in an anaesthesiological study.

Senjo Shimizu - One of the best experts on this subject based on the ideXlab platform.

  • global well posedness for the incompressible navier stokes equations in the critical Besov Space under the lagrangian coordinates
    Journal of Differential Equations, 2021
    Co-Authors: Takayoshi Ogawa, Senjo Shimizu
    Abstract:

    Abstract We consider global well-posedness of the Cauchy problem of the incompressible Navier–Stokes equations under the Lagrangian coordinates in scaling critical Besov Spaces. We prove the system is globally well-posed in the homogeneous Besov Space B ˙ p , 1 − 1 + n / p ( R n ) with 1 ≤ p ∞ . The former result was restricted for 1 ≤ p 2 n and the main reason why the well-posedness Space is enlarged is that the quasi-linear part of the system has a special feature called a multiple divergence structure and the bilinear estimate for the nonlinear terms are improved by such a structure. Our result indicates that the Navier–Stokes equations can be transferred from the Eulerian coordinates to the Lagrangian coordinates even for the solution in the limiting critical Besov Spaces.

  • necessary and sufficient condition on initial data in the Besov Space for solutions in the serrin class of the navier stokes equations
    Journal of Evolution Equations, 2020
    Co-Authors: Hideo Kozono, Akira Okada, Senjo Shimizu
    Abstract:

    The Cauchy problem of the Navier–Stokes equations in $$\mathbb {R}^n$$ with the initial data a in the Besov Space $${B}^{-1+\frac{n}{p}}_{{p},{q}}(\mathbb {R}^n)$$ for $$nSpace. Conversely, if the solution belongs to $$L^{\alpha ,q}(0,T;L^{r}(\mathbb {R}^n))$$ with $$\frac{2}{\alpha }+\frac{n}{r}=1$$ , then the initial data a necessarily belong to $${B}^{-1+\frac{n}{r}}_{{r},{q}}(\mathbb {R}^n)$$ . It implies that the initial data in the Besov Space $$B^{-1+\frac{n}{p}}_{p,q} (\mathbb {R}^n)$$ are a necessary and sufficient condition for the existence of solutions in the Serrin class.

  • characterization of initial data in the homogeneous Besov Space for solutions in the serrin class of the navier stokes equations
    Journal of Functional Analysis, 2020
    Co-Authors: Hideo Kozono, Akira Okada, Senjo Shimizu
    Abstract:

    Abstract Consider the Cauchy problem of the Navier-Stokes equations in R n with initial data a in the homogeneous Besov Space B ˙ p , q − 1 + n p ( R n ) for n p ∞ and 1 ≦ q ≦ ∞ . We show that the Stokes flow e t Δ a can be controlled in L α , q ( 0 , ∞ ; B ˙ r , 1 0 ( R n ) ) for 2 α + n r = 1 with p ≦ r ∞ , where L α , q denotes the Lorentz Space. As an application, the global existence theorem of mild solutions for the small initial data is established in the above class which is slightly stronger than Serrin's. Conversely, if the global solution belongs to the usual Serrin class L α , q ( 0 , ∞ ; L r ( R n ) ) for r and α as above with 1 q ≦ ∞ , then the initial data a necessarily belongs to B ˙ r , q − 1 + n r ( R n ) . Moreover, we prove that such solutions are analytic in the Space variables. Our method for the proof of analyticity is based on a priori estimates of higher derivatives of solutions in L p ( R n ) with Holder continuity in time.

Wei Yan - One of the best experts on this subject based on the ideXlab platform.

  • the cauchy problem for fractional camassa holm equation in Besov Space
    Nonlinear Analysis-real World Applications, 2021
    Co-Authors: Lili Fan, Hongjun Gao, Junfang Wang, Wei Yan
    Abstract:

    Abstract In this paper, we consider the fractional Camassa–Holm equation modeling the propagation of small-but-finite amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. First, we establish the local well-posedness in Besov Space B 2 , 1 s 0 with s 0 = 2 ν − 1 2 for ν > 3 2 and s 0 = 5 2 for 1 ν ≤ 3 2 . Then, with a given analytic initial data, we establish the analyticity of the solutions in both variables, globally in Space and locally in time. Finally, we give a blow-up criterion.

  • the cauchy problem for fractional camassa holm equation in Besov Space
    arXiv: Analysis of PDEs, 2020
    Co-Authors: Lili Fan, Hongjun Gao, Junfang Wang, Wei Yan
    Abstract:

    In this paper, we consider the fractional Camassa-Holm equation modelling the propagation of small-but-finite amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. First, we establish the local well-posedness in Besov Space $B^{s_0}_{2,1}$ with $s_0=2\nu-\frac 1 2$ for $\nu>\frac 3 2 $ and $s_0=\frac 5 2$ for $1<\nu\leq \frac 3 2 $. Then, with a given analytic initial data, we establish the analyticity of the solutions in both variables, globally in Space and locally in time.

  • the cauchy problem for shallow water waves of large amplitude in Besov Space
    Journal of Differential Equations, 2019
    Co-Authors: Lili Fan, Wei Yan
    Abstract:

    Abstract In this paper, we consider a nonlinear evolution equation modelling the propagation of surface waves in the shallow water regime of large amplitude, which is characterised by some cubical nonlinearities. First, we establish the local well-posedness in Besov Space B 2 , 1 3 / 2 . Then, we give a blow-up criterion. Finally, with a given analytic initial data, we establish the analyticity of the solutions in both variables, globally in Space and locally in time.

  • on well posedness of two component camassa holm system in the critical Besov Space
    Nonlinear Analysis-theory Methods & Applications, 2015
    Co-Authors: Defu Chen, Wei Yan
    Abstract:

    Abstract In this paper, we study the Cauchy problem for a two-component Camassa–Holm system in the critical Besov Space B 2 , 1 1 / 2 . We obtain the solution local existence and prove the solution mapping is Holder continuous with respect to the initial value.

  • the cauchy problem for the modified two component camassa holm system in critical Besov Space
    Annales de l'Institut Henri Poincaré C Analyse non linéaire, 2015
    Co-Authors: Wei Yan
    Abstract:

    Abstract In this paper, we are concerned with the Cauchy problem for the modified two-component Camassa–Holm system in the Besov Space with data having critical regularity. The key elements in our paper are the real interpolations and logarithmic interpolation among inhomogeneous Besov Space and Lemma 5.2.1 of [7] which is also called Osgood Lemma and the Fatou Lemma. The new ingredient that we introduce in this paper can be seen on pages 453–457.

Zachary Bradshaw - One of the best experts on this subject based on the ideXlab platform.

  • discretely self similar solutions to the navier stokes equations with Besov Space data
    Archive for Rational Mechanics and Analysis, 2018
    Co-Authors: Zachary Bradshaw, Taipeng Tsai
    Abstract:

    We construct self-similar solutions to the three dimensional Navier–Stokes equations for divergence free, self-similar initial data that can be large in the critical Besov Space $${\dot{B}_{p,\infty}^{3/p-1}}$$ where 3 < p < 6. We also construct discretely self-similar solutions for divergence free initial data in $${\dot{B}_{p,\infty}^{3/p-1}}$$ for 3   1. These results extend those of Bradshaw and Tsai (Ann Henri Poincare 2016. https://doi.org/10.1007/s00023-016-0519-0 ) which dealt with initial data in L 3 w since $${L^3_w\subsetneq \dot{B}_{p,\infty}^{3/p-1}}$$ for p > 3. We also provide several concrete examples of vector fields in the relevant function Spaces.

  • discretely self similar solutions to the navier stokes equations with Besov Space data
    arXiv: Analysis of PDEs, 2017
    Co-Authors: Zachary Bradshaw, Taipeng Tsai
    Abstract:

    We construct self-similar solutions to the three dimensional Navier-Stokes equations for divergence free, self-similar initial data that can be large in the critical Besov Space $\dot B^{-1+3/p}_{p,\infty}$ where $3 1$. These results extend those of \cite{BT1} which dealt with initial data in $L^3_w$ since $L^3_w\subsetneq \dot B^{-1+3/p}_{p,\infty}$ for $p>3$. We also provide several concrete examples of vector fields in the relevant function Spaces.