Advanced Search
Article Contents
Article Contents

An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators

The author was supported by the TJ Park Science Fellowship of POSCO TJ Park Foundation.
Abstract Full Text(HTML) Related Papers Cited by
  • In this article we prove the existence and uniqueness of a (weak) solution $u$ in $L_p\left( (0, T); Λ_{γ+m}\right)$ to the Cauchy problem

    $\begin{align}\notag&\frac{\partial u}{\partial t}(t, x) = ψ(t, i\nabla)u(t, x)+f(t, x), \;\;\;(t, x) ∈ (0, T) × {\bf{R}}^d \\& u(0, x) = 0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)\end{align}$

    where $d ∈ \mathbb{N}$, $p ∈ (1, ∞]$, $γ, m ∈ (0, ∞)$, $Λ_{γ+m}$ is the Lipschitz space on ${\bf{R}}^d$ whose order is $γ+m$, $f ∈ L_p\left( (0, T) ; Λ_{γ} \right)$, and $ψ(t, i\nabla)$ is a time measurable pseudo-differential operator whose symbol is $ψ(t, ξ)$, i.e.

    $ψ(t, i\nabla)u(t, x) = \mathcal{F}^{-1}[ψ(t, ξ){\mathcal{F}}\left[u(t, ·)\right]\left(ξ)\right](x), $

    with the assumptions

    $\begin{align*}\Re[ψ(t, ξ)] ≤ -ν|ξ|^{γ}, \end{align*}$


    $\begin{align*}|D_{ξ}^{α}ψ(t, ξ)|≤ν^{-1}|ξ|^{γ-|α|}.\end{align*}$

    Furthermore, we show

    $\begin{align}\int_0^T \|u(t, ·)\|^p_{Λ_{γ+m}} dt ≤ N \int_0^T \|f(t, ·)\|^p_{Λ_{m}} dt, \;\;\;\;\;\;\;\;\;\;(2)\end{align}$

    where $N$ is a positive constant depending only on $d$, $p$, $γ$, $ν$, $m$, and $T$,

    The unique solvability of equation (1) in $L_p$-Hölder space is also considered.More precisely, for any $f ∈ L_p((0, T);C^{n+α})$, there exists a unique solution $u ∈ L_p((0, T);C^{γ+n+α}({\bf{R}}^d))$ to equation (1) and for this solution $u$,

    $\begin{align}\int_0^T \|u(t, ·)\|^p_{C^{γ+n+α}}dt ≤N \int_0^T \|f(t, ·)\|^p_{C^{n+α}}dt, \;\;\;\;\;\;\;\;\;\;(3)\end{align}$

    where $n ∈ \mathbb{Z}_+$, $α ∈ (0, 1)$, and $γ+α \notin \mathbb{Z}_+$.

    Mathematics Subject Classification: 35K99, 47G30, 26A16.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] H. Abels, Pseudodifferential and Singular Integral Operators: An Introduction with Applications, Walter de Gruyter, 2012.
    [2] H. Dong and S. Kim, Partial schauder estimates for second-order elliptic and parabolic equations, Calculus of Variations and Partial Differential Equations, 40 (2011), 481-500.  doi: 10.1007/s00526-010-0348-9.
    [3] H. Dong and S. Kim, Partial schauder estimates for second-order elliptic and parabolic equations: a revisit, arXiv: 1502.00886, 2015. doi: 10.1007/s00526-010-0348-9.
    [4] L. Grafakos, Classical Fourier Analysis, volume 249, Springer, 2008.
    [5] L. Grafakos, Modern Fourier Analysis, volume 250, Springer, 2009. doi: 10.1007/978-0-387-09434-2.
    [6] L. Hörmander, The Analysis of Linear Partial Differential Operators III: Pseudo-differential Operators, volume 274, Springer Science & Business Media, 2007. doi: 10.1007/978-3-540-49938-1.
    [7] N. Jacob, Pseudo-Differential Operators & Markov Processes: Generators and Their Potential Theory, volume 2, Imperial College Press, 2002. doi: 10.1142/9781860949562.
    [8] I. KimK.-H. Kim and S. Lim, Parabolic BMO estimates for pseudo-differential operators of arbitrary order, Journal of Mathematical Analysis and Applications, 427 (2015), 557-580.  doi: 10.1016/j.jmaa.2015.02.065.
    [9] I. KimS. Lim and K.-H. Kim, An Lq(Lp)-theory for parabolic pseudo-differential equations: Calderón-Zygmund approach,, Potential Analysis, (2016), 1-21.  doi: 10.1007/s11118-016-9552-3.
    [10] N. V. Krylov, The Calderón-Zygmund theorem and parabolic equations in $ {L_p}\left({\mathbb{R},{C^{2 + \alpha }}} \right) $-spaces, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 1 (2002), 799-820. 
    [11] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, volume 96, American Mathematical Society Providence, RI, 2008. doi: 10.1090/gsm/096.
    [12] Y. Lin and S.Z. Lu, Pseudo-differential operators on Sobolev and Lipschitz spaces, Acta Mathematica Sinica, English Series, 16 (2010), 131-142.  doi: 10.1007/s10114-010-8109-4.
    [13] L. Lorenzi, Optimal Schauder estimates for parabolic problems with data measurable with respect to time, SIAM Journal on Mathematical Analysis, 31 (2000), 588-615.  doi: 10.1137/S0036141098342842.
    [14] R. Mikulevičius and H. Pragarauskas, On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces, Lithuanian Mathematical Journal, 32 (1992), 238-264.  doi: 10.1007/BF02450422.
    [15] R. Mikulevicius and H. Pragarauskas, On the cauchy problem for integro-differential operators in hölder classes and the uniqueness of the martingale problem, Potential Analysis, 40 (2014), 539-563.  doi: 10.1007/s11118-013-9359-4.
    [16] E. M. Stein and T. S. Murphy, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, volume 3, Princeton University Press, 1993.
  • 加载中

Article Metrics

HTML views(201) PDF downloads(194) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint