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November  2018, 17(6): 2751-2771. doi: 10.3934/cpaa.2018130

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

Ildoo Kim

Department of Mathematics, Korea University, 1 Anam-dong, Sungbuk-gu, Seoul, 136-701, Republic of Korea

Received  July 2017 Revised  March 2018 Published  June 2018

Fund Project: The author was supported by the TJ Park Science Fellowship of POSCO TJ Park Foundation.

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
$\begin{align*}\Re[ψ(t, ξ)] ≤ -ν|ξ|^{γ}, \end{align*}$
and
$\begin{align*}|D_{ξ}^{α}ψ(t, ξ)|≤ν^{-1}|ξ|^{γ-|α|}.\end{align*}$
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
$γ+α \notin \mathbb{Z}_+$
.
Keywords: Time measurable pseudo-differential operator, Lp-Lipschitz estimate, Cauchy problem.
Mathematics Subject Classification: 35K99, 47G30, 26A16.
Citation: Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2751-2771. doi: 10.3934/cpaa.2018130
References:
[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. Kim, K.-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. Kim, S. 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.

show all references

References:
[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. Kim, K.-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. Kim, S. 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.

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