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

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
$γ+α \notin \mathbb{Z}_+$
.
Citation: Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

L. Grafakos, Classical Fourier Analysis, volume 249, Springer, 2008. Google Scholar

[5]

L. Grafakos, Modern Fourier Analysis, volume 250, Springer, 2009. doi: 10.1007/978-0-387-09434-2. Google Scholar

[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. Google Scholar

[7]

N. Jacob, Pseudo-Differential Operators & Markov Processes: Generators and Their Potential Theory, volume 2, Imperial College Press, 2002. doi: 10.1142/9781860949562. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

E. M. Stein and T. S. Murphy, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, volume 3, Princeton University Press, 1993. Google Scholar

show all references

References:
[1]

H. Abels, Pseudodifferential and Singular Integral Operators: An Introduction with Applications, Walter de Gruyter, 2012. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

L. Grafakos, Classical Fourier Analysis, volume 249, Springer, 2008. Google Scholar

[5]

L. Grafakos, Modern Fourier Analysis, volume 250, Springer, 2009. doi: 10.1007/978-0-387-09434-2. Google Scholar

[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. Google Scholar

[7]

N. Jacob, Pseudo-Differential Operators & Markov Processes: Generators and Their Potential Theory, volume 2, Imperial College Press, 2002. doi: 10.1142/9781860949562. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

E. M. Stein and T. S. Murphy, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, volume 3, Princeton University Press, 1993. Google Scholar

[1]

Lanzhe Liu. Mean oscillation and boundedness of Toeplitz Type operators associated to pseudo-differential operators. Communications on Pure & Applied Analysis, 2015, 14 (2) : 627-636. doi: 10.3934/cpaa.2015.14.627

[2]

JIAO CHEN, WEI DAI, GUOZHEN LU. $L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators. Communications on Pure & Applied Analysis, 2017, 16 (3) : 883-898. doi: 10.3934/cpaa.2017042

[3]

Angkana Rüland, Eva Sincich. Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data. Inverse Problems & Imaging, 2019, 13 (5) : 1023-1044. doi: 10.3934/ipi.2019046

[4]

N. V. Krylov. Uniqueness for Lp-viscosity solutions for uniformly parabolic Isaacs equations with measurable lower order terms. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2495-2516. doi: 10.3934/cpaa.2018119

[5]

Wenxiong Chen, Congming Li. A priori estimate for the Nirenberg problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 225-233. doi: 10.3934/dcdss.2008.1.225

[6]

Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014

[7]

Andrey B. Muravnik. On the Cauchy problem for differential-difference parabolic equations with high-order nonlocal terms of general kind. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 541-561. doi: 10.3934/dcds.2006.16.541

[8]

V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731

[9]

Mauro Garavello, Paola Goatin. The Cauchy problem at a node with buffer. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1915-1938. doi: 10.3934/dcds.2012.32.1915

[10]

Adrien Dekkers, Anna Rozanova-Pierrat. Cauchy problem for the Kuznetsov equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 277-307. doi: 10.3934/dcds.2019012

[11]

Peter Giesl, Holger Wendland. Approximating the basin of attraction of time-periodic ODEs by meshless collocation of a Cauchy problem. Conference Publications, 2009, 2009 (Special) : 259-268. doi: 10.3934/proc.2009.2009.259

[12]

Mahamadi Warma. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2043-2067. doi: 10.3934/cpaa.2015.14.2043

[13]

Matteo Santacesaria. Note on Calderón's inverse problem for measurable conductivities. Inverse Problems & Imaging, 2019, 13 (1) : 149-157. doi: 10.3934/ipi.2019008

[14]

Daniel Grieser. A natural differential operator on conic spaces. Conference Publications, 2011, 2011 (Special) : 568-577. doi: 10.3934/proc.2011.2011.568

[15]

Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control & Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014

[16]

Nguyen H. Sau, Vu N. Phat. LP approach to exponential stabilization of singular linear positive time-delay systems via memory state feedback. Journal of Industrial & Management Optimization, 2018, 14 (2) : 583-596. doi: 10.3934/jimo.2017061

[17]

Zhan-Dong Mei, Jigen Peng, Yang Zhang. On general fractional abstract Cauchy problem. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2753-2772. doi: 10.3934/cpaa.2013.12.2753

[18]

Rudong Zheng, Zhaoyang Yin. The Cauchy problem for a generalized Novikov equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3503-3519. doi: 10.3934/dcds.2017149

[19]

Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409

[20]

Dorina Mitrea and Marius Mitrea. Boundary integral methods for harmonic differential forms in Lipschitz domains. Electronic Research Announcements, 1996, 2: 92-97.

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (26)
  • HTML views (93)
  • Cited by (0)

Other articles
by authors

[Back to Top]