January  2022, 21(1): 1-21. doi: 10.3934/cpaa.2021164

Green's function for second order parabolic equations with singular lower order coefficients

1. 

Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-gu, Seoul 03722, Republic of Korea

2. 

Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076

* Corresponding author

Received  June 2021 Revised  August 2021 Published  January 2022 Early access  September 2021

Fund Project: S. Kim was partially supported by the National Research Foundation of Korea under agreements NRF-2019R1A2C2002724 and NRF-20151009350

We construct Green's functions for second order parabolic operators of the form $ Pu = \partial_t u-{\rm div}({\mathbf A} \nabla u+ {\mathbf b}u)+ {\mathbf c} \cdot \nabla u+du $ in $ (-\infty, \infty) \times \Omega $, where $ \Omega $ is an open connected set in $ \mathbb{R}^n $. It is not necessary that $ \Omega $ to be bounded and $ \Omega = \mathbb{R}^n $ is not excluded. We assume that the leading coefficients $ \mathbf A $ are bounded and measurable and the lower order coefficients $ \boldsymbol{b} $, $ \boldsymbol{c} $, and $ d $ belong to critical mixed norm Lebesgue spaces and satisfy the conditions $ d-{\rm div} \boldsymbol{b} \ge 0 $ and $ {\rm div}(\boldsymbol{b}-\boldsymbol{c}) \ge 0 $. We show that the Green's function has the Gaussian bound in the entire $ (-\infty, \infty) \times \Omega $.

Citation: Seick Kim, Longjuan Xu. Green's function for second order parabolic equations with singular lower order coefficients. Communications on Pure and Applied Analysis, 2022, 21 (1) : 1-21. doi: 10.3934/cpaa.2021164
References:
[1]

D. G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc., 73 (1967), 890-896.  doi: 10.1090/S0002-9904-1967-11830-5.

[2]

D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1968), 607-694. 

[3]

S. ChoH. Dong and S. Kim, On the Green's matrices of strongly parabolic systems of second order, Indiana Univ. Math. J., 57 (2008), 1633-1677.  doi: 10.1512/iumj.2008.57.3293.

[4]

R. CoifmanP.-L. LionsY. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286. 

[5]

E. B. Davies, Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math., 109 (1987), 319-333.  doi: 10.2307/2374577.

[6]

E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, (Italian), Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., 3 (1957), 25-43. 

[7]

E. DiBenedetto, Partial Differential Equations, Second edition, Birkhäuser Boston, Ltd., Boston, MA, 2010. doi: 10.1007/978-0-8176-4552-6.

[8]

H. Dong and S. Kim, Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains, Trans. Amer. Math. Soc., 361 (2009), 3303-3323.  doi: 10.1090/S0002-9947-09-04805-3.

[9]

H. Dong and S. Kim, Fundamental solutions for second-order parabolic systems with drift terms, Proc. Amer. Math. Soc., 146 (2018), 3019-3029.  doi: 10.1090/proc/14004.

[10]

E. B. Fabes and D. W. Stroock, A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal., 96 (1986), 327-338.  doi: 10.1007/BF00251802.

[11]

M. Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1993.

[12]

M. Grüter and K.-O. Widman, The Green function for uniformly elliptic equations, Manuscripta Math., 37 (1982), 303-342.  doi: 10.1007/BF01166225.

[13]

S. Hofmann and S. Kim, Gaussian estimates for fundamental solutions to certain parabolic systems, Publ. Mat., 48 (2004), 481-496.  doi: 10.5565/PUBLMAT_48204_10.

[14]

S. Hofmann and S. Kim, The Green function estimates for strongly elliptic systems of second order, Manuscripta Math., 124 (2007), 139-172.  doi: 10.1007/s00229-007-0107-1.

[15]

S. Kim and G. Sakellaris, Green's function for second order elliptic equations with singular lower order coefficients, Commun. Partial Differ. Equ., 44 (2019), 228-270.  doi: 10.1080/03605302.2018.1543318.

[16]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.  doi: 10.1006/aima.2000.1937.

[17]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, (Russian), Translated from the Russian by S. Smith. Translations of Mathematical Monographs, American Mathematical Society, Providence, R.I. 1968.

[18]

W. LittmanG. Stampacchia and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 17 (1963), 43-77. 

[19]

J. Moser, A Harnack inequality for parabolic differential equations, Commun. Pure Appl. Math., 17 (1964), 101-134.  doi: 10.1002/cpa.3160170106.

[20]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954.  doi: 10.2307/2372841.

[21]

A. I. Nazarov and N. N. Ural'tseva, The Harnack inequality and related properties of solutions of elliptic and parabolic equations with divergence-free lower-order coefficients, (Russian), Algebra i Analiz, 23 (2011), 136–168; translation in St. Petersburg Math. J. 23 (2012), 93–115. doi: 10.1090/S1061-0022-2011-01188-4.

[22]

Z. Qian and G. Xi, Parabolic equations with singular divergence-free drift vector fields, J. Lond. Math. Soc., 100 (2019), 17-40.  doi: 10.1112/jlms.12202.

[23]

Z. Qian and G. Xi, Parabolic equations with divergence-free drift in space $L^{l}_{t}L^{q}_{x}$, Indiana Univ. Math. J., 68 (2019), 761-797.  doi: 10.1512/iumj.2019.68.7685.

[24]

Y. Semenov, Regularity theorems for parabolic equations, J. Funct. Anal., 231 (2006), 375-417.  doi: 10.1016/j.jfa.2005.04.001.

[25]

G. SereginL. SilvestreV. Šverák and A. Zlatoš, On divergence-free drifts, J. Differ. Equ., 252 (2012), 505-540.  doi: 10.1016/j.jde.2011.08.039.

[26]

Q. Zhang, A strong regularity result for parabolic equations, Commun. Math. Phys., 244 (2004), 245-260.  doi: 10.1007/s00220-003-0974-6.

show all references

References:
[1]

D. G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc., 73 (1967), 890-896.  doi: 10.1090/S0002-9904-1967-11830-5.

[2]

D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1968), 607-694. 

[3]

S. ChoH. Dong and S. Kim, On the Green's matrices of strongly parabolic systems of second order, Indiana Univ. Math. J., 57 (2008), 1633-1677.  doi: 10.1512/iumj.2008.57.3293.

[4]

R. CoifmanP.-L. LionsY. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl., 72 (1993), 247-286. 

[5]

E. B. Davies, Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math., 109 (1987), 319-333.  doi: 10.2307/2374577.

[6]

E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, (Italian), Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., 3 (1957), 25-43. 

[7]

E. DiBenedetto, Partial Differential Equations, Second edition, Birkhäuser Boston, Ltd., Boston, MA, 2010. doi: 10.1007/978-0-8176-4552-6.

[8]

H. Dong and S. Kim, Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains, Trans. Amer. Math. Soc., 361 (2009), 3303-3323.  doi: 10.1090/S0002-9947-09-04805-3.

[9]

H. Dong and S. Kim, Fundamental solutions for second-order parabolic systems with drift terms, Proc. Amer. Math. Soc., 146 (2018), 3019-3029.  doi: 10.1090/proc/14004.

[10]

E. B. Fabes and D. W. Stroock, A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal., 96 (1986), 327-338.  doi: 10.1007/BF00251802.

[11]

M. Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1993.

[12]

M. Grüter and K.-O. Widman, The Green function for uniformly elliptic equations, Manuscripta Math., 37 (1982), 303-342.  doi: 10.1007/BF01166225.

[13]

S. Hofmann and S. Kim, Gaussian estimates for fundamental solutions to certain parabolic systems, Publ. Mat., 48 (2004), 481-496.  doi: 10.5565/PUBLMAT_48204_10.

[14]

S. Hofmann and S. Kim, The Green function estimates for strongly elliptic systems of second order, Manuscripta Math., 124 (2007), 139-172.  doi: 10.1007/s00229-007-0107-1.

[15]

S. Kim and G. Sakellaris, Green's function for second order elliptic equations with singular lower order coefficients, Commun. Partial Differ. Equ., 44 (2019), 228-270.  doi: 10.1080/03605302.2018.1543318.

[16]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.  doi: 10.1006/aima.2000.1937.

[17]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, (Russian), Translated from the Russian by S. Smith. Translations of Mathematical Monographs, American Mathematical Society, Providence, R.I. 1968.

[18]

W. LittmanG. Stampacchia and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 17 (1963), 43-77. 

[19]

J. Moser, A Harnack inequality for parabolic differential equations, Commun. Pure Appl. Math., 17 (1964), 101-134.  doi: 10.1002/cpa.3160170106.

[20]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954.  doi: 10.2307/2372841.

[21]

A. I. Nazarov and N. N. Ural'tseva, The Harnack inequality and related properties of solutions of elliptic and parabolic equations with divergence-free lower-order coefficients, (Russian), Algebra i Analiz, 23 (2011), 136–168; translation in St. Petersburg Math. J. 23 (2012), 93–115. doi: 10.1090/S1061-0022-2011-01188-4.

[22]

Z. Qian and G. Xi, Parabolic equations with singular divergence-free drift vector fields, J. Lond. Math. Soc., 100 (2019), 17-40.  doi: 10.1112/jlms.12202.

[23]

Z. Qian and G. Xi, Parabolic equations with divergence-free drift in space $L^{l}_{t}L^{q}_{x}$, Indiana Univ. Math. J., 68 (2019), 761-797.  doi: 10.1512/iumj.2019.68.7685.

[24]

Y. Semenov, Regularity theorems for parabolic equations, J. Funct. Anal., 231 (2006), 375-417.  doi: 10.1016/j.jfa.2005.04.001.

[25]

G. SereginL. SilvestreV. Šverák and A. Zlatoš, On divergence-free drifts, J. Differ. Equ., 252 (2012), 505-540.  doi: 10.1016/j.jde.2011.08.039.

[26]

Q. Zhang, A strong regularity result for parabolic equations, Commun. Math. Phys., 244 (2004), 245-260.  doi: 10.1007/s00220-003-0974-6.

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