doi: 10.3934/cpaa.2021164
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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 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 & Applied Analysis, 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.  Google Scholar

[2]

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

[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.  Google Scholar

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

[5]

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

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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.   Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[26]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[5]

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

[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.   Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[26]

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

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