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Extra regularity for parabolic equations with drift terms

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Abstract

We consider a parabolic equation with a drift term Δu+b∇u−u t =0. Under some natural conditions on the vector valued function b, we prove that solutions possess extra regularity and better qualitative behavior than those provided by standard theory. For example, we show that the fundamental solution has global Gaussian upper bound even for some b with a large singularity in the form of c/|x|. We also show that bounded solutions are Hölder continuous when |b|2 is just form bounded and divergence free, a case where not even continuity is expected. A Liouville type theorem is also proven.

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References

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

    MATH  Google Scholar 

  2. Aizenman, M., Simon, B.: Brownian motion and Harnack inequality for Schrdinger operators. Commun. Pure Appl. Math. 35, 209–273 (1982)

    MathSciNet  MATH  Google Scholar 

  3. Chen, Z.Q., Zhao, Z.: Diffusion processes and second order elliptic operators with singular coefficients for lower order terms. Math. Ann. 302, 323–357 (1995)

    MathSciNet  MATH  Google Scholar 

  4. Cranston, M., Zhao, Z.: Conditional transformation of drift formula and potential theory for ½Δ+b()∇, Commun. Math. Phys. 112, 613–625 (1987)

    MathSciNet  MATH  Google Scholar 

  5. Davies, E.B.: Heat kernels and spectral theory. Cambridge Univ. Press, Cambridge, 1989

  6. Fabes, E.B., Stroock, D.W.: A new proof of Moser’s parabolic Harnack inequality using the old idea of Nash. Arch. Rational Mech. Anal. 96, 327–338 (1986)

    MathSciNet  MATH  Google Scholar 

  7. Gerhard, W.D.: The probabilistic solution of the Dirichlet problem for \(\frac12\Delta +\langle a,\nabla\rangle+b\) with singular coefficients. J. Theoret. Probab. 5, 503–520 (1992)

    MathSciNet  MATH  Google Scholar 

  8. Grigoryan, A., Hansen, W.: A Liouville property for Schrdinger operators. Math. Ann. 312, 659–716 (1998)

    MathSciNet  MATH  Google Scholar 

  9. Kato, T.: Perturbation Theory for Linear Operators. Springer, 1980

  10. Kovalenko, V.F., Semenov, Yu.A.: C o -semigroups in the spaces L p(R d) and \(\widehat C(R^ d)\) generated by Δ+b. ∇. (Russian) Teor. Veroyatnost. i Primenen. 35, 449–458 (1990); translation in Theory Probab. Appl. 35, 443–453 (1990)

    MATH  Google Scholar 

  11. Kondratiev, V., Liskevich, V., Sobol, Z., Us, O.: Heat kernel estimates with applications to an exterior problem for a class of semi-linear equation. J. London Math. Soc., to appear

  12. Kurata, K.: An estimate on the heat kernel of magnetic Schrdinger operators and uniformly elliptic operators with non-negative potentials. J. London Math. Soc. 62, 885–903 (2000)

    Google Scholar 

  13. Liskevich, V., Semenov, Yu.: Estimates for fundamental solutions of second-order parabolic equations. J. London Math. Soc. 62 (2), 521–543 (2000)

    Article  MATH  Google Scholar 

  14. Liskevich, V., Sobol, Z.: Estimates of integral kernels for semigroups associated with second order elliptic operators with singular coefficients. Potential Anal. 18, 359–390 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Moser, J.: On Harnack’s theorem for elliptic differential equations. Commun. Pure Appl. Math.14, 577–591 (1961)

    Google Scholar 

  16. Norris, J.R.: Large time behavior of heat flow: global estimates and exact asymptotics. Arch. Rat. Mech. Anal. 140, 161–195 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Osada, H.: Diffusion processes with generators of generalized divergence form. J. Math. Kyoto Univ. 72, 597–619 (1987)

    MathSciNet  Google Scholar 

  18. Ouhabaz, E.-M.: L ∞-contractivity of semigroups generated by sectorial forms. J. London Math. Soc 46, 529–542 (1992)

    MathSciNet  MATH  Google Scholar 

  19. Semenov, Yu.A.: Stability of L p-spectrum of generalized Schrödinger operators and equivalence of Green’s functions. IMRN 12, 573–593 (1997)

    Article  MathSciNet  Google Scholar 

  20. Semenov, Yu.A.: Hölder continuity of bounded solutions of parabolic equations. Preprint 1998

  21. Simon, B.: Schrödinger semigroups. Bull. AMS 7, 447–526 (1982)

    MathSciNet  MATH  Google Scholar 

  22. Simon, B.: Large time behavior of the L p norm of Schrdinger semigroups. J. Funct. Anal. 40, 66–83 (1981)

    MathSciNet  MATH  Google Scholar 

  23. Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. (French) Ann. Inst. Fourier (Grenoble) 15 (1), 189–258 (1965)

    MATH  Google Scholar 

  24. Voigt, J.: Absorption semigroups, their generators and Schrödinger semigroups. J. Funct. Anal. 67, 167–205 (1986)

    MathSciNet  MATH  Google Scholar 

  25. Zhang, Qi S.: A Harnack inequality for the equation ∇(a∇ u)+b∇ u=0, when | b| ∈K n+1. Manuscripta Math. 89, 61–77 (1996)

    MathSciNet  MATH  Google Scholar 

  26. Zhang, Qi S.: An optimal parabolic estimate and its applications in prescribing scalar curvature on some open manifolds with Ricci ≥0. Math Ann. 316, 703–731 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang, Qi S.: On a parabolic with a singular lower order term. II. Indiana U. Math. J. 46, 989–1020 (1997)

    MATH  Google Scholar 

  28. Zhang, Qi S.: Gaussian bounds for the fundamental solutions of ∇ (A∇ u)+B∇ u-u t =0. Manuscripta Math. 93, 381–390 (1997)

    MathSciNet  MATH  Google Scholar 

  29. Zhang, Qi S., Zhao, Z.: Estimates of global bounds for some Schrdinger heat kernels on manifolds. Illinois J. Math. 44, 556–557 (2000)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    Vitali Liskevich

  2. Department of Mathematics, University of California, Riverside, CA, 92521, USA

    Qi S. Zhang

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  1. Vitali Liskevich
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  2. Qi S. Zhang
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Correspondence to Vitali Liskevich.

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Liskevich, V., Zhang, Q. Extra regularity for parabolic equations with drift terms. manuscripta math. 113, 191–209 (2004). https://doi.org/10.1007/s00229-003-0433-x

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