$L^p$-maximal regularity of nonlocal parabolic equations and applications

Zhang, Xicheng

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 4, pp. 573-614.

By using Fourierʼs transform and Fefferman–Steinʼs theorem, we investigate the $L^p$-maximal regularity of nonlocal parabolic and elliptic equations with singular and non-symmetric Lévy operators, and obtain the unique strong solvability of the corresponding nonlocal parabolic and elliptic equations, where the probabilistic representation plays an important role. As a consequence, a characterization for the domain of pseudo-differential operators of Lévy type with singular kernels is given in terms of the Bessel potential spaces. As a byproduct, we also show that a large class of non-symmetric Lévy operators generates an analytic semigroup in $L^p$-spaces. Moreover, as applications, we prove Krylovʼs estimate for stochastic differential equations driven by Cauchy processes (i.e. critical diffusion processes), and also obtain the global well-posedness for a class of quasi-linear first order parabolic systems with critical diffusions. In particular, critical Hamilton–Jacobi equations and multidimensional critical Burgerʼs equations are uniquely solvable and the smooth solutions are obtained.

[1] H. Amann, Linear and Quasilinear Parabolic Problems, vol. I. Abstract Linear Theory, Monogr. Math. vol. 89, Birkhäuser, Boston, MA (1995) | MR | Zbl

[2] D. Applebaum, Lévy Processes and Stochastic Calculus, Camb. Stud. Adv. Math. vol. 93, Cambridge University Press (2004) | MR | Zbl

[3] G. Barles, E. Chasseigne, A. Ciomaga, C. Imbert, Lipschitz regularity of solutions for mixed integro-differential equations, arXiv:1107.3228v1 | MR | Zbl

[4] G. Barles, E. Chasseigne, C. Imbert, Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations, J. Eur. Math. Soc. 13 (2011), 1-26 | EuDML | MR | Zbl

[5] F.E. Benth, K.H. Karlsen, K. Reikvam, Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: a viscosity solution approach, Finance Stoch. 5 no. 3 (2001), 275-303 | MR | Zbl

[6] J. Bergh, J. Löfström, Interpolation Spaces, Grundlehren Math. Wiss. vol. 223, Springer-Verlag (1976) | Zbl

[7] P. Biler, T. Funaki, W.A. Woyczynski, Fractal Burgers equations, J. Differ. Equ. 148 (1998), 9-46 | MR | Zbl

[8] L. Caffarelli, L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Commun. Pure Appl. Math. 62 no. 5 (2009), 597-638 | MR | Zbl

[9] L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasigeostrophic equation, Ann. Math. 171 no. 3 (2010), 1903-1930 | MR | Zbl

[10] P. Constantin, Euler equations, Navier–Stokes equations and turbulence, Mathematical Foundation of Turbulent Flows, Lect. Notes Math. vol. 1871 (2006), 1-43 | MR | Zbl

[11] H. Dong, D. Kim, On $L_p$-estimates for a class of nonlocal elliptic equations, J. Funct. Anal. 262 no. 3 (2012), 1166-1199 | MR | Zbl

[12] J. Droniou, C. Imbert, Fractal first order partial equations, Arch. Ration. Mech. Anal. 182 no. 2 (2006), 299-331 | MR | Zbl

[13] W. Farkas, N. Jacob, R. Schilling, Function spaces related to continuous negative definite functions: ψ-Bessel potential spaces, Diss. Math. 393 (2001) | EuDML | MR

[14] A. Friedman, Stochastic Differential Equations and Applications, vol. 1, Academic Press, New York (1975) | MR | Zbl

[15] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. vol. 44, Springer-Verlag (1983) | MR | Zbl

[16] E. Priola, Pathwise uniqueness for singular SDEs driven by stable processes, Osaka J. Math. 49 no. 2 (2012), 421-447 | MR | Zbl

[17] N. Jacob, Pseudo Differential Operators, Markov Processes, vol. I. Fourier Analysis and Semigroups, Imperical College Press, World Scientific Publishing, Singapore (2001)

[18] N. Jacob, A. Potrykus, J.L. Wu, Solving a nonlinear pseudo-differential equation of Burgers type, Stoch. Dyn. 8 no. 4 (2008), 613-624 | MR | Zbl

[19] O. Kallenberg, Foundations of Modern Probability, Springer-Verlag, New York (1997) | MR | Zbl

[20] A. Kiselev, F. Nazarov, R. Schterenberg, Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ. 5 no. 3 (2008), 211-240 | MR | Zbl

[21] A. Kiselev, F. Nazarov, A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math. 167 (2007), 445-453 | MR | Zbl

[22] N.V. Krylov, A generalization of the Littlewood–Paley inequality and some other results related to stochastic partial differential equations, Ulam Quaterly 2 no. 4 (1994), 16-26 | MR | Zbl

[23] N.V. Krylov, The heat equation in $L_q((0,T),L_p)$-spaces with weights, SIAM J. Math. Anal. 32 no. 5 (2001), 1117-1141 | MR | Zbl

[24] N.V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Grad. Stud. Math. vol. 96, AMS (2008) | MR | Zbl

[25] V.P. Kurenok, A note on $L_2$-estimates for stable integrals with drift, Trans. Am. Math. Soc. 360 (2008), 925-938 | MR | Zbl

[26] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Uraltceva, Linear and Quasi-Linear Parabolic Equations, Nauka, Moscow (1967), AMS (1968) | Zbl

[27] R. Mikulevicius, H. Pragarauskas, On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces, Lith. Math. J. 32 no. 2 (1992) | MR | Zbl

[28] R. Mikulevicius, H. Pragarauskas, On $L_p$ theory for Zakai equation with discontinuous observation process, arXiv:1012.5816v1 | Zbl

[29] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press (1999) | MR | Zbl

[30] L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana Univ. Math. J. 55 no. 3 (2006), 1155-1174 | MR | Zbl

[31] L. Silvestre, On the differentiability of the solution to the Hamilton–Jacobi equation with critical fractional diffusion, Adv. Math. 226 no. 2 (2011), 2020-2039 | MR | Zbl

[32] L. Silvestre, Hölder estimates for advection fractional-diffusion equations, arXiv:1009.5723v2 (2011), Ann. Sc. Norm. Super. Pisa, Cl. Sci. (2011) | MR

[33] L. Silvestre, On the differentiability of the solution to an equation with drift and fractional-diffusion, arXiv:1012.2401 (2011), Indiana Univ. Math. J. (2011) | MR

[34] H.M. Soner, Optimal control with state-space constraint II, SIAM J. Control Optim. 24 no. 6 (2006), 1110-1122 | MR | Zbl

[35] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ (1970) | MR | Zbl

[36] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press (1993) | MR

[37] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publishing Company, Amsterdam (1978) | MR | Zbl

[38] W.A. Woyczyński, Lévy processes in the physical sciences, Lévy Processes, Birkhäuser, Boston, MA (2001), 241-266 | MR | Zbl

[39] X. Zhang, Stochastic differential equations with Sobolev drifts and driven by α-stable processes, Ann. Inst. Henri Poincaré B, Probab. Stat., http://dx.doi.org/10.1214/12-AIHP476. | EuDML | Numdam | MR

[40] X. Zhang, Stochastic functional differential equations driven by Lévy processes and quasi-linear partial integro-differential equations, Ann. Appl. Probab., http://dx.doi.org/10.1214/12-AAP851. | MR