Abstract
We review results on the existence and uniqueness for a surface growth model with or without space–time white noise. If the surface is a graph, then this model has striking similarities to the three dimensional Navier-Stokes equations in terms of energy estimates and scaling properties, and in both models the question of uniqueness of global weak solutions remains open.
In the physically relevant dimension \(d=2\) and with the physically relevant space–time white noise driving the equation, the direct fixed-point argument for mild solutions fails, as there is not sufficient regularity for the stochastic forcing. The situation is the simplest case where the method of regularity structures introduced by Martin Hairer can be applied, although we follow here a significantly simpler approach to highlight the key problems. Using spectral Galerkin method or any other type of regularization of the noise, one can give a rigorous meaning to the stochastic PDE and show existence and uniqueness of local solutions in that setting. Moreover, several types of regularization seem to yield all the same solution.
We finally comment briefly on possible blow up phenomena and show with a simple argument that many complex-valued solutions actually do blow up in finite time. This shows that energy estimates alone are not enough to verify global uniqueness of solutions. Results in this direction are known already for the 3D-Navier Stokes by Li and Sinai, treating complex valued solutions, and more recently by Tao by constructing an equation of Navier-Stokes type with blow up.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Academic Press, San Diego (2003)
Agélas, L.: Global regularity of solutions of equation modeling epitaxy thin film growth in \(\mathbb{R}^{d}\), \(d=1,2\). J. Evol. Equ. 15(1), 89–106 (2015). doi:10.1007/s00028-014-0250-6
Barabasi, A.L., Stanley, H.E.: Fractal Concepts in Surface Growth. Cambridge University Press, Cambridge (1995)
Barbato, D., Morandin, F., Romito, M.: Smooth solutions for the dyadic model. Nonlinearity 24(11), 3083–3097 (2011) (featured article)
Barbato, D., Morandin, F., Romito, M.: Global regularity for a logarithmically supercritical hyperdissipative dyadic equation. Dyn. Partial Differ. Equ. 11(1), 39–52 (2014)
Barbato, D., Morandin, F., Romito, M.: Global regularity for a slightly supercritical hyperdissipative Navier-Stokes system. Anal. PDE 7(8), 2009–2027 (2014)
Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94(1), 61–66 (1984)
Blömker, D.: Nonhomogeneous noise and \(Q\)-Wiener processes on bounded domains. Stoch. Anal. Appl. 23(2), 255–273 (2005)
Blömker, D., Flandoli, F., Romito, M.: Markovianity and ergodicity for a surface growth PDE. Ann. Probab. 37(1), 275–313 (2009)
Blömker, D., Gugg, C.: Thin-film-growth-models: on local solutions. In: Albeverio, S., et al. (eds.) Recent Developments in Stochastic Analysis and Related Topics. Proceedings of the First Sino-German Conference on Stochastic Analysis (A Satellite Conference of ICM 2002), Beijing, China, August 29–September 3, 2002, pp. 66–77. World Scientific, River Edge (2004)
Blömker, D., Gugg, C.: On the existence of solutions for amorphous molecular beam epitaxy. Nonlinear Anal., Real World Appl. 3(1), 61–73 (2002)
Blömker, D., Gugg, C., Raible, M.: Thin-film-growth models: roughness and correlation functions. Eur. J. Appl. Math. 13(4), 385–402 (2002)
Blömker, D., Hairer, M.: Stationary solutions for a model of amorphous thin-film growth. Stoch. Anal. Appl. 22(4), 903–922 (2004)
Blömker, D., Maier-Paape, S., Wanner, T.: Roughness in surface growth equations. Interfaces Free Bound. 3(4), 465–484 (2001)
Blömker, D., Nolde, C., Robinson, J.C.: Rigorous numerical verification of uniqueness and smoothness in a surface growth model. J. Math. Anal. Appl. 429(1), 311–325 (2015)
Blömker, D., Romito, M.: Regularity and blow up in a surface growth model. Dyn. Partial Differ. Equ. 6(3), 227–252 (2009)
Blömker, D., Romito, M.: Local existence and uniqueness in the largest critical space for a surface growth model. Nonlinear Differ. Equ. Appl. 19(3), 365–381 (2012)
Blömker, D., Romito, M.: Local existence and uniqueness for a two-dimensional surface growth equation with space-time white noise. Stoch. Anal. Appl. 31(6), 1049–1076 (2013)
Blömker, D., Romito, M., Tribe, R.: A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees. Ann. Inst. Henri Poincaré Probab. Stat. 43(2), 175–192 (2007)
Bogachev, V.I.: Gaussian Measures. Mathematical Surveys and Monographs, vol. 62. Am. Math. Soc., Providence (1998)
Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun. Pure Appl. Math. 35(6), 771–831 (1982)
Cannone, M.: Harmonic analysis tools for solving the incompressible Navier-Stokes equations. In: Friedlander, S., Serre, D. (eds.) Handbook of Mathematical Fluid Dynamics, vol. 3, pp. 161–244. North-Holland, Amsterdam (2004)
Castro, M., Cuerno, R., Vázquez, L., Gago, R.: Self-organized ordering of nanostructures produced by ion-beam sputtering. Phys. Rev. Lett. 94, 016102 (2005)
Chernyshenko, S.I., Constantin, P., Robinson, J.C., Titi, E.S.: A posteriori regularity of the three-dimensional Navier-Stokes equations from numerical computations. J. Math. Phys. 48(6), 065204 (2007), 15 pp.
Chow, P.-L.: Stochastic Partial Differential Equations. Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series. Chapman & Hall/CRC, Boca Raton (2007)
Collet, P., Eckmann, J.-P., Epstein, H., Stubbe, J.: A global attracting set for the Kuramoto-Sivashinsky equation. Commun. Math. Phys. 152(1), 203–214 (1993)
Cuerno, R., Barabási, A.-L.: Dynamic scaling of ion-sputtered surfaces. Phys. Rev. Lett. 74, 4746–4749 (1995)
Da Prato, G., Debussche, A.: Stochastic Cahn-Hilliard equation. Nonlinear Anal., Theory Methods Appl. 26(2), 1–263 (1996)
Da Prato, G., Debussche, A.: Two-dimensional Navier-Stokes equations driven by a space-time white noise. J. Funct. Anal. 196(1), 180–210 (2002)
Da Prato, G., Debussche, A.: Ergodicity for the 3D stochastic Navier-Stokes equations. J. Math. Pures Appl. (9) 82(8), 877–947 (2003)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications, vol. 44. Cambridge University Press, Cambridge (1992)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, 2nd edn. Encyclopedia of Mathematics and Its Applications, vol. 152. Cambridge University Press, Cambridge (2014)
Dashti, M., Robinson, J.C.: An a posteriori condition on the numerical approximations of the Navier-Stokes equations for the existence of a strong solution. SIAM J. Numer. Anal. 46(6), 3136–3150 (2008)
Debussche, A., Odasso, C.: Markov solutions for the 3D stochastic Navier-Stokes equations with state dependent noise. J. Evol. Equ. 6(2), 305–324 (2006)
Escudero, C., Gazzola, F., Peral, I.: Global existence versus blow-up results for a fourth order parabolic PDE involving the Hessian. J. Math. Pures Appl. 103(4), 924–957 (2015). doi:10.1016/j.matpur.2014.09.007
Escudero, C., Hakl, R., Peral, I., Torres, P.J.: On radial stationary solutions to a model of non-equilibrium growth. Eur. J. Appl. Math. 24(3), 437–453 (2013)
Es-Sarhir, A., Stannat, W.: Improved moment estimates for invariant measures of semilinear diffusions in Hilbert spaces and applications. J. Funct. Anal. 259(5), 1248–1272 (2010)
Flandoli, F.: An introduction to 3D stochastic fluid dynamics. In: Proceedings of the CIME Course on SPDE in Hydrodynamics: Recent Progress and Prospects. Lecture Notes in Math., vol. 1942, pp. 51–150. Springer, Berlin (2008)
Flandoli, F., Romito, M.: Statistically stationary solutions to the 3-D Navier-Stokes equations do not show singularities. Electron. J. Probab. 6(5) (2001), 15 pp. (electronic)
Flandoli, F., Romito, M.: Markov selections and their regularity for the three-dimensional stochastic Navier-Stokes equations. C. R. Math. Acad. Sci. Paris, Ser. I 343, 47–50 (2006)
Flandoli, F., Romito, M.: Regularity of transition semigroups associated to a 3D stochastic Navier-Stokes equation. In: Baxendale, P.H., Lototski, S.V. (eds.) Stochastic Differential Equations: Theory and Applications. Interdiscip. Math. Sci., vol. 2, pp. 263–280. World Scientific, Singapore (2007)
Flandoli, F., Romito, M.: Markov selections for the three-dimensional stochastic Navier-Stokes equations. Probab. Theory Relat. Fields 140(3–4), 407–458 (2008)
Fujita, H., Kato, T.: On the Navier-Stokes initial value problem. I. Arch. Ration. Mech. Anal. 16, 269–315 (1964)
Frisch, T., Verga, A.: Effect of step stiffness and diffusion anisotropy on the meandering of a growing vicinal surface. Phys. Rev. Lett. 96, 166104 (2006)
Friz, P., Hairer, M.: A Course on Rough Paths: With an Introduction to Regularity Structures. Springer, Berlin (2014)
Germain, P., Pavlović, N., Staffilani, G.: Regularity of solutions to the Navier-Stokes equations evolving from small data in \(\mbox{BMO}^{-1}\). Int. Math. Res. Not. 21 (2007), Art. ID rnm087, 35 pp.
Goldys, B., Röckner, M., Zhang, X.: Martingale solutions and Markov selections for stochastic partial differential equations. Stoch. Process. Appl. 119(5), 1725–1764 (2009)
Gubinelli, M., Imkeller, P., Perkowski, N.: Paraproducts, rough paths and controlled distributions (2014). arXiv:1210.2684v3 [math.PR]
Hairer, M.: An Introduction to Stochastic PDEs. Lecture Notes (2009). arXiv:0907.4178 [math.PR]
Hairer, M.: Solving the KPZ equation. Ann. Math. 178(2), 559–664 (2013)
Hairer, M.: A theory of regularity structures. Invent. Math. 198(2), 269–504 (2014)
Hairer, M., Weber, H.: Rough Burgers-like equations with multiplicative noise. Probab. Theory Relat. Fields 155(1–2), 71–126 (2013)
Hairer, M., Weber, H.: Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions. Ann. Fac. Sci. Toulouse 24(1), 55–92 (2015). doi:10.5802/afst.1442
Halpin-Healy, T., Zhang, Y.C.: Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Phys. Rep. 254, 215–414 (1995)
Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)
Hoppe, R., Linz, S., Litvinov, W.: On solutions of certain classes of evolution equations for surface morphologies. Nonlinear Phenom. Complex Syst. 6, 582–591 (2003)
Hoppe, R., Nash, E.: A combined spectral element/finite element approach to the numerical solution of a nonlinear evolution equation describing amorphous surface growth of thin films. J. Numer. Math. 100(2), 127–136 (2002)
Hoppe, R.H., Nash, E.: Numerical solution of a nonlinear evolution equation describing amorphous surface growth of thin films. In: Feistauer, M., et al. (eds.) Numerical Mathematics and Advanced Applications. Proceedings of ENUMATH 2003, the 5th European Conference on Numerical Mathematics and Advanced Applications, Prague, Czech Republic, August 18–22, 2003, pp. 440–448. Springer, Berlin (2004)
Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889–892 (1986)
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Graduate Texts in Mathematics, vol. 113. Springer, Berlin (1991)
Kato, T.: Nonstationary flows of viscous and ideal fluids in \(\mathbf{R}^{3}\). J. Funct. Anal. 9, 296–305 (1972)
Koch, H., Tataru, D.: Well-posedness for the Navier-Stokes equations. Adv. Math. 157(1), 22–35 (2001)
Koch, H., Lamm, T.: Geometric flows with rough initial data. Asian J. Math. 16(2), 209–235 (2012)
Krylov, N.V.: The selection of a Markov process from a Markov system of processes, and the construction of quasi-diffusion processes. Izv. Akad. Nauk SSSR, Ser. Mat. 37, 691–708 (1973) (Russian)
Lai, Z.-W., Das Sarma, S.: Kinetic growth with surface relaxation: continuum versus atomistic models. Phys. Rev. Lett. 66, 2348–2351 (1991)
Lemarié-Rieusset, P.G.: Recent Developments in the Navier-Stokes Problem. Chapman & Hall/CRC Research Notes in Mathematics Series, vol. 431. CRC Press, Boca Raton (2002)
Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63(1), 193–248 (1934)
Li, D., Sinai, Ya.G.: Blow ups of complex solutions of the 3D Navier-Stokes system and renormalization group method. J. Eur. Math. Soc. 10(2), 267–313 (2008)
Li, D., Sinai, Ya.G.: Complex singularities of solutions of some 1D hydrodynamic models. Physica D 237, 1945–1950 (2008)
Liu, K.: Stability of Infinite Dimensional Stochastic Differential Equations with Applications. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Math., vol. 135. CRC Press, Boca Raton (2006)
Liu, W., Röckner, M.: Local and global well-posedness of SPDE with generalized coercivity conditions. J. Differ. Equ. 254(2), 725–755 (2013)
Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser Verlag, Basel (1995)
Morosi, C., Pizzocchero, L.: On approximate solutions of semilinear evolution equations. II: Generalizations, and applications to Navier-Stokes equations. Rev. Math. Phys. 20(6), 625–706 (2008)
Morosi, C., Pizzocchero, L.: An \(H^{1}\) setting for the Navier-Stokes equations: quantitative estimates. Nonlinear Anal. 74(6), 2398–2414 (2011)
Morosi, C., Pizzocchero, L.: On approximate solutions of the incompressible Euler and Navier-Stokes equations. Nonlinear Anal. 75(4), 2209–2235 (2012)
Mörters, P., Peres, Y.: Brownian Motion. With an Appendix by Oded Schramm and Wendelin Werner. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 30. Cambridge University Press, Cambridge (2010)
Muñoz-García, J., Cuerno, R., Castro, M.: Coupling of morphology to surface transport in ion-beam-irradiated surfaces: normal incidence and rotating targets. J. Phys. Condens. Matter 21(22), 224020 (2009)
Muñoz-García, J., Gago, R., Vázquez, L., Sánchez-García, J.A., Cuerno, R.: Observation and modeling of interrupted pattern coarsening: Surface nanostructuring by ion erosion. Phys. Rev. Lett. 104, 026101 (2010)
Nelson, E.: The free Markoff field. J. Funct. Anal. 12, 211–227 (1973)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
Politi, P., ben-Avraham, D.: From the conserved Kuramoto-Sivashinsky equation to a coalescing particles model. Physica D 238, 156–161 (2009)
Prévot, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics. Springer, Berlin (2007)
Prodi, G.: Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. 48(4), 173–182 (1959) (Italian)
Raible, M., Linz, S.J., Hänggi, P.: Amorphous thin film growth: minimal deposition equation. Phys. Rev. E 62, 1691–1705 (2000)
Raible, M., Linz, S., Hänggi, P.: Amorphous thin film growth: modeling and pattern formation. Adv. Solid State Phys. 41, 391–403 (2001)
Raible, M., Mayr, S.G., Linz, S.J., Moske, M., Hänggi, P., Samwer, K.: Amorphous thin film growth: theory compared with experiment. Europhys. Lett. 50, 61–67 (2000)
Robinson, J.C., Sadowski, W.: Decay of weak solutions and the singular set of the three-dimensional Navier-Stokes equations. Nonlinearity 20(5), 1185–1191 (2007)
Röckner, M., Zhu, R., Zhu, X.: Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise. Stoch. Process. Appl. 124(5), 1974–2002 (2014)
Romito, M.: Analysis of equilibrium states of Markov solutions to the 3D Navier-Stokes equations driven by additive noise. J. Stat. Phys. 131(3), 415–444 (2008)
Romito, M.: Existence of martingale and stationary suitable weak solutions for a stochastic Navier-Stokes system. Stochastics 82(3), 327–337 (2010)
Romito, M.: An almost sure energy inequality for Markov solutions to the 3D Navier-Stokes equations. In: Stochastic Partial Differential Equations and Applications. Quad. Mat., vol. 25, pp. 243–255. Dept. Math., Seconda Univ. Napoli, Caserta (2010)
Romito, M.: Critical strong Feller regularity for Markov solutions to the Navier–Stokes equations. J. Math. Anal. Appl. 384(1), 115–129 (2011)
Romito, M.: The Martingale problem for Markov solutions to the Navier-Stokes equations. In: Seminar on Stochastic Analysis, Random Fields and Applications VI. Progr. Probab., vol. 63, pp. 227–244. Birkhäuser/Springer Basel AG, Basel (2011)
Romito, M.: Uniqueness and blow-up for a stochastic viscous dyadic model. Probab. Theory Relat. Fields 158(3–4), 895–924 (2014)
Romito, M., Xu, L.: Ergodicity of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noise. Stoch. Process. Appl. 121(4), 673–700 (2011)
Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. de Gruyter, Berlin (1996)
Scheffer, V.: Turbulence and Hausdorff Dimension, Proc. Conf, Univ. Paris-Sud. Orsay, 1975. Lecture Notes in Math., vol. 565, pp. 174–183. Springer, Berlin (1976)
Schilling, R., Partzsch, L.: Brownian Motion. An Introduction to Stochastic Processes. De Gruyter Textbook, de Gruyter, Berlin (2014)
Serrin, J.: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 9, 187–195 (1962)
Siegert, M., Plischke, M.: Solid-on-solid models of molecular-beam epitaxy. Phys. Rev. E 50, 917–931 (1994)
Simon, B.: The \(P(\phi )_{2}\) Euclidean (Quantum) Field Theory. Princeton Series in Physics. Princeton University Press, Princeton (1974)
Stannat, W.: Stochastic partial differential equations: Kolmogorov operators and invariant measures. Jahresber. Dtsch. Math.-Ver. 113(2), 81–109 (2011)
Stein, O., Winkler, M.: Amorphous molecular beam epitaxy: global solutions and absorbing sets. Eur. J. Appl. Math. 16(6), 767–798 (2005)
Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer, Berlin (1979)
Sun, T., Guo, H., Grant, M.: Dynamics of driven interfaces with a conservation law. Phys. Rev. A 40, R6763–R6766 (1989)
Swann, H.S.G.: The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in \(R_{3}\). Trans. Am. Math. Soc. 157, 373–397 (1971)
Tao, T.: Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation. Anal. PDE 2(3), 361–366 (2009)
Tao, T.: Finite time blowup for an averaged three-dimensional Navier-Stokes equation. J. Am. Math. Soc. (2015). doi:10.1090/jams/838
Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and Its Applications, vol. 2. North-Holland, Amsterdam (1984)
Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68. Springer, New York (1988)
van Neerven, J.: Stochastic Evolution Equations. ISEM Lecture Notes (2007/2008)
Villain, J.: Continuum models of crystal growth from atomic beams with and without desorption. J. Phys. I France 1, 19–42 (1991)
Walsh, J.B.: An introduction to stochastic partial differential equations. In: École d’été de probabilités de Saint-Flour, XIV—1984. Lecture Notes in Math., vol. 1180, pp. 265–439. Springer, Berlin (1986)
Wang, C.: Well-posedness for the heat flow of biharmonic maps with rough initial data. J. Geom. Anal. 22(1), 223–243 (2012)
Winkler, M.: Global solutions in higher dimensions to a fourth-order parabolic equation modeling epitaxial thin-film growth. Z. Angew. Math. Phys. 62(4), 575–608 (2011)
Zhu, R., Zhu, X.: Three-dimensional Navier-Stokes equations driven by space-time white noise. J. Differ. Equ. 259(9), 4443–4508 (2015). doi:10.1016/j.jde.2015.06.002
Acknowledgements
We thank Christian Nolde for the code used in the simulations presented in Fig. 1. We also like to thank Franco Flandoli for joint work on the inviscid surface growth and valuable comments that improved the manuscript, Martin Hairer for many interesting discussions, and Michael Winkler for discussions on possible blow up.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Blömker, D., Romito, M. Stochastic PDEs and Lack of Regularity. Jahresber. Dtsch. Math. Ver. 117, 233–286 (2015). https://doi.org/10.1365/s13291-015-0123-0
Published:
Issue Date:
DOI: https://doi.org/10.1365/s13291-015-0123-0
Keywords
- Local existence and uniqueness
- Surface growth model
- Regularization of noise
- Fixed point argument
- Mild solution