Research Statement Zachary Bradshaw October 20, 2016 1 Overview My research is in the field of partial differential equations. I am primarily interested in the three dimensional non-stationary Navier-Stokes equations. This system describes the evolution of an incompressible, viscous fluid s velocity field u starting from some initial state u 0 at time t = 0. In R 3 [0, ), the unknowns are the velocity field u and the pressure p which solve t u ν u + (u )u + p = f in R 3 (0, T ), (1) u = 0 for a prescribed viscosity coefficient ν and body force f. This model provides the foundation for our mathematical understanding of fluids in the real world and has many important applications. Remarkably, fundamental mathematical questions about this system remain unanswered. A notable example is the problem of global regularity, which is one of the Clay Mathematics Institute s Millennium Problems. My research has developed in three directions: (1) In [11], T.-P. Tsai and I develop a new construction of self-similar solutions (and other scaling invariant solutions) to (1) for large, possibly rough initial data. The previous best results were for smooth data. In [12, 13], we extend our result to more general contexts and introduce a new class solutions satisfying a rotated scaling invariance. (2) In [4, 5, 7], Z. Grujić and I establish regularity criteria for solutions to (1). I also study the regularity of solutions to the magnetohydrodynamic equations in [3]. (3) I work on turbulence in fluids and related media aiming to provide a rigorous explanation of physical and numerical phenomena. In [6, 9, 8], Z. Grujić and I studied the turbulent energy cascades and flux locality in atmospheric and magnetohyrdorynamic flows. In [10], Z. Grujić, I. Kukavica, and I establish estimates for the small length scales in turbulent solutions to (1). In this statement I will elaborate on these directions. I will also discuss current and future research objectives in Section 5. 1
2 Self-similar solutions to the Navier-Stokes Equations The Navier-Stokes equations have a natural scaling: If u is a solution for the initial data u 0 and with associated pressure distribution p, then, for any λ > 0, u λ (x, t): = λu(λx, λ 2 t) is also a solution with associated pressure p λ (x, t): = λ 2 p(λx, λ 2 t) and initial data u λ 0(x): = λu 0 (λx). A solution is self-similar (SS) if it is scaling invariant, i.e., if u λ (x, t) = u(x, t) for all λ > 0. If this holds for a particular λ > 1, then u is discretely self-similar with factor λ (λ-dss). Similarly u 0 can be self-similar (a.k.a. 1-homogeneous) or λ-dss. Self-similar solutions were introduced in 1934 by J. Leray in the seminal work [26] and are historically important (see, e.g., [27, 30, 16, 15, 19]). Self-similar solutions evolving from 1- homogeneous data were first constructed for small data (in an appropriate function space; see [19, 15, 1]). H. Jia and V. Šverák gave the first construction for large initial data in [21] assuming the initial data is Hölder continuous on R 3 \ {0}. M. Korobkov and T.-P. Tsai gave a second construction in [24]. T.-P. Tsai constructed DSS solutions in [31] provided the scaling factor λ is close to one. In all cases the data is Hölder continuous on R 3 \ {0}. T.-P. Tsai and I develop a new approach to the existence problem in [11]. This paper was recently accepted to Annales Henri Poincaré. We extend these ideas in two submitted papers [12] and [13]. The details of these papers are as follows: (1) In [11], we construct SS/λ-DSS solutions for any divergence free, SS/λ-DSS data in the weak Lebesgue space L 3 w(r 3 ) (this space is slightly larger than L 3 and functions like x 1 ). In contrast to the existence theorems in [21, 31, 24], no smoothness is assumed u 0 can even be singular away from the origin and λ is allowed to be any value greater than 1. (2) In [12], we introduce a new class of solutions which are rotational corrections of SS or DSS solutions which generalize the SS and DSS solutions. We also construct such solutions for any divergence free L 3 w data on either the whole or half-space. This is the first existence result for large, rough initial data on the half-space. (3) In [13], we construct SS or 2-DSS solutions for any divergence free, SS or 2-DSS data in the homogeneous Besov space Ḃ3/p 1 p, where 3 p < 6 (for a definition of Ḃp, 3/p 1 see [25]). This improves upon [11] because L 3 w Ḃ3/p 1 p, when p > 3. It is currently the most general existence result for SS or DSS solutions. 3 Turbulence Despite a chaotic appearance, regular processes and coherent structures are features of fluid turbulence. I am interested in understanding these dynamics via the governing equations. For example, consider the direct energy cascade for 3D fluids. This refers to the net inertial transport of energy from larger to smaller scale structures between a macroscale where energy is injected and microscale i.e., dissipative scale where inertial effects are outweighed by viscous forces and energy is primarily lost as heat. An important mathematical goal is to demonstrate how these dynamics emerge from the Navier-Stokes equations. 2
My contributions to the mathematical theory of turbulence are the following: (1) In 3D magnetohydrodynamic turbulence (e.g. turbulence in the solar wind) the total energy, i.e., the sum of magnetic, kinetic, and potential energies, exhibits a direct cascade. In [6], Z. Grujić and I establish conditions for a direct cascade where the energy flux is scalelocal. A related dynamical process is the concentration of enstrophy. Here, the intermittent regions of high spatial complexity shrink as turbulence evolves. In [9], Z. Grujić and I establish conditions triggering this concentration effect. (2) In [8], Z. Grujić and I confirm the existence of a forward temperature variance cascade for solutions to the surface quasi-geostrophic equations (which model some atmospheric flows) under conditions consistent with qualitative properties of turbulence. The triggering condition accommodates the inherent non-locality introduced by fractional dissipation. (3) The dissipative scale of a smooth flow coincides with it s analyticity radius. Generally this can be estimated in terms of global quantities on the whole space or the torus but existing methods break down on bounded domains. In [10], Z. Grujić, I. Kukavica, and I establish sharp lower bounds for the local analyticity radii of solutions to (1) in terms of purely local quantities. These can be formulated on bounded domains. I also establish a lower bound for the analyticity radius for solutions to the 3D MHD equations in [3]. 4 Regularity criteria for fluid models The smoothness of solutions to (1) is an open problem. Presently, only conditional regularity criteria are available. These should be physically motivated or reveal important properties about the scaling dynamics of solutions. My work in this area is as follows: (1) In a paper recently accepted to Arch. Rat. Mech. Anal., Z. Grujić and I develop regularity criteria involving finite windows of frequencies (in the Littlewood-Paley sense) of a fluid [7]. These results highlight the essential frequencies in singularity formation in a Leray-Hopf weak solution. One of the criteria is a refinement of a Besov space version of the classical non-endpoint Prodi-Serrin condition. (2) In [5] and [4], Z. Grujić and I develop L log L bounds for the vorticity under mild conditions on the flow. This logarithmic gain is significant in the context of a physically motivated argument due to Z. Grujić [20]. Roughly, the filamentary structure of the regions of high vorticity is consistent with critical decay of transverse length scales of the superlevel sets of the L norm of the vorticity. In [20], a critical decay rate is given in terms of these transverse length scales if the transverse length scales are smaller than this critical rate then the fluid must be regular. The logarithmic gain in [5] and [4] ensures that transversal length scales will become small enough prior to a singular time to trigger the regularity criteria of [20]. (3) In [3], I establish several regularity criteria for solutions to the magnetohydrodynamic equations. These state that if the superlevel sets of both the velocity field and the magnetic 3
field, or either in certain cases, are sparse in a local, one-dimensional sense, then neither field blows up at an initial possible blow-up time. 5 Ongoing and future research directions Self-similar solutions. It is important to seek SS/DSS solutions in larger scaling invariant spaces than T.-P. Tsai and I consider in [11, 12, 13] (a space X is scaling invariant if u X = u λ X ). In particular we have the inclusions, L 3 w Ḃ3/p 1 p (3, ), BMO 1 Ḃ 1,, where BMO 1 denotes the Koch-Tataru space and Ḃs p,q denotes a homogeneous Besov space. These spaces are historically important (see [23] and [25] for more details). Self-similar and discretely self-similar solutions are known to exist for small initial data in all these spaces except Ḃ 1, but the best large data result is for Ḃ3/p 1 p, where 3 p < 6 [13]. I am currently working on constructing solutions in larger spaces. Infinite energy solutions. The Leray-Hopf weak solutions preserve energy globally: Beginning with divergence free initial data in L 2 (R 3 ) and any T > 0, a Leray-Hopf weak solution is known to exist in L (0, T ; L 2 ) L 2 (0, T ; H 1 ) which satisfies the global energy inequality [26]. If we start with divergence free initial data in the space of uniformly locally square integrable functions L 2 u loc, then it is possible to construct a local Leray solution (see [25, 22]) which preserves energy locally. The local Leray solutions are intended to be local analogues to the Leray-Hopf solutions. Some properties are not shared. In an ongoing project with T.-P. Tsai, we are constructing solutions in intermediate spaces to better understand the differences between Leray-Hopf and local Leray solutions and why things break down. Data assimilation. Going forward, I would like to study determining structures for fluid equations. In certain scenarios data can be obtained about flow continuously at finitely many nodes (e.g. measurements of atmospheric data). A comprehensive understanding of the flow cannot be obtained from such a discrete sampling. But, the long time behavior of solutions to (1) in 2D can be characterized by a finite number of determining parameters [17]. So, a fine enough sample of data can be used to fully understand the long-time evolution of the flow. I have recently begun to work on several topics in this area. Computational project on turbulence for undergraduates. Part of my thesis work dealt with a new methodology for detecting turbulent cascades i.e., the statistically uniform transport of energy from larger to smaller scales as a feature of the governing model (see [6, 8, 9]). This approach works directly with a physical solution and, in contrast to other approached, does not involve the Fourier transform. The underlying structures are approachable given basic knowledge of multi-variable calculus and could be understood by a strong undergraduate student. Presently, this methodology has not been examined from a computational standpoint. It is my intention to address this working with motivated undergraduate students who are familiar with computer programming. 4
References [1] Barraza, O., Self-similar solutions in weak L p -spaces of the Navier-Stokes equations. Rev. Mat. Iberoamericana 12 (1996), 411-439. [2] Bourgain, J. and Pavlović, Natasa Ill-posedness of the Navier-Stokes equations in a critical space in 3D. J. Funct. Anal. 255 (2008), no. 9, 2233-2247. [3] Bradshaw, Z., Geometric measure-type regularity criteria for the 3D magnetohydrodynamical system. Nonlinear Anal. 75 (2012), no. 16, 6180-6190. [4] Bradshaw, Z. and Grujić, Z., A spatially localized LlogL estimate on the vorticity in the 3D NSE. Indiana Univ. Math. J. 64 (2015), no. 2, 433-440. [5] Bradshaw, Z. and Grujić, Z., Blow-up scenarios for the 3D Navier-Stokes equations exhibiting sub-criticality with respect to the scaling of one-dimensional local sparseness. J. Math. Fluid Mech. 16 (2014), no. 2, 321-334. [6] Bradshaw, Z. and Grujić, Z. Energy cascades in physical scales of 3D incompressible magnetohydrodynamic turbulence. J. Math. Phys. 54 (2013), no. 9, 093503, 18 pp. [7] Bradshaw, Z. and Grujić, Z. Frequency localized regularity criteria for the 3D Navier- Stokes equations. submitted, accepted to Arch. Rat. Math. Anal. (arxiv:1501.01043). [8] Bradshaw, Z. and Grujić, Z. A note on the surface quasi-geostrophic temperature variance cascade. Commun. Math. Sci. 13 (2015), no. 2, 557-564. [9] Bradshaw, Z. and Grujić, Z. On the transport and concentration of enstrophy in 3D magnetohydrodynamic turbulence. Nonlinearity 26 (2013), no. 8, 2373-2390. [10] Bradshaw, Z., Grujić, Z., and Kukavica, I., Local analyticity radii of solutions to the 3D Navier-Stokes equations with locally analytic forcing. J. Differential Equations 259 (2015), no. 8, 3955-3975. [11] Bradshaw, Z. and Tsai, T.-P., Forward discretely self-similar solutions of the Navier- Stokes equations II. Accepted to Anal. Henri Poincaré (arxiv:1510.07504). [12] Bradshaw, Z. and Tsai, T.-P., Rotationally corrected scaling invariant solutions to the Navier-Stokes equations, submitted (arxiv:1610.05680). [13] Bradshaw, Z. and Tsai, T.-P., Self-similar solutions to the Navier-Stokes equations in critical Besov spaces, in preparation. [14] Caffarelli, L., Kohn, R. and Nirenberg, L. Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982), no. 6, 771-831. [15] Cannone, M., Meyer, Y., and Planchon, F., Solutions auto-similaires des équations de Navier-Stokes dans R 3, Exposé n. VIII, Séminaire X-EDP, Ecole Polytechnique, 1994. [16] Cannone, M. and Planchon, F. Self-similar solutions for Navier-Stokes equations in R3. Comm. Partial Differential Equations 21 (1996), no. 1-2, 179-193. 5
[17] Foias, C., Manley, O., Rosa, R., and Temam, R., Navier-Stokes equations and turbulence. Encyclopedia of Mathematics and its Applications, 83. Cambridge University Press, Cambridge, 2001. [18] Galdi, G. P., An introduction to the mathematical theory of the Navier-Stokes equations. Vol. II. Nonlinear steady problems. Springer Tracts in Natural Philosophy, 39. Springer-Verlag, New York, 1994. [19] Giga, Y. and Miyakawa, T., Navier-Stokes flows in R 3 with measures as initial vorticity and the Morrey spaces, Comm. Partial Differential Equations 14 (1989), 577-618. [20] Grujić, Z. A geometric measure-type regularity criterion for solutions to the 3D Navier- Stokes equations, Nonlinearity 26 (2013), 289-296. [21] Jia, H. and Šverák, V., Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions. Invent. Math. 196 (2014), no. 1, 233-265. [22] Kikuchi, N. and Seregin, G., Weak solutions to the Cauchy problem for the Navier- Stokes equations satisfying the local energy inequality. Nonlinear equations and spectral theory, 141-164, Amer. Math. Soc. Transl. Ser. 2, 220, Amer. Math. Soc., Providence, RI, 2007. [23] Koch, H., Tataru, D., Well-posedness for the Navier-Stokes equations. Adv. Math. 157 (1), 22 35 (2001) [24] Korobkov, M. and Tsai, T.-P., Forward self-similar solutions of the Navier-Stokes equations in the half space, submitted. [25] Lemarié-Rieusset, P. G., Recent developments in the Navier-Stokes problem. Chapman Hall/CRC Research Notes in Mathematics, 431. Chapman Hall/CRC, Boca Raton, FL, 2002. [26] Leray, J., Sur le mouvement d un liquide visqueux emplissant l espace. (French) Acta Math. 63 (1934), no. 1, 193-248. [27] Nečas, J., Růžička, M., and Šverák, V., On Leray s self-similar solutions of the Navier- Stokes equations, Acta Math. 176 (1996), 283 294. [28] Solonnikov, V. A., Estimates for solutions of the nonstationary Stokes problem in anisotropic Sobolev spaces and estimates for the resolvent of the Stokes operator. (Russian) Uspekhi Mat. Nauk 58 (2003), no. 2(350), 123 156; translation in Russian Math. Surveys 58 (2003), no. 2, 331-365. [29] Šverák, V., On Landau s solutions of the Navier-Stokes equations. J. Math. Sci., 179 (2011), no. 1, 208-228. [30] Tsai, T.-P., On Leray s self-similar solutions of the Navier-Stokes equations satisfying local energy estimates, Archive for Rational Mechanics and Analysis 143 (1998), 29 51. [31] Tsai, T.-P., Forward discretely self-similar solutions of the Navier-Stokes equations. Comm. Math. Phys. 328 (2014), no. 1, 29-44. 6