My doctoral research concentrated on elliptic quasilinear partial differential equations of the form

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1 1 Introduction I am interested in applied analysis and computation of non-linear partial differential equations, primarily in systems that are motivated by the physics of fluid dynamics. My Ph.D. thesis concentrated on answering questions of well posedness of free boundary problems, which involved the study of existence and uniqueness of solutions. The problems addressed were a vortex sheet problem and the Boussinesq equations. In the following discussion a more detailed overview of my Ph.D. thesis, directed under the supervision of Prof. David Ambrose at Drexel University, is provided. Some of the current and future problems of interest to me are described. Several are direct consequences of Ph.D. research and others are from related areas I am interested in exploring. 2 Ph.D. Research My doctoral research concentrated on elliptic quasilinear partial differential equations of the form v y + Aw = F (v, w) x, (2.1) w y + Av = G(v, w) x, (2.2) where v, w are functions of (x, y), A is a linear operator which acts as a multiplier in Fourier space and F, G are some non-linearities. These types of equations were studied on two types of domains: First, an infinite strip where (x, y) (, ) [0, Y ], for some Y > 0; and second, for periodic problems where (x, y) [0, 2π] [0, Y ]. In each case Dirichelet, Neumann and mixed types of boundary conditions, are considered on the boundary {y = 0} {y = Y }. The existence of solutions of a boundary value problem is usually treated with a maximal principal argument such as in Gilbarg and Trudinger [7]. Instead, I used a contraction mapping argument, which is typically used for initial value problems. The question of the existence of a solution to equations (2.1) and (2.2) is formulated as a contraction mapping problem by using the Duhamel s formula to construct a weak solution representation, in the form of T (v, w) = (v, w). Such a formulation allows us to treat the original system as a fixed point problem of the mapping T. Since A is a multiplier in Fourier space we define σ A by the Fourier transform of Ah(x) with respect to x: Âh(x) = σ A ĥ(ξ). My thesis work shows that if σ A is of order ξ and if F and G are Lipschitz continuous on an analytic function space, with the boundary conditions satisfying a smallness property, then the mapping T will be a contraction on the given function space, and its fixed point will be the unique solution to equations (2.1) and (2.2) with appropriate 1

2 boundary conditions. An application of these results is a vortex sheet problem and the Bona-Chen-Saut type Boussinesq equations. The result of this work was accepted for publication in Applicable Analysis [10]. 2.1 Vortex Sheet A vortex sheet is an interface between two fluids which have a discontinuity in the tangential component of velocity. For the particular vortex sheet considered, the two fluids have the same density, and are incompressible and irrotational away from the interface. Since there is a discontinuity in the velocity of the fluid, if we define the velocity by u, then along the interface the vorticity ω = curl(u) is analogous to a Dirac measure. It is possible to model the evolution of a vortex sheet using the Euler equations with the discontinuity condition on velocity. There is a fair amount of analytical and computational work in the literature on the vortex sheet problem. Specifically the work of Sulem, Sulem, Bardos, and Frisch [12] and Duchon and Robert [6] directly relates to my contribution. In [12] the initial value problem is reformulated and the existence of an analytic solution in finite time is proven. In [6] the result is extended to existence of an analytic solution for infinite time to the initial value problem formulated in [12]. My work is largely motivated by [6]; I generalized their methodology to treat this vortex sheet boundary value problem where the boundary is taken to be {t = 0} {t = T }. The vortex sheet formulation in [12] is of the form of (2.1) and (2.2) where y becomes the temporal variable t, A is the Hilbert transform with a derivative in the spatial variable x and F, G are principal value integrals that are Lipschitz continuous in an analytic function space. When applied to the general results of my work the vortex sheet has a non-periodic analytic solutions for finite time. The difference between these solutions and the work in [12] is the final time can be taken arbitrarily large. Also the smallness property of the boundary conditions is not time dependent, hence as the time is taken to go to infinity the results are consistent with the work of [6]. There are a number of extensions to this work I intend to consider in the future. First, it would be interesting to treat the problem in 3D. The theory used depends on a formulation of a weak solution to equations of the form (2.1) and (2.2). In [12] they derive the full 3D model for the vortex sheet. I would like to generalize my current work to formulate a weak solution to the 3D model and use a similar type of a contraction argument. In that case, not only would we be able to show existence of solutions to the boundary value problem, but we would also hope to be able to extend the work of [6] to show existence of solutions for infinite time of the initial value problem. Second, in the vortex sheet problem studied, the density of the two fluids is the same. Baker, Meiron, and Orszag [3] derive a generalized model for vortex sheet problem with the fluids having different densities. It is interesting to explore if it is possible to reformulate the generalized model into the form similar to (2.1) and (2.2). 2

3 2.2 Boussinesq Equations The Boussinesq equations are an approximation of the Euler equations with specific assumptions that model small amplitude long surface waves. The type of Boussinesq equation studied in my thesis are of the form η t + w x + (wη) x + aw xxx bη xxt = 0, (2.3) w t + η x + ww x + cη xxx dη xxt = 0, (2.4) where w is related to the velocity of the flow, η is related to the position of the free surface and a, b, c, d are parameters governed by a specific physical system. This model was first studied by Bona, Chen, and Saut [4], [5]; in their work they have answered the question of existence of a solution to the initial value problem for finite time, for certain values of the parameters. My Ph.D. thesis treats (2.3) and (2.4) as a boundary value problem with the boundary being {t = 0} {t = T }. It is possible to rewrite (2.3) and (2.4) into the form of (2.1) and (2.2). In the case of a > 0, b > 0, c < 0 and d sufficiently large, the corresponding operators A, F, G will satisfy the theory outlined in the previous section. This results in the existence of periodic solutions to the Boussinesq equations. The main advantage of the new solutions is that existence of solutions is proved for a different set of values of the parameters a, b, c, and d; as for the vortex sheet, the time interval can be taken to be as large as desired, without requiring the size of the data to go to zero. Some of the direct future work I will explore is analyzing (2.3) and (2.4) for more general values of a, b, c, and d. Currently these values are chosen so that A satisfies desired properties. In general, if the desired properties do not hold for a finite number of Fourier modes, it would be of interest to study those modes individually so that results could be generalized. Another direction of interest is to analyze higher order Boussinesq approximations. Since the Boussinesq equations are an approximation of the Euler equations, answers to questions of well posedness for higher order approximation may give us better understanding of the Euler equations. 3 Current Research Direction In order to complement the analysis work I have done during Ph.D. studies. I am interested in performing computational research. In particular I am interested in carrying out research similar to the work of Lopes Filho et al. [9]; there, they show evidence of nonuniqueness of weak solutions to the Euler equations. They do this by computing solutions of the initial value problem for vortex sheets. I am interested in addressing the same question on nonuniqueness of solutions, but by computing solutions of the vortex sheet boundary value problems. The evolution of the interface in the vortex sheet problem is given by the Birkhoff- Rott integral, which is a singular integral. One way to compute the Birkhoff-Rott 3

4 integral, is by regularizing the singularity using the vortex blob method as in [2], [8], [11]. My goal is to use a blob method along with a modified shooting method to compute the boundary value problem. I am currently working on a computation using Python numpy and scipy packages to compute a vortex sheet boundary value problem. The main challenge I am faced with is solving the resulting optimization problem. I intend to do solve the optimization problem using a quasi-newton line search algorithm such as the Broyden-Fletcher-Goldfarb-Shanno method, which was successfully done by Ambrose and Wilkening [1] for computing symmetric, timeperiodic solutions of a vortex sheet with surface tension. References [1] D.M. Ambrose and J. Wilkening. Computation of symmetric, time-periodic solutions of the vortex sheet with surface tension. Proc. Natl. Acad. Sci. USA, 107: , [2] G.R. Baker and J.T. Beale. Vortex blob methods applied to interfacial motion. Journal of Computational Physics, 196(1): , [3] G.R. Baker, D. Meiron, and S. Orszag. Generalized vortex methods for freesurface flow problems. J. of Fluid Mech., 123: , [4] J.L. Bona, M. Chen, and J.C. Saut. Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. i: Derivation and linear theory. Journal of Nonlinear Science, 12: , [5] J.L. Bona, M. Chen, and J.C. Saut. Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. ii: The nonlinear theory. Nonlinearity, 17: , [6] J. Duchon and R. Robert. Global vortex solutions of euler equations in the plane. Comm. Partial Differential Equations, 73: , [7] D. Gilbarg and N.S. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer Classics in Mathematics, [8] R. Krasny. A study of singularity formation in a vortex sheet by the point-vortex approximation. Journal of Fluid Mechanics, 167:65 93, [9] M.C. Lopes Filho, J. Lowengrub, H.J. Nussenzveig Lopes, and Y. Zheng. Numerical evidence of nonuniqueness in the evolution of vortex sheets. M2AN, 40: , [10] T. Milgrom and D.M. Ambrose. Temporal boundary value problems in interfacial fluid dynamics. Applicable Analysis. 4

5 [11] M.J. Shelley. A study of singularity formation in vortex-sheet motion by a spectrally accurate vortex method. Journal of Fluid Mechanics, 244: , [12] C. Sulem, P.L. Sulem, C. Bardos, and U. Frisch. Finite time analyticity for two and three dimensional kelvin-helmholtz instability. Comm. Math. Phys., 80: ,