Teaching: Course webpages are available at learning.drexel.edu

Conferences: The First Drexel Waves Workshop was a success, and we look forward to hosting the Second Drexel Waves Workshop in 2023.

Refereed Journal Publications

Traveling and time-periodic waves for dispersive PDE and interfacial fluid dynamics

• B.F. Akers and D.M. Ambrose. Internal capillary-gravity Wilton ripples. Submitted, 2024.
• D.M. Ambrose and J.D. Wright. Nonexistence of small, smooth, time-periodic, spatially periodic solutions for nonlinear Schrodinger equations. Quart. Appl. Math., 77:579-590, 2019. [Preprint.]
• B.F. Akers, D.M. Ambrose, and D.W. Sulon. Periodic traveling interfacial hydroelastic waves with or without mass II: Multiple bifurcations and ripples. European J. Appl. Math., 30:756-790, 2019. [Arxiv.]
• B.F. Akers, D.M. Ambrose, and D.W. Sulon. Periodic traveling interfacial hydroelastic waves with or without mass. Z. Angew. Math. Phys., 68: 141, 2017. [Arxiv.]
• D.M. Ambrose, W.A. Strauss, and J.D. Wright. Global bifurcation theory for periodic traveling interfacial gravity-capillary waves. Ann. Inst. H. Poincare Anal. Non Lineaire, 33:1081-1101, 2016. [Arxiv.]
• B.F. Akers, D.M. Ambrose, K. Pond, and J.D. Wright. Overturned internal capillary-gravity waves. Eur. J. Mech. B Fluids, 57:143-151, 2016. [Preprint.]
• D.M. Ambrose and J.D. Wright. Nonexistence of small doubly periodic solutions for dispersive equations. Analysis & PDE, 9:15-42, 2016. [Arxiv.]
• D.M. Ambrose, M. Kondrla, and M. Valle. Computing time-periodic solutions of a model for the vortex sheet with surface tension. Quart. Appl. Math., 73:317-329, 2015. [Preprint.]
• D.M. Ambrose and J.D. Wright. Non-existence of small-amplitude doubly periodic waves for dispersive equations. C. R. Math. Acad. Sci. Paris, 352:597-602, 2014. [Preprint.]
• B.F. Akers, D.M. Ambrose, and J.D. Wright. Gravity perturbed Crapper waves. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470: 20130526, 2014. [Preprint.]
• B. Akers, D.M. Ambrose, and J.D. Wright. Traveling waves from the arclength parameterization: Vortex sheets with surface tension. Interfaces Free Bound., 15:359-380, 2013. [Preprint.]
• D.M. Ambrose and J. Wilkening. Computation of time-periodic solutions of the Benjamin-Ono equation. J. Nonlinear Sci., 20:277-308, 2010. [Open access.]
• D.M. Ambrose and J. Wilkening. Computation of symmetric, time-periodic solutions of the vortex sheet with surface tension. Proc. Natl. Acad. Sci. USA, 107:3361-3366, 2010. [Open access.]
• D.M. Ambrose and J. Wilkening. Global paths of time-periodic solutions of the Benjamin-Ono equation connecting pairs of traveling waves. Commun. Appl. Math. Comput. Sci., 4:177-215, 2009. [Arxiv.]

Analysis for the Euler and Navier-Stokes equations and related models

• D.M. Ambrose, M.C. Lopes Filho, and H.J. Nussenzveig Lopes. Improved regularity and analyticity of Cannone-Karch solutions of the three-dimensional Navier-Stokes equations on the torus. Accepted, Monatsh. Math., 2024. [Arxiv.]
• D.M. Ambrose, F. Hadadifard, and J.P. Kelliher. Contour dynamics and global regularity for periodic vortex patches and layers. SIAM J. Math. Anal., 56:2286-2311, 2024. [Arxiv.]
• D.M. Ambrose, M.C. Lopes Filho, and H.J. Nussenzveig Lopes. Existence and analyticity of the Lei-Lin solution of the Navier-Stokes equations on the torus. Proc. Amer. Math. Soc., 152:781-795, 2024. [Arxiv.]
• D.M. Ambrose, E. Cozzi, D. Erickson, and J.P. Kelliher. Existence of solutions to fluid equations in Hölder and uniformly local Sobolev spaces. J. Differential Equations, 364:107-151, 2023. [Arxiv.]
• D.M. Ambrose, M.C. Lopes Filho, and H.J. Nussenzveig Lopes. Confinement of vorticity for the 2D Euler-alpha equations. J. Differential Equations, 265:5472-5489, 2018. [Arxiv.]
• D.M. Ambrose, J.P. Kelliher, M.C.Lopes Filho, and H.J. Nussenzveig Lopes. Serfati solutions to the 2D Euler equations on exterior domains. J. Differential Equations, 259:4509-4560, 2015. [Arxiv.]

Equations with nonlinear/degenerate dispersion and related problems

• D.M. Ambrose and J. Woods. Well-posedness and ill-posedness for linear fifth-order dispersive equations in the presence of backwards diffusion. J. Dynam. Differential Equations, 34:897-917, 2022. [Preprint.]
• T. Akhunov, D.M. Ambrose, and J.D. Wright. Well-posedness of fully nonlinear KdV-type evolution equations. Nonlinearity, 32:2914-2954, 2019. [Arxiv.]
• D.M. Ambrose, G.R. Simpson, J.D. Wright, and D.G. Yang. Existence theory for magma equations in dimension two and higher. Nonlinearity, 31:4724-4745, 2018. [Arxiv.]
• D.M. Ambrose and J.D. Wright. Dispersion vs. anti-diffusion: Well-posedness in variable coefficient and quasilinear equations of KdV-type. Indiana U. Math. J., 62:1237-1281, 2013. [Arxiv.]
• D.M. Ambrose and J.D. Wright. Traveling waves and weak solutions for an equation with degenerate dispersion. Proc. Amer. Math. Soc., 141:3825-3838, 2013.
• D.M. Ambrose, G. Simpson, J.D. Wright, and D.G. Yang. Ill-posedness of degenerate dispersive equations. Nonlinearity, 25: 2655-2680, 2012. [Arxiv.]
• D.M. Ambrose and J.D. Wright. Preservation of support and positivity for solutions of degenerate evolution equations. Nonlinearity, 23:607-620, 2010.

Mean field games

• D.M. Ambrose, M. Griffin-Pickering, and A.R. Mészáros. Kinetic-type mean field games with non-separable local Hamiltonians. Submitted, 2024. [Arxiv.]
• L.C. Brown and D.M. Ambrose. Equilibria in the large-scale competition for market share in a commodity with resource-buying. Accepted, Dyn. Games Appl., 2024.
• J. Sin, J.W. Bonnes, L.C. Brown, and D.M. Ambrose. Existence and computation of stationary solutions for congestion-type mean field games via bifurcation theory and forward-forward problems. J. Dyn. Games, 11:48-62, 2024.
• D.M. Ambrose and A.R. Mészáros. Well-posedness of mean field games master equations involving non-separable local Hamiltonians. Trans. Amer. Math. Soc., 376:2481-2523, 2023. [Arxiv.]
• D.M. Ambrose. Existence theory for non-separable mean field games in Sobolev spaces. Indiana U. Math. J., 71:611-647, 2022. [Arxiv.]
• D.M. Ambrose. Existence theory for a time-dependent mean field games model of household wealth. Appl. Math. Optim., 83:2051-2081, 2021. [Arxiv.]
• D.M. Ambrose. Strong solutions for time-dependent mean field games with non-separable Hamiltonians. J. Math. Pures Appl., 113:141-154, 2018. [Arxiv.]
• D.M. Ambrose. Small strong solutions for time-dependent mean field games with local coupling. C. R. Math. Acad. Sci. Paris, 354:589-594, 2016. [Preprint.]

Analysis and computation for models of flame fronts

• D.M. Ambrose, M.C. Lopes Filho, and H.J. Nussenzveig Lopes. Existence and analyticity of solutions of the Kuramoto-Sivashinsky equation with singular data. Accepted, Proc. Roy. Soc. Edinburgh Sect. A, 2024. [Arxiv.]
• S. Liu and D.M. Ambrose. Well-posedness of a two-dimensional coordinate-free model for the motion of flame fronts. Phys. D, 447:133682, 2023. [Preprint.]
• D.M. Ambrose and A.L. Mazzucato. Global solutions of the two-dimensional Kuramoto-Sivashinsky equation with a linearly growing mode in each direction. J. Nonlinear Sci., 31, paper no. 96, 2021 [Arxiv.]
• D.M. Ambrose, F. Hadadifard, and J.D. Wright. Well-posedness and asymptotics of a coordinate-free model of flame fronts. SIAM J. Appl. Dyn. Syst., 20:2261-2294, 2021. [Arxiv.]
• B.F. Akers and D.M. Ambrose. Efficient computation of coordinate-free models of flame fronts. ANZIAM J., 63:58-69, 2021. [Preprint.]
• D.M. Ambrose and A.L. Mazzucato. Global existence and analyticity for the 2D Kuramoto-Sivashinksy equation. J. Dynam. Differential Equations, 31:1525-1547, 2019.[Arxiv.]

Analysis for other nonlinear partial differential equations

• D.M. Ambrose, P.M. Lushnikov, M. Siegel, and D.A. Silantyev. Global existence and singularity formation for the generalized Constantin-Lax-Majda equation with dissipation: The real line vs. periodic domains. Nonlinearity, 37:025004, 2024. [Arxiv.]
• D.M. Ambrose. The radius of analyticity for solutions to a problem in epitaxial growth on the torus. Bull. Lond. Math. Soc., 51:877-886, 2019. [Arxiv.]
• D.M. Ambrose and G. Simpson. Local existence theory for derivative nonlinear Schrödinger equations with non-integer power nonlinearities. SIAM J. Math. Anal., 47:2241-2264, 2015. [Arxiv.]

Analysis and computing for waves in electromagnetics

• D.M. Ambrose, F. Cakoni, and S. Moskow. A perturbation problem for transmission eigenvalues. Res. Math. Sci., 9, paper no. 11, 2022. [Preprint.]
• D.M. Ambrose, E. Das Gupta, S. Moskow, V. Ozornina, and G. Simpson. Detection of thin high contrast dielectrics from boundary measurements. J. Phys. Comm., 3:115016, 2019. [Open access.]
• D.M. Ambrose, J. Gopalakrishnan, S. Moskow, and S. Rome. Scattering of electromagnetic waves by thin high contrast dielectrics II: Asymptotics of the electric field and a method for inversion. Comm. Math. Sci., 15:1041-1053, 2017. [Preprint.]
• D.M. Ambrose and D.P. Nicholls. Fokas integral equations for three dimensional layered-media scattering. J. Comp. Phys., 276:1-25, 2014. [Preprint.]
• D.M. Ambrose and S. Moskow. Scattering of electromagnetic waves by thin high contrast dielectrics: Effects of the object boundary. Comm. Math. Sci., 11: 293-314, 2013.

Numerical methods for initial value problems in interfacial fluid dynamics

• D.M. Ambrose, M. Siegel, and K. Zhang. Convergence of the boundary integral method for interfacial Stokes flow. Math. Comp., 92:695-748, 2023. [Arxiv.]
• D.M. Ambrose, R. Camassa, J.L. Marzuola, R.M. McLaughlin, Q. Robinson, and J. Wilkening. Numerical algorithms for water waves with background flow over obstacles and topography. Adv. Comput. Math., 48, paper no. 46, 2022. [Open access.]
• D.M. Ambrose, Y. Liu, and M. Siegel. Convergence of a boundary integral method for 3D interfacial Darcy flow with surface tension. Math. Comp., 86:2745-2775, 2017. [Preprint.]
• D.M. Ambrose, M. Siegel, and S. Tlupova. A small-scale decomposition for 3D boundary integral computations with surface tension. J. Comp. Phys., 247:168-191, 2013. [Preprint.]
• D.M. Ambrose and M. Siegel. A non-stiff boundary integral method for 3D porous media flow with surface tension. Math. Comput. Simulation, 82:968-983, 2012.

Existence theory in interfacial fluid dynamics

• S. Liu and D.M. Ambrose. Asymptotics of two-dimensional hydroelastic waves: The zero mass, zero bending limit. Submitted, 2024. [Arxiv.]
• D.M. Ambrose. The velocity field and Birkhoff-Rott integral for non-decaying, non-periodic vortex sheets. Submitted, 2024. [Arxiv.]
• S. Liu and D.M. Ambrose. Well-posedness of a model equation for water waves in fluids with odd viscosity. Accepted, J. Dynam. Differential Equations, 2023.
• H. Kim and D.M. Ambrose. Well-posedness, ill-posedness, and traveling waves for models of pulsatile flow in viscoelastic vessels. Z. Angew. Math. Phys., 73, paper no. 247, 2022. [Arxiv.]
• S. Liu and D.M. Ambrose. The zero surface tension limit of three-dimensional interfacial Darcy flow. J. Differential Equations, 268:3599-3645, 2020. [Preprint.]
• S. Liu and D.M. Ambrose. Sufficiently strong dispersion removes ill-posedness in truncated series models of water waves. Discrete Contin. Dyn. Syst., 39:3123-3147, 2019. [Preprint.]
• D.M. Ambrose, J.L. Bona, and T. Milgrom. Global solutions and ill-posedness for the Kaup system and related Boussinesq systems. Indiana U. Math. J., 68:1173-1198, 2019. [Preprint.]
• D.M. Ambrose and M. Siegel. Well-posedness of two-dimensional hydroelastic waves. Proc. Roy. Soc. Edinburgh Sect. A, 147:529-570, 2017. [Preprint.]
• S. Liu and D.M. Ambrose. Well-posedness of two-dimensional hydroelastic waves with mass. J. Differential Equations, 262:4656-4699, 2017. [Preprint.]
• D.M. Ambrose, J.L. Bona, and D.P. Nicholls. On ill-posedness of truncated series models for water waves. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470:20130849, 2014. [Preprint.]
• D.M. Ambrose. The zero surface tension limit of two-dimensional interfacial Darcy flow. J. Math. Fluid Mech., 16:105-143, 2014. [Preprint.]
• T. Milgrom and D.M. Ambrose. Temporal boundary value problems in interfacial fluid dynamics. Appl. Anal., 92:922-948, 2013.
• D.M. Ambrose, J.L. Bona, and D.P. Nicholls. Well-posedness of a model for water waves with viscosity. Discrete Contin. Dyn. Syst. Ser. B, 17:1113-1137, 2012.
• D.M. Ambrose, M.C. Lopes Filho, H.J. Nussenzveig Lopes, and W.A. Strauss. Transport of interfaces with surface tension by 2D viscous flows. Interfaces Free Bound., 12:23-44, 2010.
• D.M. Ambrose. Singularity formation in a model for the vortex sheet with surface tension. Math. Comput. Simulation, 80:102-111, 2009.
• D.M. Ambrose and N. Masmoudi. The zero surface tension limit of three-dimensional water waves. Indiana U. Math. J., 58:479-522, 2009.
• D.M. Ambrose and N. Masmoudi. Well-posedness of 3D vortex sheets with surface tension. Comm. Math. Sci., 5:391-430, 2007. [Open access.]
• D.M. Ambrose. Well-posedness of two-phase Darcy flow in 3D. Quart. Appl. Math., 65:189-203, 2007.
• D.M. Ambrose and N. Masmoudi. The zero surface tension limit of two-dimensional water waves. Comm. Pure Appl. Math, 58:1287-1315, 2005.
• D.M. Ambrose. Well-posedness of two-phase Hele-Shaw flow without surface tension. European J. Appl. Math., 15:597-607, 2004.
• D.M. Ambrose. Well-posedness of vortex sheets with surface tension. SIAM J. Math. Anal., 35:211-244, 2003.

Book Chapters and Other Expository Articles

• D.M. Ambrose. Vortex sheets, Boussinesq equations, and other problems in the Wiener algebra. SIAM DSWeb, 2019. [Link.] [Download.]
• D.M. Ambrose. Vortex sheet formulations and initial value problems: Analysis and computing. Lectures on the theory of water waves, 140-170, London Math. Soc. Lecture Note Ser., 426, Cambridge Univ. Press, Cambridge, 2016.

Refereed Conference Proceedings

• D.M. Ambrose and J. Wilkening. Dependence of time-periodic vortex sheets with surface tension on mean vortex sheet strength. Procedia IUTAM, 11:15-22, 2014. [Preprint.]
• D.M. Ambrose and J. Wilkening. Computation of time-periodic solutions of nonlinear systems of partial differential equations. Proceedings of Hyperbolic Problems: Theory, Numerics, and Applications. Beijing, China (2010). 2012, 273-280, Higher Education Press.
• D.M. Ambrose. Short-time well-posedness of irrotational free-surface problems in 3D fluids. Proceedings of Hyperbolic Problems: Theory, Numerics, and Applications. Lyon, France (2006). 2008, 307-314, Springer-Verlag.
• D.M. Ambrose. Regularization of the Kelvin-Helmholtz instability by surface tension. Phil. Trans. R. Soc. A, 365:2253-2266, 2007. Proceedings of the Semester on Wave Motion, Institute Mittag-Leffler (2005).
• D.M. Ambrose. Short-time well-posedness of free-surface problems in 2D fluids. Proceedings of Hyperbolic Problems: Theory, Numerics, and Applications. Osaka, Japan (2004). 2006, 247-254, Yokohama Publishers.

External Funding

• PI for NSF grant DMS-2307638, Well-Posedness and Singularity Formation in Applied Free Boundary Problems. $300,000. August 1, 2023 -- July 31, 2026.
• PI for NSF grant DMS-1907684, Partial Differential Equation Methods for Mean Field Games. $316,981. August 1, 2019 -- July 31, 2023.
• PI for NSF grant DMS-1515849, Dynamics of Dispersive PDE. $269,987. August 15, 2015 -- July 31, 2019.
• PI for NSF grant DMS-1016267, Collaborative Research: Efficient Surface-Based Numerical Methods for 3D Interfacial Flow with Surface Tension. $269,989. October 1, 2010 -- September 30, 2015. [This is a collaborative grant with Michael Siegel of NJIT.]
• PI for NSF grant DMS-1008387, Dispersive PDE and Interfacial Fluid Dynamics. $159,000. September 15, 2010 -- August 31, 2014.
• PI for NSF grant DMS-0707807, Long-Time Behavior of Free-Surface Problems in Fluid Dynamics. $119,999. June 15, 2007 -- May 31, 2010. [Renumbered as DMS-0926378.]
• PI for NSF grant DMS-0406130, Analytical and Computational Approaches to Free-Surface Problems in Fluid Dynamics. $81,143. June 1, 2004 -- May 31, 2007. [Renumbered as DMS-0610898.]