Marcelo Mendes Disconzi
Department of Mathematics, Vanderbilt University

email: marcelo.disconzi at vanderbilt.edu
office: Stevenson Center 1515
phone: (615) 322 7147   fax: (615) 343 0215
mail to: 1326 Stevenson Center, Vanderbilt University, Nashville TN 37240

Marcelo Mendes Disconzi
General
Here you find information about my academic work and education, some notes I have written, and some links.

My interests are partial differential equations, geometric analysis, and mathematical physics. Currently the main topics of my research are the study of compressible and incompressible Euler equations (including generalizations such as free boundary problems), and the evolution problem for Einstein's equations coupled to matter (especially, Einstein's equations coupled to the equations of fluid dynamics). Other topics I have worked on are: geometric constraints induced by Einstein's equations on three-dimensional slices (for instance, problems related to the Penrose inequality and the positive mass theorem); variational and analytic aspects of effective potentials arising in compactifications of string theory (for example, the study of equations of motion derived from effective actions); and conformal deformations of Riemannian metrics (mainly, the Yamabe problem).

I organize the Partial Differential Equations Seminar. Click here for the Partial Differential Equations research group at Vanderbilt. Click here for past events organized at Vanderbilt.

Selected papers
Below is a sample of my papers, together with a short description of each of them. For a complete list of my publications, see my CV.

The free boundary Euler equations with large surface tension. (with David G. Ebin.) Journal of Differential Equations, Vol. 261, Issue 2, pp. 821-889 (2016).
We study the free boundary Euler equations with surface tension in three spatial dimensions, showing that the equations are well-posed if the coefficient of surface tension is positive. Then we prove that under natural assumptions, the solutions of the free boundary motion converge to solutions of the Euler equations in a domain with fixed boundary when the coefficient of surface tension tends to infinity.

On the well-posedness of relativistic viscous fluids. Nonlinearity 27, no. 8, pp. 1915-1935 (2014).
Using a simple and well-motivated modification of the stress-energy tensor for a viscous fluid proposed by Lichnerowicz, we prove that Einstein's equations coupled to a relativistic version of the Navier-Stokes equations are well-posed in a suitable Gevrey class if the fluid is incompressible and irrotational. These last two conditions are given an appropriate relativistic interpretation. The solutions enjoy the domain of dependence or finite propagation speed property. We also derive a full set of equations, describing a relativistic fluid that is not necessarily incompressible or irrotational, which is well-suited for comparisons with the system of an inviscid fluid.

Motion of slightly compressible fluids in a bounded domain, II. (with David G. Ebin.) Communications in Contemporary Mathematics, Vol. 19, No. 4, pp. 1650054, 57 pages (2017).
We study the problem of inviscid slightly compressible fluids in a bounded domain. We find a unique solution to the initial-boundary value problem and show that it is near the analogous solution for an incompressible fluid provided the initial conditions for the two problems are close. In particular, the divergence of the initial velocity of the compressible flow at time zero is assumed to be small. Furthermore we find that solutions to the compressible motion problem in Lagrangian coordinates depend differentiably on their initial data, an unexpected result for this type of non-linear equations.

Compactness and Non-Compactness for Yamabe Problem on Manifolds with Boundary. (with Marcus A. Khuri.) J. Reine Angew. Math. (Crelle's Journal), Vol. 2017, Issue 724, pp. 145-201 (2017).
We study the problem of conformal deformation of Riemannian structure to constant scalar curvature with zero mean curvature on the boundary. We prove compactness for the full set of solutions when the boundary is umbilic and the dimension $n \leq 24$. The Weyl Vanishing Theorem is also established under these hypotheses, and we provide counter-examples to compactness when $n \geq 25$. Lastly, our methods point towards a vanishing theorem for the umbilicity tensor, which is anticipated to be fundamental for a study of the nonumbilic case.

On the boundedness of effective potentials arising from string compactifications. (with Michael R. Douglas and Vamsi P. Pingali.) Communications in Mathematical Physics,  Vol. 325, Issue 3, pp. 847-878 (2014).
We study effective potentials coming from compactifications of string theory. We show that, under mild assumptions, such potentials are bounded from below in four dimensions, giving an affirmative answer to a conjecture proposed  by the second author in arXiv:0911.3378v4 [hep-th]. We also derive some sufficient conditions for the existence of critical points. All proofs and mathematical hypotheses are discussed in the context of their relevance to the physics of the problem.

Selected pre-prints
Below is a sample of my pre-prints, together with a short description of each of them. For a complete list, see my CV.

A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. (with Igor Kukavica.) arXiv:1708.00861 [math.AP], 40 pages (2017).
We derive a priori estimates for the compressible free-boundary Euler equations with surface tension in three spatial dimensions in the case of a liquid. These are estimates for local existence in Lagrangian coordinates when the initial velocity and initial density belong to H^3, with an extra regularity condition on the moving boundary, thus lowering the regularity of the initial data. Our methods are direct and involve two key elements: the boundary regularity provided by the mean curvature, and a new compressible Cauchy invariance.

Notes
Here are some notes that I (and other people) have written.

Some advanced techniques on PDE's - we review how the negative norm Sobolev spaces can be used to derive a necessary and sufficient condition for existence of weak solutions of any linear PDE. Using this, to show Egorov's example of a PDE that is not locally solvable at the origin. Some further applications are derived (pdf file).

Holographic renormalization - notes of a talk I gave in the RTG Seminar in Geometry and Physics at Stony Brook (pdf file).

Correlation functions in QFT - (handwritten). The basic ideas and concepts of quantum field theory are discussed with the intent of making physics books and papers on the subject more accessible to a mathematical audience. The focus is on correlation functions for the scalar field: what they are, how to compute them, their Feynman diagrams and renormalization properties. For a more details, see the table of contents.

Elementary realization of of BRST symmetry and gauge fixing - notes of a series of lectures given by Martin Rocek. All ideas of BRST symmetry and BV formalism are developed at a very basic level using finite dimensional integrals instead of path integrals. Excellent for those interested in the general idea of the formalism (pdf file).

Some algebraic structures in physics - notes from a series of informal meetings that I and some other students organized with the goal of sharing our different background in physics and mathematics (pdf file).

Some ideas in Conformal Field Theory - (handwritten) notes from a talk I gave in the RTG Seminar in Geometry and Physics at Stony Brook (.zip file with a bunch of .jpg files, or click here to access each file separetely).

Topics in Differential Topology - notes by Somnath Basu of a course taught by Blaine Lawson (pdf file).

Spontaneous symmetry breaking - introductory notes on the Higgs mechanism (pdf file).

Mathematical Foundations of Classical and Quantum Field Theory - notes of two summer courses I took on the subject (pdf  file).

The Comprehensive LaTex symbol list - excellent material by Scott Pakin (pdf file).

Simons Center for Geometry and Physics - the intellectual focus of the Center is at the interface of Mathematics, in particular Geometry, and Theoretical Physics.

Media and outreach

Here is a department news story on the occasion of my 2018 Sloan Fellowship award. The announcement of the 2018 Sloan Fellows also appeared in the New York Times and in the American Mathematical Society website. Here the Vanderbilt news story.

The three-dimensional free boundary Euler equations with surface tension. Video of a talk I gave in the workshop Recent Advances in Hydrodynamics that took place at the Banff International Research Station in Banff, Canada.

The "sticky" universe. A news story on a paper that I wrote with Robert Scherrer and Thomas Kephart. In 2015, the year the paper was published, it received widespread media coverage, including from The Guardian, Redorbit, New Statesman, The Huffington Post,  among many others. This unexpected media attention led me to write some reflections on science and the media. In 2016, the paper was again in the news with stories in the Wired, BBC Brazil (in Portuguese), Revista Piauí (in Portuguese), and TV Cultura (in Portuguese). This story was in the cover of the 2017 department newsletter, Spectrum.

The Einstein system for inviscid and viscid relativistic fluids (.flv file). Video of a talk I presented at the Colloquium of the Department of Applied Mathematics at USP (Brazil). The talk was in Ensligh, although the introduction and Q&A were in Portuguese.

I am occasionally a guest in the radio program Fronteiras da Ciencia, a radio program (in Portuguese) dedicated to science discussions for the general public. I participated in the episodes Gravitacao quantica, Teoria de supercordas, A grande ruptura cosmica (big rip), and a discussion about the work of Stephen Hawking on the occasion of his passing (mp3 files).