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 Ln, Vanderbilt University, Nashville TN 37240

Marcelo Mendes Disconzi
Research

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, mathematical fluid dynamics, mathematical general relativity, geometric analysis, and mathematical physics. Currently, the main focus of my research are relativistic fluids, including the relativistic Euler equations, relativistic fluids with viscosity, and their coupling to Einstein's equations. I strive to establish results under realistic physical assumptions. This involves considering relativistic fluids in three spatial dimensions, with vorticity, without symmetry assumptions, and possibly allowing for the presence of free-boundaries. Not only is such treatment essential for applications, but it also involves a great deal of rich mathematics. I am also interested in the mathematical study of fluids more broadly, including the free-boundary classical compressible and incompressible Euler equations. 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 am happy to be a member of the Vanderbilt Initiative for Gravity, Waves, and Fluids (VandyGWAF), which is an interdisciplinary research venture providing mathematicians, physicists, and astrophysicists with the resources and space to connect and collaboratively work on problems of outstanding scientific merit.

I gratefully acknowledge support from a Sloan Research Fellowship provided by the Alfred P. Sloan foundation, from NSF grant DMS-2107701, and from a Dean's Faculty Fellowship.

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 relativistic Euler equations: Remarkable null structures and regularity properties. (with Jared Speck.) Annales Henri Poincare, Vol. 20, Issue 7, pp. 2173-2270 (2019).
We derive a new formulation of the relativistic Euler equations that exhibits remarkable properties. This new formulation consists of a coupled system of geometric wave, transport, and transport-div-curl equations, sourced by nonlinearities that are null forms relative to the acoustical metric. Our new formulation is well-suited for various applications, in particular for the study of stable shock formation, as it is surveyed in the paper. Moreover, using the new formulation presented here, we establish a local well-posedness result showing that the vorticity and the entropy of the fluid are one degree more differentiable compared to the regularity guaranteed by standard estimates (assuming that the initial data enjoy the extra differentiability). This gain in regularity is essential for the study of shock formation without symmetry assumptions. Our results hold for an arbitrary equation of state, not necessarily of barotropic type.

On the existence of solutions and causality for relativistic viscous conformal fluids. Communications in Pure and Applied Analysis, Vol. 18, No. 4, pp. 1567-1599 (2019).
We consider a stress-energy tensor describing a pure radiation viscous fluid with conformal symmetry introduced in arXiv:1708.06255 [gr-qc]. We show that the corresponding equations of motions are causal in Minkowski background and also when coupled to Einstein's equations, and solve the associated initial-value problem.

On the Incompressible Limit for the Compressible Free-Boundary Euler Equations with Surface Tension in the Case of a Liquid. (with Chenyun Luo.) Archive for Rational Mechanics and Analysis, 58 pages (to appear)
In this paper we establish the incompressible limit for the compressible free-boundary Euler equations with surface tension in the case of a liquid. Compared to the case without surface tension treated recently, the presence of surface tension introduces severe new technical challenges, in that several boundary terms that automatically vanish when surface tension is absent now contribute at top order. Combined with the necessity of producing estimates uniform in the sound speed in order to pass to the limit, such difficulties imply that neither the techniques employed for the case without surface tension, nor estimates previously derived for a liquid with surface tension and fixed sound speed, are applicable here. In order to obtain our result, we devise a suitable weighted energy that takes into account the coupling of the fluid motion with the boundary geometry. Estimates are closed by exploiting the full non-linear structure of the Euler equations and invoking several geometric properties of the boundary in order to produce some remarkable cancellations. We stress that we do not assume the fluid to be irrotational.

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.

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.

Rough sound waves in 3D compressible Euler flow with vorticity. (with Chenyun Luo, Giusy Mazzone, and Jared Speck.) arXiv:1909.02550 [math.AP], 100 pages (2019).
We prove a series of results tied to the regularity and geometry of solutions to the 3D compressible Euler equations with vorticity and entropy. Our framework exploits and reveals additional virtues of a recent new formulation of the equations, which decomposed the flow into a geometric "(sound) wave-part" coupled to a "transport-div-curl-part" (transport-part for short), with both parts exhibiting remarkable properties. Our main result is that the time of existence can be controlled in terms of the H2+(?3)-norm of the wave-part of the initial data and various Sobolev and Hölder norms of the transport-part of the initial data, the latter comprising the initial vorticity and entropy. The wave-part regularity assumptions are optimal in the scale of Sobolev spaces: shocks can instantly form if one only assumes a bound for the H2(?3)-norm of the wave-part of the initial data. Our proof relies on the assumption that the transport-part of the initial data is more regular than the wave-part, and we show that the additional regularity is propagated by the flow, even though the transport-part of the flow is deeply coupled to the rougher wave-part. To implement our approach, we derive several results of independent interest: i) sharp estimates for the acoustic geometry, i.e., the geometry of sound cones; ii) Strichartz estimates for quasilinear sound waves coupled to vorticity and entropy; and iii) Schauder estimates for the transport-div-curl-part. Compared to previous works on low regularity, the main new features of the paper are that the quasilinear PDE systems under study exhibit multiple speeds of propagation and that elliptic estimates for various components of the fluid are needed, both to avoid loss of regularity and to gain space-time integrability.

The relativistic Euler equations with a physical vacuum boundary: Hadamard local well-posedness, rough solutions, and continuation criterion. (with Daniel Tataru and Mihaela Ifrim.) arXiv:2007.05787 [math.AP], 48 pages (2020).
In this paper we provide a complete local well-posedness theory for the free boundary relativistic Euler equations with a physical vacuum boundary on a Minkowski background. Specifically, we establish the f ollowing results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and continuous dependence on the data; (ii) low regularity solutions: our uniqueness result holds at the level of Lipschitz velocity and density, while our rough solutions, obtained as unique limits of smooth solutions, have regularity only a half derivative above scaling; (iii) stability: our uniqueness in fact follows from a more general result, namely, we show that a certain nonlinear functional that tracks the distance between two solutions (in part by measuring the distance between their respective boundaries) is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions; (v) a sharp continuation criterion, at the level of scaling, showing that solutions can be continued as long as the the velocity is in L1tLip and a suitable weighted version of the density is at the same regularity level. Our entire approach is in Eulerian coordinates and relies on the functional framework developed in the companion work of the second and third authors corresponding to the non relativistic problem. All our results are valid for a general equation of state p(ϱ)=ϱγ, γ>1.

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

Recent developments in the theory of relativistic fluids - Notes (.pdf) of a series of lectures I gave at the Summer School on Recent Advances in Mathematical Fluid Dynamics, May 20-24, at USC. Here is a .zip file with the .tex, .bbl, and pictures of the notes. People are welcome to download, modify, and use the .tex file as they see fit (and here are the handwritten notes, just in case). Here is some extra material that complements the notes. Other topics covered in the summer school were Non-uniqueness of weak solutions to the Euler and Navier-Stokes equations, by Tristan Buckmaster, Recent developments on water waves, by Yu Deng, and Nonlinear dynamics of the Schrodinger equations with periodic boundary conditions, by Emanuele Haus.

Recent advances in classical and relativistic fluids - Notes (.pdf) of a series of lectures I gave at the summer school Boston City Limits 2018, June 11-21, at MIT. Here is a .zip file with the .tex, .bbl, and pictures of the notes. People are welcome to download, modify, and use the .tex file as they see fit (and here are the handwritten notes, just in case). Other topics covered in the summer school were mathematical general relativity, by Stefanos Aretakis (notes here), the formation of singularities in general relativity, by Jared Speck (notes TBP), and solitions, bubbling, and blow-up for semilinear PDEs, by Andrew Lawrie (notes here).

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 - 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).

Postdocs and PhD students
Chenyun Luo, postdoc (2017-2020).

Media and outreach

Here is Vanderbilt news story about a NSF Research Trainee award for which I am a Co-Principla Investigator together with Principal Investigator Kelly Holley-Bockeklmann.. The goal is to establish a graduate certificate program in the emerging field of multimessenger astronomy.

General-relativistic viscous fluids. Video of a talk I gave at the Princeton Gravity Initiative, Princeton University.

General-relativistic viscous fluids. Video of a talk I gave in the Workshop on QGP Phenomenology.

General relativistic viscous fluids. Video of a talk I gave at the Colloquium of the Zu Chongzhi Center for Mathematics and Computational Sciences, Duke Kunshan University.

The relativistic Euler equations with a physical vacuum boundary. Video of a talk I gave in the Center for Nonlinear Analysis at Carnegie Mellon University.

General-relativistic viscous fluids. Video of a talk I gave in the workshop Relativistic Hydrodynamics: Foundations, Novel Applications and Interdisciplinary Connections that was jointly organized by the Gravity, Quantum Fields and Information group at the Max Planck Institute for Gravitational Physics (Albert Einstein Institute) and the Illinois Center for Advanced Studies of the Universe. Click here for the slides of the talk. Here is a discussion session of the workshop (the discussion of my talk starts at 21:57min).

General-relativistic viscous fluids. Video of a talk I gave in the workshop Mathematical and Computational Approaches for the Einstein Field Equations with Matter Fields that took place at the Institute for Computational and Experimental Research in Mathematics (ICERM) in Providence, RI. Click here for the slides of the talk.

General-relativistic viscous fluids. Video of a talk I gave in the Webinar on Quark Matter and Relativistic Hydrodynamics (Physics Department, Sharif University of Technology, Tehran, Iran). The talk is followed by nearly an hour of discussion with the audience.

Strichartz estimates for the compressible Euler equation with vorticity and low-regularity solutions. Video of a talk I gave in the workshop Dynamics in Geometric Dispersive Equations and the Effects of Trapping, Scattering and Weak Turbulence that took place at the Banff International Research Station in Banff, Canada (2020).

Here is a Vanderbilt news story of a Robert Noyce Scholarship grant (2019-2024) for which I am a Co-Principal Investigator together with Principal Investigators Heather Johnson and Teresa Dunleavy, of the Department of Teaching and Learning, and Co-Principal Investigators David Weintraub of the Physics Department, and Isaac Thompson from Fisk University.

Here is a news story with a short video description (in Portuguese) by the Sociedade Brasileira de Fisica (Brazilian Physical Society) describing my work Causality of the Einstein-Israel-Stewart Theory with Bulk Viscosity (2019), written with Fabio S. Bemfica and Jorge Noronha. Click here to access the video directly (.mov file). The follow-up work, Nonlinear Constraints on Relativistic Fluids Far From Equilibrium (2021), that I wrote with Fabio S. Bemfica, Vu Hoang, Jorge Noronha, and Maria Radosz, was also featured (in Portuguese) by the Sociedade Brasileira de Fisica. (Click here to access the video directly, .mov file.)

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.

Here is a department news story on the occasion of my 2018 Dean's Faculty Fellowship award.

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 (2016).

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, 2013). 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 (2013), Teoria de supercordas (2013), A grande ruptura cosmica (big rip) (2017), and a discussion about the work of Stephen Hawking (2018) on the occasion of his passing (mp3 files).