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

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

Research

General

Here you will 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 (VandyGRAF), 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 Vanderbilt's Seeding Success Grant. I also acknowledge past support from the NSF and Vanderbilt's internal grants and fellowship programs.

Click here for my CV.

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.

Here you will 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 (VandyGRAF), 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 Vanderbilt's Seeding Success Grant. I also acknowledge past support from the NSF and Vanderbilt's internal grants and fellowship programs.

Click here for my CV.

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 and pre-prints

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.

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.

Rough sound waves in 3D compressible Euler flow with vorticity. (with Chenyun
Luo,
Giusy Mazzone, and Jared
Speck.) Selecta Mathematica, Vol. 28, No. 2, Paper No. 41, 153 pages (2022).

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
H^{2+}(?^{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 H^{2}(?^{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: 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.

The relativistic Euler equations with a physical vacuum boundary: Hadamard local well-posedness, rough solutions, and continuation criterion. (with Daniel Tataru
and Mihaela Ifrim.)
Archive for Rational Mechanics and Analysis, Vol. 245, pp. 127-182 (2022).

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 L^{1}_{t}Lip
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)=ϱ^{γ}, γ>1.

First-order General Relativistic Viscous Fluid Dynamics. (with Fabio Bemfica
and Jorge Noronha.)
Physical Review X, Vol. 12, Issue 2, pp. 021044, 42 pages (2022).

We present the first generalization of Navier-Stokes theory to relativity that satisfies all of the following properties: (a) the system coupled to Einstein's equations is causal and strongly hyperbolic; (b) equilibrium states are stable; (c) all leading dissipative contributions are present, i.e., shear viscosity, bulk viscosity, and thermal conductivity; (d) non-zero baryon number is included; (e) entropy production is non-negative in the regime of validity of the theory; (f) all of the above holds in the nonlinear regime without any simplifying symmetry assumptions. These properties are accomplished using a generalization of Eckart's theory containing only the hydrodynamic variables, so that no new extended degrees of freedom are needed as in Müller-Israel-Stewart theories. Property (b), in particular, follows from a more general result that we also establish, namely, sufficient conditions that when added to stability in the fluid's rest frame imply stability in any reference frame obtained via a Lorentz transformation. All our results are mathematically rigorously established. The framework presented here provides the starting point for systematic investigations of general-relativistic viscous phenomena in neutron star mergers.

Notes

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

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

Recent developments in relativistic fluids. - Handwritten notes (Latexed notes to come...)
of an eight-hours series of lectures at the Advanced Studies Institutes (ASI) in Uzbekistan,
part of the USA-Uzbekistan Collaboration for Research, Education and Student Training, July 25-August 4.

A brief overview of recent developments in relativistic fluids. - Handwritten notes (Latexed notes to come...)
of a six-hours series of lectures at the semester program General Relativity, Center of Mathematical Sciences and Applications, Harvard University, May 16-17, 2022. See below for videos of the lectures.

The Classical Gauge Theoretic Structure of the Standard
Model - This is an undergraduate senior thesis (.pdf) which I supervised by
Elijah Sheridan. It provides a helpful "dictionary" for mathematicians
to understand the basic constructions and concepts of classical gauge theory
used by physicists in the standard model.

Introduction to nonlinear
wave equations - Short introduction to the topic by Jonathan Luk.

General relativity - Introduction to mathematical general relativity from a modern perspective by Stefanos Aretakis.

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), Lorenzo Gavassino (2022-2025).

Brian Luczak, graduate student (current), Runzhang Zhong, graduate student (current).

Brian Luczak, graduate student (current), Runzhang Zhong, graduate student (current).

Media and outreach

A brief overview of recent developments in relativistic fluids.
Video of
a six-hours series of lectures at the semester program General Relativity, Center of Mathematical Sciences and Applications, Harvard University (2022). Click here for the second video.

General-relativistic viscous fluids. Video of a
talk I gave in
General Relativity Conference, Center of Mathematical Sciences and Applications, Harvard University (2022).

Here is a news story by
the Illinois Center for Advanced Studies of the Universe at UIUC,
describing my work First-order General Relativistic Viscous Fluid Dynamics (2022), written with
Fabio S. Bemfica and Jorge
Noronha
(2022).

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-Bockelmann. The goal is to establish a graduate certificate program in the emerging field of multimessenger astronomy (2022).

General-relativistic viscous fluids. Video of a
talk I gave at the
Princeton Gravity Initiative,
Princeton University (2021).

General-relativistic viscous fluids. Video of a
talk I gave in the Workshop on QGP Phenomenology (2021).

The relativistic Euler equations with a physical vacuum boundary. Video of a talk I gave at the
Rocky Mountain Mathematical Physics Seminar,
University of Colorado Boulder (2021).

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 (2021).

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 (2021).

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 (2020).

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 (2020).

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 (2020).

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 (2019).

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 (2018).

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

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 "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 (2013).

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).