Marcelo Mendes Disconzi

General
Here you will find information about my academic work. Click here for research-related notes and videos.

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 NSF grant DMS-2107701, from DOE grant DE-SC0024711, from a Chancellor's Faculty Fellowship, and from a Vanderbilt's Seeding Success Grant. I also acknowledge past support from a Sloan Research Fellowship, the NSF, and Vanderbilt's internal grants and fellowship programs.

Click here for my CV.

Click here for some notes, videos of talks, useful links, and other research-related materials.

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.

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 Holder 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: 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 L¹_tLip 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 Mueller-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.