Faculty of Science
Faculty of Science
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Inhomogeneous and compressible fluids: statistical solutions and dissipative anomalies
Research Project  | 1 Project Members

This project addresses several physically motivated questions on the analysis of equations of fluid dynamics. The project has a foundational character and aims at providing a rigorous analytical understanding which will expand our knowledge in the following two directions:

  • From homogeneous incompressible fluids to inhomogeneous and compressible fluids.
  • From classical notions of weak solutions to generalized notions of weak solutions.

In the last ten years, major progress has been achieved on several long-standing questions rooted in the Kolmogorov-Onsager theory of turbulence, most notably the proof via convex integration of the existence of nonconservative solutions of the Euler equations with regularity below 1/3, in which energy is dissipated through creation of small spatial scales. The transport structure of the vorticity equation in two spatial dimensions and the presence of an inverse cascade have been exploited to prove energy conservation and further properties in some Onsager-supercritical regimes. Refined numerical simulations and the manifested insufficiency of the entropy conditions to select a unique solution in many hyperbolic problems led the way to generalized notions of weak solutions, including statistical and measure-valued solutions, with an extended effort in their analysis and computation. 


In spite of all this exciting recent progress, most of the theory is presently limited to homogeneous incompressible fluids and to classical weak solutions. In this project, we aim to go significantly beyond the state of the art, addressing in particular the following topics:

  1. Energy conservation and further properties of Onsager-supercritical statistical solutions for homogeneous incompressible fluids originating from vanishing viscosity or from numerical approximation schemes.
  2. Statistical solutions for inhomogeneous and compressible fluids: foundations.
  3. Energy conservation and further properties of Onsager-supercritical weak solutions for inhomogeneous incompressible fluids.
  4. Conservation anomalies in statistical solutions and for inhomogeneous incompressible fluids.
  5. The same topics as in (C)-(D) for inhomogeneous compressible fluids.


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Nonlocal equations in mathematical fluid dynamics
Research Project  | 1 Project Members

In this project, the PI aims to study partial differential equations (PDEs) arising from fluid dynamics. In a mathematical perspective, the fluid PDEs typically display nonlinear/nonlocal structure, which is the main source of difficulty in the analysis. Although such nonlinear/nonlocal structure is a ubiquitous feature in fluid PDEs, it reveals different phenomena in different situations, therefore different strategies must be taken accordingly in the analysis. For this reason, the PI proposes several research directions in which distinct behaviors of fluids will be investigated.Direction A) Steady state flow: A steady state flow is often given as a solution to an overdetermined elliptic PDE, which immediately implies that not every state can be a steady state. Steady fluids sometimes exhibit trivial structure but not always, and this feature strongly depends on the property of the nonlocal operator inherent to the equation. This sub-project aims to establish a necessary/sufficient condition for a steady fluid to enjoy non-trivial structure and discover its distinctive features. In particular, this research component will be focused on the two-dimensional incompressible Euler equation (2D Euler).Direction B) Fluids near a stable steady state: Tautologically, a smooth, steady fluid remains smooth for all time since it does not evolve non-trivially in time. A natural question is whether a small perturbation of such a smooth steady state can also remain smooth for all time. Especially, when the well-posedness of the governing equation is in question, the investigation of initial data near stable steady states can give insight into potential global-in-time solutions. In the second part of this proposal, the construction of global solutions to the generalized surface quasi-geostrophic equations (gSQG) is studied by means of KAM theory.Direction C) Long-time behavior of fluids: One of the fundamental questions in PDEs is whether general smooth initial data can develop a finite-time singularity or remain smooth for all time. Even if the solution remains smooth, it is often observed that its long-time behavior can be highly complicated. The main objective in this direction is to understand the long- time behavior of global solutions generated by the nonlinearity of the equations, especially instability of an equilibrium and a finite time singularity formation. The research project in this direction is centered around the two-dimensional incompressible Boussinesq system (2D Boussinesq), and its one-dimensional toy-model.The mathematical tools in this project vary from variational flavor techniques to microlocal analysis. The unifying goal of the whole project is to understand the role of nonlinearity/nonlocality in various situations of fluids, and develop/extend modern techniques to address related open problems.

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FLUTURA: Fluids, Turbulence, Advection
Research Project  | 1 Project Members

The Euler and the Navier-Stokes equations are basic mathematical models for the dynamics of incompressible fluids. Despite being among the first partial differential equations (PDEs) ever written, to a very large extent their mathematical theory remains a riddle. One of the main reasons is the intrinsic lack of regularity of solutions of problems in fluid dynamics. The theory of turbulence advances quantitative predictions for the "typical" properties of "chaotic" fluids, in particular with regard to universal behaviors at small spatial scales. These predictions are well supported by experimental observations, but very few is understood from the point of view of rigorous analysis. Understanding the correct framework validating and predicting the experimental evidence is an exciting and ambitious mathematical question.

In this project we address several challenging open problems in mathematical fluid dynamics. The common feature of all these problems is the appearance of irregular behaviors and the cascade of information to small spatial scales. The irregularity is the main source of difficulty in the mathematical analysis of these questions, since it precludes the use of most classical PDE methods, from energy estimates to the theory of characteristics.

We aim to derive a quantitative analysis and to provide detailed insight of the properties of solutions in physically relevant regimes. This project will significantly expand our theoretical understanding of several phenomena involving irregular flows, addressing in particular the following topics:

  • Selection and nonselection by vanishing viscosity.
  • Anomalous dissipation in the Obukhov-Corrsin theory of scalar turbulence.
  • Conservation and dissipation of the kinetic energy and of the enstrophy in fluid flows.
  • Mixing, regularity, and irregularity for fluid flows.
  • Existence, uniqueness, and structure of solutions for nonlinear problems with diffuse/singular vorticity.

The unity of the project stems from the concepts and the methods in the background, all related to the theory of advection (both deterministic and stochastic) in a nonsmooth setting and relying in an essential way on techniques of geometric measure theory as a crucial tool in the analysis of the fine properties of irregular flows.

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FLIRT: Fluid Flows and Irregular Transport
Research Project  | 9 Project Members
Several important partial differential equations (PDEs) arising in the mathematical description of physical phenomena exhibit transport features: physical quantities are advected by velocity fields that drive the dynamics of the system. This is the case for instance for the Euler equation of fluid dynamics, for conservation laws, and for kinetic equations. medskip An ubiquitous feature of these phenomena is their intrinsic lack of regularity. From the mbox{mathematical} point of view this stems from the nonlinearity and/or nonlocality of the PDEs. Moreover, the lack of regularity also encodes actual properties of the underlying physical systems: conservation laws develop shocks (discontinuities that propagate in time), solutions to the Euler equation exhibit rough and ``disordered'' behaviors. This irregularity is the major difficulty in the mathematical analysis of such problems, since it prevents the use of many standard methods, foremost the classical (and powerful) theory of characteristics. medskip For these reasons, the study in a non smooth setting of transport and continuity equations, and of flows of ordinary differential equations, is a fundamental tool to approach challenging important questions concerning these PDEs. medskip This project aims at establishing: medskip (1) deep insight into the structure of solutions of nonlinear PDEs, in particular the Euler equation and multidimensional systems of conservation laws, medskip (2) rigorous bounds for mixing phenomena in fluid flows, phenomena for which giving a precise mathematical formulation is extremely challenging. medskip The unifying factor of this proposal is that the analysis will rely on major advances in the theory of flows of ordinary differential equations in a non smooth setting, thus providing a robust formulation via characteristics for the PDEs under consideration. The guiding thread is the crucial role of geometric measure theory techniques, which are extremely efficient to describe and investigate irregular phenomena. Several important partial differential equations (PDEs) arising in the mathematical description of physical phenomena exhibit transport features: physical quantities are advected by velocity fields that drive the dynamics of the system. This is the case for instance for the Euler equation of fluid dynamics, for conservation laws, and for kinetic equations. An ubiquitous feature of these phenomena is their intrinsic lack of regularity. From the mathematical point of view this stems from the nonlinearity and/or nonlocality of the PDEs. Moreover, the lack of regularity also encodes actual properties of the underlying physical systems: conservation laws develop shocks (discontinuities that propagate in time), solutions to the Euler equation exhibit rough and "disordered" behaviors. This irregularity is the major difficulty in the mathematical analysis of such problems, since it prevents the use of many standard methods, foremost the classical (and powerful) theory of characteristics. For these reasons, the study in a non smooth setting of transport and continuity equations, and of flows of ordinary differential equations, is a fundamental tool to approach challenging important questions concerning these PDEs. This project aims at establishing: (1) deep insight into the structure of solutions of nonlinear PDEs, in particular the Euler equation and multidimensional systems of conservation laws, (2) rigorous bounds for mixing phenomena in fluid flows, phenomena for which giving a precise mathematical formulation is extremely challenging. The unifying factor of this proposal is that the analysis will rely on major advances in the theory of flows of ordinary differential equations in a non smooth setting, thus providing a robust formulation via characteristics for the PDEs under consideration. The guiding thread is the crucial role of geometric measure theory techniques, which are extremely efficient to describe and investigate irregular phenomena.
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Kinetic description of emerging challenges in multiscale problems of natural sciences
Research Project  | 1 Project Members
The ultimate goal of this network is the development, analysis and computation of novel kinetic descriptions with particular focus on Quantum dynamics with applications to chemistry; Network dynamics with applications to social sciences; Kinetic models of biological processes. KI-Net is offering a unique platform to carry out these objectives by fostering cross-fertilization between mathematics and other scientific disciplines. It is centered around three hubs: the Center for Scientific Computation & Math Modeling ( CSCAMM ) in the University of Maryland, the Institute for Computational and Engineering Science ( ICES ) at UT Austin, and the Department of Mathematics at the University of Wisconsin-Madison.
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Continuity equations with non smooth velocity: quantitative estimates and applications to nonlinear problems
Research Project  | 3 Project Members
We investigate well-posedness and further properties for the continuity equation and for the ordinary differential equation out of the classical smooth (Lipschitz regular) framework. The leading theme is the search for quantitative estimates: stability, compactness and regularity statements in which we give an explicit control of the quantities under analysis in terms of natural bounds on the data. This quantitative analysis allows us to study several applications to nonlinear partial differential equations: we address various questions about existence, continuity with respect to initial data, convergence of singular approximations and convergence of nonlocal approximations for the Euler equation, the Vlasov-Poisson equation, and the Burgers' equation. We investigate well-posedness and further properties for the continuity equation and for the ordinary differential equation out of the classical smooth (Lipschitz regular) framework. The leading theme is the search for quantitative estimates: stability, compactness and regularity statements in which we give an explicit control of the quantities under analysis in terms of natural bounds on the data. This quantitative analysis allows us to study several applications to nonlinear partial differential equations: we address various questions about existence, continuity with respect to initial data, convergence of singular approximations and convergence of nonlocal approximations for the Euler equation, the Vlasov-Poisson equation, and the Burgers' equation.
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Continuity equations with non smooth velocity: fluid dynamics and further applications
Research Project  | 2 Project Members
We will address various open problems related to the behaviour of the continuity equation and of the associated ordinary differential equation when the vector field governing the transport process lacks the usual (Lipschitz) regularity properties. The motivations come from the applications of such results to nonlinear problems, originating in fluid dynamics or in the theory of conservation laws. Besides exploiting hyperbolic PDEs techniques, the analysis requires new tools from geometric measure theory, properly adapted in order to describe and control the irregular behaviours under consideration. One first line of work regards a precise understanding of further suitable weak settings in which the continuity equation and the ordinary differential equation are well-posed and enjoy additional properties (compactness or regularity of solutions, for instance). A second line will address some questions on two-dimensional incompressible nonviscous fluids, mainly in the framework of measure-valued vorticity. We will address various open problems related to the behaviour of the continuity equation and of the associated ordinary differential equation when the vector field governing the transport process lacks the usual (Lipschitz) regularity properties. The motivations come from the applications of such results to nonlinear problems, originating in fluid dynamics or in the theory of conservation laws. Besides exploiting hyperbolic PDEs techniques, the analysis requires new tools from geometric measure theory, properly adapted in order to describe and control the irregular behaviours under consideration. One first line of work regards a precise understanding of further suitable weak settings in which the continuity equation and the ordinary differential equation are well-posed and enjoy additional properties (compactness or regularity of solutions, for instance). A second line will address some questions on two-dimensional incompressible nonviscous fluids, mainly in the framework of measure-valued vorticity.