Studies on singular flows in which either the velocity fields or the vorticity fields change dramatically on small regions are of considerable interests in both the mathematical theory and applications. Important examples of such flows include supersonic shock waves, boundary layers, and motions of vortex sheets, whose studies pose many outstanding challenges in both theoretical and numerical analysis. The aim of this talk is to discuss some of the key issues in studying such flows and to present some recent progress. First we deal with a supersonic flow past a perturbed cone, and prove the global existence of a shock wave for the stationary supersonic gas flow past an infinite curved and symmetric cone. For a general perturbed cone, a local existence theory for both steady and unsteady is also established. We then present a result on global existence and uniqueness of weak solutions to the 2-D Prandtl's system for unsteady boundary layers. Finally, we will discuss some new results on the analysis of the vortex sheets motions which include the existence of 2-D vortex sheets with reflection symmetry; and no energy concentration for steady 3-D axisymmetric vortex sheets.
We review recent advances in understanding singularity and small scales formation in solutions of fluid dynamics equations. The focus is on the Euler and surface quasi-geostrophic (SQG) equations and associated models.
We consider a system of two incompressible fluids separated by a free interface. The first fluid is inviscid, governed by the Euler system, while the second fluid has positive viscosity and is governed by the Navier-Stokes system. We formulate a notion of splash-type self-intersection singularities, and show that this system cannot form such a singularity in finite time. The main obstacle in our analysis is to handle the effect of bulk vorticity induced by the Navier-Stokes flow. To the best of our knowledge this is the first result on preclusion of splash-type singularities in the presence of…
We consider a singular limit problem for the Navier-Stokes system of a rotating compressible fluid, where the Rossby and Mach numbers tend simultaneously to zero. The limit problem is identified as the 2-D Navier-Stokes system in the ``horizontal'' variables containing an extra term that accounts for compressibility in the original system.
We study singular limit for scaled barotropic Euler system modelling a rotating, compressible and inviscid fluid, where Mach and Rossby numbers are proportional to a small parameter $ε$. If the fluid is confined to an infinite slab, the limit behaviour is identified as a horizontal motion of an incompressible inviscid system that is analogous to the Euler system. We consider a very general class of solutions, named dissipative solution for the scaled compressible Euler systems and will show that it converges to a strong solution of that incompressible inviscid system.
We consider a singular limit problem for the complete compressible Euler system in the low Mach and strong stratification regime. We identify the limit problem - the anelastic Euler system - in the case of well prepared initial data. The result holds in the large class of the dissipative measure-valued solutions of the primitive system. Applications are discussed to the driven shallow water equations.
In bounded smooth domains $Ω\subset\mathbb{R}^N$, $N\in\{2,3\}$, considering the chemotaxis--fluid system \[ \begin{cases} \begin{split} & n_t + u\cdot \nabla n &= Δn - χ\nabla \cdot(\frac{n}{c}\nabla c) &\\ & c_t + u\cdot \nabla c &= Δc - c + n &\\ & u_t + κ(u\cdot \nabla) u &= Δu + \nabla P + n\nabla Φ& \end{split}\end{cases} \] with singular sensitivity, we prove global existence of classical solutions for given $Φ\in C^2(\barΩ)$, for $κ=0$ (Stokes-fluid) if $N=3$ and $κ\in\{0,1\}$ (Stokes- or Navier--Stokes fluid) if $N=2$ and under the condition that \[ 0<χ<\sqrt{\frac{2}{N}}. \]
We study the singular limit of a rotating compressible fluid described by a scaled barotropic Navier-Stokes system, where the Rossby number, the Mach number and the Froude number tend to 0 in a particular mutual rate while the Reynolds number tends to infinity. The inviscid planar Euler system is identified as the limit problem. The proof is based on the application of the method of relative entropies and careful analysis of oscillatory integrals describing the propagation of Rossby-acoustic waves.
In this paper we analyze a 2D free-boundary viscoelastic fluid model of Oldroyd-B type at infinite Weissenberg number. Our main goal is to show the existence of the so-called splash singularities, namely points where the boundary remains smooth but self-intersects. The combination of existence and stability results allows us to construct a special class of initial data, which evolve in time into self-intersecting configurations. To this purpose we apply the classical conformal mapping method and later we move to the Lagrangian framework, as a consequence we deduce the existence of splash singu…
In bounded smooth domains $Ω\subset\mathbb{R}^N$, $N\in\{2,3\}$, we consider the Keller-Segel-Stokes system \begin{align*} n_t + u\cdot \nabla n &= Δn - χ\nabla \cdot(\frac{n}{c}\nabla c),\\ c_t + u\cdot \nabla c &= Δc - c + n,\\ u_t &= Δu + \nabla P + n\nabla ϕ, \qquad \nabla \cdot u=0, \end{align*} and prove global existence of generalized solutions if \[ χ<\begin{cases} \infty,&N=2,\\ \frac{5}{3},&N=3. \end{cases} \] These solutions are such that blow-up into a persistent Dirac-type singularity is excluded.
In the present paper we study a singular perturbation problem for a Navier-Stokes-Korteweg model with Coriolis force. Namely, we perform the incompressible and fast rotation asymptotics simultaneously, while we keep the capillarity coefficient constant in order to capture surface tension effects in the limit. We consider here the case of variable rotation axis: we prove the convergence to a linear parabolic-type equation with variable coefficients. The proof of the result relies on compensated compactness arguments. Besides, we look for minimal regularity assumptions on the variations of the a…
In this article our goal is to study the singular limits for a scaled barotropic Euler system modelling a rotating, compressible and inviscid fluid, where Mach number $=ε^m $, Rossby number $=ε$ and Froude number $=ε^n $ are proportional to a small parameter $ε\rightarrow 0$. The fluid is confined to an infinite slab, the limit behaviour is identified as the incompressible Euler system. For \emph{well--prepared} initial data, the convergence is shown on the life span time interval of the strong solutions of the target system, whereas a class of generalized \emph{dissipative solutions} is consi…
In this paper we discuss the existence of stationary incompressible fluids with splash singularities. Specifically, we show that there are stationary solutions to the Euler equations with two fluids whose interfaces are arbitrarily close to a splash, and that there are stationary water waves with splash singularities.
We study the low Mach low Freude numbers limit in the compressible Navier-Stokes equations and the transport equation for evolution of an entropy variable -- the potential temperature $Θ$. We consider the case of well-prepared initial data on "flat" tours and Reynolds number tending to infinity, and the case of ill-prepared data on an infinite slab. In both cases, we show that the weak solutions to the primitive system converge to the solution to the anelastic Navier-Stokes system and the transport equation for the second order variation of $Θ$
A two-dimensional steady problem of a potential free-surface flow of an ideal incompressible fluid caused by a singular sink is considered. The sink is placed at the horizontal bottom of the fluid layer. With the help of the Levi-Civita technique, the problem is rewritten as an operator equation in a Hilbert space. It is proven that there exists a unique solution of the problem provided that the Froude number is greater than some particular value. The free boundary corresponding to this solution is investigated. It has a cusp over the sink and decreases monotonically when going from infinity t…
The singular limit of the thin film Muskat problem is performed when the density (and possibly the viscosity) of the lighter fluid vanishes and the porous medium equation is identified as the limit problem. In particular, the height of the denser fluid is shown to converge towards the solution to the porous medium equation and an explicit rate for this convergence is provided in space dimension d $\le$ 4. Moreover, the limit of the height of the lighter fluid is determined in a certain regime and is given by the corresponding initial condition.
We consider a model of steady, incompressible non-Newtonian flow with neglected convective term under external forcing. Our structural assumptions allow for certain non-degenerate power-law or Carreau-type fluids. We provide the full-range theory, namely existence, optimal regularity and uniqueness of solutions, not only with respect to forcing belonging to Lebesgue spaces, but also with respect to their refinements, namely the weighted Lebesgue spaces, with weights in a respective Muckenhoupt class. The analytical highlight is derivation of existence and uniqueness theory for forcing with its…
We discuss the regularity of solutions of 2D incompressible Navier-Stokes equations forced by singular forces. The problem is motivated by the study of complex fluids modeled by the Navier-Stokes equations coupled to a nonlinear Fokker-Planck equation describing microscopic corpora embedded in the fluid. This leads naturally to bounded added stress and hence to $W^{-1,\infty}$ forcing of the Navier-Stokes equations.
In the analysis of PDEs, regularity of often measured in terms of Sobolev, H{ö}lder, Besov or Lipschitz spaces, etc. However, sometimes a gain of regularity can also be expressed just in terms of Lebesgue spaces, by passing from a singular setting to a less singular one. In this article we will obtain a gain of integrability for weak solutions of the micropolar fluid equations using as general framework Morrey spaces, which is a very useful language to study regularity in PDEs. An interesting point is that the two variables of the micropolar fluid equations can be studied separately.
We consider self-gravitating fluids in cosmological spacetimes with Gowdy symmetry on the torus $T^3$ and, in this class, we solve the singular initial value problem for the Einstein-Euler system of general relativity, when an initial data set is prescribed on the hypersurface of singularity. We specify initial conditions for the geometric and matter variables and identify the asymptotic behavior of these variables near the cosmological singularity. Our analysis of this class of nonlinear and singular partial differential equations exhibits a condition on the sound speed, which leads us to the…
We explain the construction of some solutions of the Stokes system with a given set of singular points, in the sense of Caffarelli, Kohn and Nirenberg. By means of a partial regularity theorem (proved elsewhere), it turns out that we are able to show the existence of a suitable weak solution to the Navier-Stokes equations with a singular set of positive one dimensional Hausdorff measure.
We prove that for large enough data, the life span of smooth solutions to the Cauchy problem for the following two quasilinear hyperbolic systems is finite: (1) equations of relativistic compressible fluid dynamics, (2) equations of plasma dynamics (Euler-Maxwell equations with background charge) in the spherically symmetric case.
We show that certain singular structures (Hölderian cusps and mild divergences) are transported by the flow of homeomorphisms generated by an Osgood velocity field. The structure of these singularities is related to the modulus of continuity of the velocity and the results are shown to be sharp in the sense that slightly more singular structures cannot generally be propagated. For the 2D Euler equation, we prove that certain singular structures are preserved by the motion, e.g. a system of $\log\log_+(1/|x|)$ vortices (and those that are slightly less singular) travel with the fluid in a nonli…
In this paper, we establish a local theory, i.e., existence, uniqueness and blow-up criterion, for a general family of singular SDEs in some Hilbert space. The key requirement is an approximation property that allows us to embed the singular drift and diffusion mappings into a hierarchy of regular mappings that are invariant with respect to the Hilbert space and enjoy a cancellation property. Various nonlinear models in fluid dynamics with transport noise belong to this type of singular SDEs. With a cancellation estimate for generalized Lie derivative operators, we can construct such regular a…
This paper addresses the mathematical models for the heat-conduction equations and the Navier-Stokes equations via fractional derivatives without singular kernel.
We investigate the long-term behavior, as a certain regularization parameter vanishes, of the three-dimensional Navier-Stokes-Voigt model of a viscoelastic incompressible fluid. We prove the existence of global and exponential attractors of optimal regularity. We then derive explicit upper bounds for the dimension of these attractors in terms of the three-dimensional Grashof number and the regularization parameter. Finally, we also prove convergence of the (strong) global attractor of the 3D Navier-Stokes-Voigt model to the (weak) global attractor of the 3D Navier-Stokes equation. Our analysis…
We consider the incompressible Euler and Navier-Stokes equations in a three-dimensional moving thin domain. Under the assumption that the moving thin domain degenerates into a two-dimensional moving closed surface as the width of the thin domain goes to zero, we give a heuristic derivation of singular limit equations on the degenerate moving surface of the Euler and Navier-Stokes equations in the moving thin domain and investigate relations between their energy structures. We also compare the limit equations with the Euler and Navier-Stokes equations on a stationary manifold, which are describ…
This review article offers a survey of the research program focused on a systematic computational search for extreme and potentially singular behavior in hydrodynamic models motivated by open questions concerning the possibility of a finite-time blow-up in the solutions of the Navier-Stokes system. Inspired by the seminal work of Lu & Doering (2008), we sought such extreme behavior by solving PDE optimization problems with objective functionals chosen based on certain conditional regularity results and a priori estimates available for different models. No evidence for singularity formation was…
