Serfati solutions to the incompressible 2D Euler equations in an exterior domain

Speaker: 

Helena Nussenzveig Lopes

Institution: 

University of Campinas, Brazil

Time: 

Thursday, January 12, 2012 - 3:00pm

Location: 

RH 306

In 1963 V. I. Yudovich proved the existence and uniqueness of weak solutions of the incompressible 2D Euler equations assuming that the vorticity, which is the curl of velocity, is bounded and integrable in the full plane. A few extensions of this result have been established, most notably by Yudovich himself and, also, by M. Vishik, always assuming some decay of vorticity at infinity. Paradoxically, if the vorticity is doubly-periodic then there is no difficulty in establishing well-posedness of weak solutions, as long as the vorticity is also bounded. In this talk I will report on work in progress aimed at extending, for 2D flows in a domain exterior to an island, well-posedness of weak solutions to include all vorticities which are bounded and are the curl of a bounded velocity field. This work is related to recent results by Taniuchi, Tashiro and Yoneda and it builds on previous, albeit incomplete, work due to Ph. Serfati, where flow in the full plane was considered.

This is joint work with J. P. Kelliher (UCR) and M. C. Lopes Filho (UNICAMP).

Vacuum in Gas and Fluid Dynamics

Speaker: 

Juhi Jang

Institution: 

The University of California - Riverside

Time: 

Friday, May 27, 2011 - 4:00pm

Location: 

RH 306

An interesting problem in gas and fluid dynamics is to understand the behavior of vacuum states, namely the behavior of the system in the presence of vacuum. A particular interest is so called physical vacuum which naturally arises in physical problems. I'll report on a recent progress in a rigorous study of a physical vacuum. If time permits, I'll also discuss stability theory of Lane-Emden equilibrium stars under Euler-Poisson or Navier-Stokes-Poisson system.

Discrete Data Assimilation in the 2D Navier-Stokes Equations

Speaker: 

Eric Oslon

Institution: 

University of Nevada - Reno

Time: 

Monday, May 16, 2011 - 4:00pm

Location: 

RH 306

Consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times. Discrete data assimilation inserts these observational measurements of the reference dynamical system into an approximate solution by means of an impulsive forcing. In this way the approximating solution is coupled to the reference solution at a discrete sequence of points in time. In this lecture I will discuss the discrete data assimilation for the incompressible two-dimensional Navier-Stokes
equations. In both cases we obtain bounds on the time interval between subsequent observations which guarantee the convergence of the approximating solution obtained by discrete data assimilation to the reference solution.

Strong Unique Continuation for the Navier-Stokes Equation with Non-analytic Forcing

Speaker: 

Mihaela Ignatova

Institution: 

University of Southern California

Time: 

Thursday, May 19, 2011 - 3:00pm

Location: 

RH 440R

The Navier-Stokes system is the classical model for the motion of a viscous incompressible homogeneous fluid. Physically, the equations
express Newton's second law of motion and the conservation of mass. The unknowns are the velocity vector field u and the scalar pressure field p, while the volume forces f and the kinematic viscosity are given. In this talk, we address the spatial complexity and the local behavior of solutions to the three dimensional (NSE) with general non-analytic forcing. Motivated by a result of Kukavica and Robinson in [4], we consider a system of elliptic-parabolic type for a diference of two solutions (u1; p1) and (u2; p2) of (NSE) with the same Gevrey forcing f. By proving delicate Carleman estimates with the same singular weights for the Laplacian and the heat operator (cf. [1, 2, 3]), we establish a quantitive estimate of unique continuation leading to the strong unique continuation property for solutions of the coupled elliptic-parabolic system. Namely, we obtain that if the velocity vector fields u1 and u2 are not identically equal, then their diference
u1-u2 has finite order of vanishing at any point. Moreover, we establish a polynomial estimate on the rate of vanishing, provided the forcing f lies in the Gevrey class for certain restricted range of the exponents. In particular, the necessary condition for the result in [4] is satisfied; thus a finite-dimensional family
of smooth solutions can be distinguished by comparing a finite number of their point values.
This is a joint work with Igor Kukavica.

References
[1] M. Ignatova and I. Kukavica, Unique continuation and complexity of solutions to parabolic partial diferential equations with Gevrey coeficients, Advances in Diferential Equations 15 (2010), 953-975.
[2] M. Ignatova and I. Kukavica, Strong unique continuation for higher order elliptic equations with Gevrey
coeficients, Journal of Diferential Equations (submitted in August, 2010).
[3] M. Ignatova and I. Kukavica, Strong unique continuation for the Navier-Stokes equation with non-analytic forcing, Journal of Dynamics and Diferential Equations (submitted in January, 2011).
[4] I. Kukavica and J.C. Robinson, Distinguishing smooth functions by a finite number of point values, and a version of the Takens embedding theorem, Physica D 196 (2004), 45-66.

Parabolic approximation of the 3D incompressible Navier-Stokes equations.

Speaker: 

Walter Rusin

Institution: 

University of Southern California

Time: 

Thursday, May 26, 2011 - 3:00pm

Location: 

RH 440R

Solutions of the Navier-Stokes equations (NSE) satisfy the same scaling invariance as the solutions of the heat equation. However, as opposed to the exponential decay of the heat kernel, the kernel of the solution operator of the linear problem associated with NSE (the Stokes system), has only polynomial decay. We consider a parabolic system that shares many features with NSE (scaling, energy estimate) and show that it may be thought of as an approximation of the Navier-Stokes equations. In particular, we address the problem of convergence of solutions to solutions of NSE and partial regularity questions.

On the Loss of Regularity for the Three-Dimensional Euler Equations

Speaker: 

Edriss S. Titi

Institution: 

University of Califonia-Irvine and Weizmann Institute of Science

Time: 

Thursday, April 7, 2011 - 3:00pm

Location: 

RH 440R

A basic example of shear flow was introduced by DiPerna and Majda to study the weak limit of oscillatory solutions of the Euler equations of incompressible ideal fluids. In particular, they proved by means of this example that weak limit of solutions of Euler equations may, in some cases, fail to be a solution of Euler equations. We use this shear flow example to provide non-generic, yet nontrivial, examples concerning the immediate loss of smoothness and ill-posedness of solutions of the three-dimensional Euler equations, for initial data that do not belong to $C^{1,\alpha}$. Moreover, we show by means of this shear flow example the existence of weak solutions for the three-dimensional Euler equations with vorticity that is having a nontrivial density concentrated on non-smooth surface. This is very different from what has been proven for the two-dimensional Kelvin-Helmholtz problem where a minimal regularity implies the real analyticity of the interface. Eventually, we use this shear flow to provide explicit examples of non-regular solutions of the three-dimensional Euler equations that conserve the energy, an issue which is related to the Onsager conjecture.

This is a joint work with Claude Bardos.

Lp and Schauder estimates for non-local elliptic equations.

Speaker: 

Prof. Hongjie Dong

Institution: 

Brown University

Time: 

Thursday, April 21, 2011 - 3:00pm

Location: 

RH 440R

I will discuss some recent results about Lp and Schauder estimates for a class of non-local elliptic equations. Compared to previous known results, the novelty of our results is that the kernels of the operators are not necessarily to be homogeneous, regular, or symmetric.

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