90 research outputs found
Young Measures Generated by Ideal Incompressible Fluid Flows
In their seminal paper "Oscillations and concentrations in weak solutions of
the incompressible fluid equations", R. DiPerna and A. Majda introduced the
notion of measure-valued solution for the incompressible Euler equations in
order to capture complex phenomena present in limits of approximate solutions,
such as persistence of oscillation and development of concentrations.
Furthermore, they gave several explicit examples exhibiting such phenomena. In
this paper we show that any measure-valued solution can be generated by a
sequence of exact weak solutions. In particular this gives rise to a very
large, arguably too large, set of weak solutions of the incompressible Euler
equations.Comment: 35 pages. Final revised version. To appear in Arch. Ration. Mech.
Ana
On the stability of travelling waves with vorticity obtained by minimisation
We modify the approach of Burton and Toland [Comm. Pure Appl. Math. (2011)]
to show the existence of periodic surface water waves with vorticity in order
that it becomes suited to a stability analysis. This is achieved by enlarging
the function space to a class of stream functions that do not correspond
necessarily to travelling profiles. In particular, for smooth profiles and
smooth stream functions, the normal component of the velocity field at the free
boundary is not required a priori to vanish in some Galilean coordinate system.
Travelling periodic waves are obtained by a direct minimisation of a functional
that corresponds to the total energy and that is therefore preserved by the
time-dependent evolutionary problem (this minimisation appears in Burton and
Toland after a first maximisation). In addition, we not only use the
circulation along the upper boundary as a constraint, but also the total
horizontal impulse (the velocity becoming a Lagrange multiplier). This allows
us to preclude parallel flows by choosing appropriately the values of these two
constraints and the sign of the vorticity. By stability, we mean conditional
energetic stability of the set of minimizers as a whole, the perturbations
being spatially periodic of given period.Comment: NoDEA Nonlinear Differential Equations and Applications, to appea
Steady nearly incompressible vector fields in 2D: chain rule and renormalization
Given bounded vector field , scalar field and a smooth function we study the characterization of the distribution in terms of and . In the case of vector fields (and under some further assumptions) such characterization was obtained by L. Ambrosio, C. De Lellis and J. Mal'y, up to an error term which is a measure concentrated on so-called emph{tangential set} of . We answer some questions posed in their paper concerning the properties of this term. In particular we construct a nearly incompressible vector field and a bounded function for which this term is nonzero.
For steady nearly incompressible vector fields (and under some further assumptions) in case when we provide complete characterization of in terms of and . Our approach relies on the structure of level sets of Lipschitz functions on obtained by G. Alberti, S. Bianchini and G. Crippa.
Extending our technique we obtain new sufficient conditions when any bounded weak solution of is emph{renormalized}, i.e. also solves for any smooth function . As a consequence we obtain new uniqueness result for this equation
Asymptotic Stability of the Relativistic Boltzmann Equation for the Soft Potentials
In this paper it is shown that unique solutions to the relativistic Boltzmann
equation exist for all time and decay with any polynomial rate towards their
steady state relativistic Maxwellian provided that the initial data starts out
sufficiently close in . If the initial data are continuous then
so is the corresponding solution. We work in the case of a spatially periodic
box. Conditions on the collision kernel are generic in the sense of
(Dudy{\'n}ski and Ekiel-Je{\.z}ewska, Comm. Math. Phys., 1988); this resolves
the open question of global existence for the soft potentials.Comment: 64 page
Isometric Immersions and Compensated Compactness
A fundamental problem in differential geometry is to characterize intrinsic
metrics on a two-dimensional Riemannian manifold which can be
realized as isometric immersions into . This problem can be formulated as
initial and/or boundary value problems for a system of nonlinear partial
differential equations of mixed elliptic-hyperbolic type whose mathematical
theory is largely incomplete. In this paper, we develop a general approach,
which combines a fluid dynamic formulation of balance laws for the
Gauss-Codazzi system with a compensated compactness framework, to deal with the
initial and/or boundary value problems for isometric immersions in . The
compensated compactness framework formed here is a natural formulation to
ensure the weak continuity of the Gauss-Codazzi system for approximate
solutions, which yields the isometric realization of two-dimensional surfaces
in . As a first application of this approach, we study the isometric
immersion problem for two-dimensional Riemannian manifolds with strictly
negative Gauss curvature. We prove that there exists a isometric
immersion of the two-dimensional manifold in satisfying our prescribed
initial conditions. TComment: 25 pages, 6 figue
Regularizing effect and local existence for non-cutoff Boltzmann equation
The Boltzmann equation without Grad's angular cutoff assumption is believed
to have regularizing effect on the solution because of the non-integrable
angular singularity of the cross-section. However, even though so far this has
been justified satisfactorily for the spatially homogeneous Boltzmann equation,
it is still basically unsolved for the spatially inhomogeneous Boltzmann
equation. In this paper, by sharpening the coercivity and upper bound estimates
for the collision operator, establishing the hypo-ellipticity of the Boltzmann
operator based on a generalized version of the uncertainty principle, and
analyzing the commutators between the collision operator and some weighted
pseudo differential operators, we prove the regularizing effect in all (time,
space and velocity) variables on solutions when some mild regularity is imposed
on these solutions. For completeness, we also show that when the initial data
has this mild regularity and Maxwellian type decay in velocity variable, there
exists a unique local solution with the same regularity, so that this solution
enjoys the regularity for positive time
The Orbital Stability of the Ground States and the Singularity Formation for the Gravitational Vlasov Poisson System
International audienceWe study the gravitational Vlasov Poisson system where , , \rho(x)=\int_{\RR^N} f(x,v)dxdv, in dimension . In dimension where the problem is subcritical, we prove using concentration compactness techniques that every minimizing sequence to a large class of minimization problems attained on steady states solutions are up to a translation shift relatively compact in the energy space. This implies in particular the orbital stability {\it in the energy space} of the spherically symmetric polytropes what improves the nonlinear stability results obtained for this class in \cite{Guo,GuoRein,Dol}. In dimension where the problem is critical, we obtain the polytropic steady states as best constant minimizers of a suitable Sobolev type inequality relating the kinetic and the potential energy. We then derive using an explicit pseudo-conformal symmetry the existence of critical mass finite time blow up solutions, and prove more generally a mass concentration phenomenon for finite time blow up solutions. This is the first result of description of a singularity formation in a Vlasov setting. The global structure of the problem is reminiscent to the one for the focusing non linear Schrödinger equation in the energy space
On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials
The paper concerns - convergence to equilibrium for weak solutions of
the spatially homogeneous Boltzmann Equation for soft potentials (-4\le
\gm<0), with and without angular cutoff. We prove the time-averaged
-convergence to equilibrium for all weak solutions whose initial data have
finite entropy and finite moments up to order greater than 2+|\gm|. For the
usual -convergence we prove that the convergence rate can be controlled
from below by the initial energy tails, and hence, for initial data with long
energy tails, the convergence can be arbitrarily slow. We also show that under
the integrable angular cutoff on the collision kernel with -1\le \gm<0, there
are algebraic upper and lower bounds on the rate of -convergence to
equilibrium. Our methods of proof are based on entropy inequalities and moment
estimates.Comment: This version contains a strengthened theorem 3, on rate of
convergence, considerably relaxing the hypotheses on the initial data, and
introducing a new method for avoiding use of poitwise lower bounds in
applications of entropy production to convergence problem
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