interaction kernels

ABSTRACT. We consider the family of long-wave unstable lubrication equations ht =−(hhxxx)x − (h m hx)x with m ≥ 3. Given a fixed m ≥ 3, we prove the existence of a weak solution that becomes singular in finite time. Specifically, given compactly supported nonnegative initial data with negative energy, there is a time T ∗ < ∞, determined by m and the H1 norm of the initial data, and a compactly supported nonnegative weak solution such that lim supt→T ∗ ‖h(·,t)‖L ∞ = lim supt→T ∗ ‖h(·,t)‖H1 =∞. We discuss the relevance of these singular solutions to an earlier conjecture [Comm. Pure. Appl. Math. 51 (1998), 625-661] on when finite-time singularities are possible for long-wave unstable lubrication equations. 1.

with mildly singular interaction kernels

This paper studies the transport of a mass µ in R d, d ≥ 2, by a flow field v = −∇K ∗µ. We focus on kernels K = |x | α /α for 2 − d ≤ α < 2. For this range we prove the existence for all time of radially symmetric measure solutions that are monotone decreasing as a function of the radius. The monotonicity is preserved for all time, in contrast to the case α> 2 where radially symmetric solutions are known to lose monotonicity. In the case of the Newtonian potential (α = 2 − d) we show that under the assumption of radial symmetry the equation can be transformed into the inviscid Burgers equation on a half line. It follows that there exists a unique classical solution for all time in the case of monotone data, and a solution defined by a choice of a jump condition in the case of general radially symmetric data. In the case 2 − d < α < 2 and at the critical exponent p we exhibit initial data in L p for which the solution immediately develops a Dirac mass singularity. This extends recent work on the local ill-posedness of solutions at the critical exponent.

We construct a two parameter family of collapsing solutions to the 4+1 Yang-Mills equations and derive the dynamical law of the collapse. Our arguments indicate that this family of solutions is stable. The latter fact is also supported by numerical simulations.

We consider well-posedness of the aggregation equation ∂tu + div(uv) = 0, v = −∇K ∗ u with initial data in P2(R d)∩L p (R d), in dimensions two and higher. We consider radially symmetric kernels where the singularity at the origin is of order |x | α, α> 2 − d, and prove local well-posedness in P2(R d) ∩ L p (R d) for sufficiently large p> ps. In the special case of K(x) = |x|, the exponent ps = d/(d − 1) is sharp for local well-posedness, in that solutions can instantaneously concentrate mass for initial data in P2(R d)∩L p (R d) with p < ps. We also give an Osgood condition on the potential K(x) which guaranties global existence and uniqueness in P2(R d) ∩ L p (R d).

An Eulerian-Lagrangian approach to incompressible fluids that is convenient for both analysis and physics is presented. Bounds on burning rates in combustion and heat transfer in convection are discussed, as well as results concerning spectra. Incompressible fluids are described by the Navier-Stokes equation. Turbulence ([1], [2], [3]) experiments provide measurements that correspond to averages of certain quantities associated to the variables appearing in the Navier-Stokes equation. The present mathematical knowledge about the Navier-Stokes equations is incomplete. Some of the quantities measured in experiments are accessible to mathematical theory. They are usually low order, one-point bulk averages like the time average of integrals of squares of gradients. Most other measured quantities are not amenable to rigorous quantitative a priori analysis. Turbulence is concerned with statistical or collective properties of fluids. Nevertheless, the main impediment to progress in the rigorous analysis of turbulence is the present lack of understanding of possible blow up in individual solutions of the Euler and Navier-Stokes systems. I will discuss briefly the blow up problem and present an Eulerian-Lagrangian approach to fluids. I will also give examples of low order one-point bulk quantities that can be treated with present knowledge and discuss results on certain twopoint quantities. 1 An Eulerian-Lagrangian Approach to Fluids I will start by recalling that the Navier-Stokes-Euler system can be written as an evolution equation for the three-component velocity vector u = u(x, t), ∂u + u · ∇u + ∇p = ν∆u + f;