March 3, 2014.

If one considers the 2d Euler equation of incompressible inviscid fluids on the plane in the vorticity form and takes the initial data as the characteristic function of a certain domain, then the Yudovich theory guarantees that the solution will exist globally and will be equal to the characteristic function of a time-dependent domain which is homeomorphic to the original one for all times. The numerical experiments dating back to the works of P. Saffman and Zabusky et al. indicate the existence of the centrally-symmetric V-states, i.e. a symmetric pair of patches that rotates with constant angular velocity around the origin without changing shape. If the distance between the patches in the pair equals to \lambda and \lambda>0, then the boundary of the V-state seems to be smooth. However, when \lambda=0, the both patches form a 90 degrees angle at the point of contact. The analytical proof for the existence of these V-states has never been obtained and this is an interesting problem. In the recent preprint, I considered the analogous equation with the cut-off. Loosely speaking, this corresponds to looking at the window around the origin where the contact of the patches is supposed to happen. Mathematically, the model with cut-off is important as it possesses the explicit singular solution: y_0(x)=|x| . Then, I addressed the problem of existence of the curve of smooth solutions that converge to y_0 in the uniform metric when the parameter \lambda\to 0.  Technically, this boils down to application of the implicit function theorem and is somewhat tedious. This technique might be important to better understand the mechanism of the merging and the sharp corner formation in the Euler dynamics. Another important problem is to prove that the merging in finite time is possible for the \alpha-model when \alpha<1 and is close to 1.

May 2011. Consider H_0=-\Delta, x\in \mathbb{R}^d. Going on the Fourier side one can see that H_0 is equivalent to multiplication by |\omega|^2 and so the spectrum of H_0 is purely absolutely continuous (a.c.). From the physics perspective, the presence of a.c. spectrum is an indication that the wave propagation governed by

\displaystyle i\psi_t=H_0\psi

does have a transport effect (though without much specifics).

Perturb the Laplacian as follows H=H_0+V(x) where V(x) is potential and ask the question what is a minimal assumptions on V to guarantee that the a.c. spectrum is preserved. Make it a perturbation theory question, assume that V is in some weighted Lebesgue space L^p_w. Then, what are the critical p and w? In one-dimensional case, one answer is L^2(\mathbb{R}). This is a critical space. This result was proved for Schrodinger in a great paper by Deift and Killip but for Dirac it was known for at least half a century and dates back to the works by Mark Krein. Krein’s result on Dirac, however, is only a continuous analog of the classical results for polynomials orthogonal on the unit circle (Szego case).

For d>1, the conjecture by Barry Simon is that

\displaystyle \int_{\mathbb{R}^d} \frac{V^2(x)}{|x|^{d-1}+1}dx <\infty

is sufficient for the preservation of a.c. spectrum. Very little is known so far. Only the case of Schrodinger on the Cayley tree is well-understood. Take a rooted Cayley tree \mathbb{B} with the origin at O and assume that each vertex has exactly three neighbors while O has only two. Consider the Laplacian on \mathbb{B} defined at each point as the sum over the neighbors and then a simple calculation shows that the spectrum of H_0 is [-2\sqrt 2, 2\sqrt 2] and it is purely a.c. Then, perturb by V. The multidimensional L^2 result reads as follows. Consider all paths that go from O to infinity without self-intersections (rays). Put the probability measure on them by tossing a Bernoulli coin at each vertex. Then, the claim is that the a.c. spectrum contains the a.c. spectrum of unperturbed operator if with positive probability the potential V(X_n)\in \ell^2 where X_n denotes the path from the origin. There is more quantitative version, of course, which implies Simon’s conjecture if the Jensen inequality is applied. The condition we have here is more general and more physically appealing: it says that we only need enough directions were the potential is small for the particle to propagate.

For d>1, sparse or slowly decaying and oscillating potentials can be handled. If the potential does not oscillate, then the scattering process is quite complicated, it is governed by very intricate evolution equation (that captures semiclassical WKB correction as very special case). This evolution equation is poorly studied and much work is needed in this direction. Soft one-dimensional methods seem to be of little help.

In Euclidean case, what would be the analog of the probability space on the set of paths that escape to infinity? This question was addressed here. It turns out that there is a natural Ito’s stochastic equation that describes these paths. The statement we have is somewhat weak though. It says that the a.c. spectrum contains the positive half-line if V(X_t)\in L^1(\mathbb{R}^+) with positive probability and we can not yet replace summability by the square summability over the path X_t. Nevertheless, even this result gives rise to interesting questions like how one computes probabilities given by this Ito’s calculus? That conventionally can be reduced to the analysis of the corresponding potential theory and the modified harmonic measure. The potential theory one encounters in this case is somewhat in between elliptic and the parabolic one: on the large scale it is parabolic and on the small scale it is elliptic. The estimates on the harmonic measure in terms of the geometric properties of support can be found in my paper with Kupin.

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May 2012. I recently finished writing a survey for the Nikolskii conference volume, it contains more details.

January 2012. This problem is motivated by the multidimensional L^2 conjecture posed by Barry Simon. It deserves a post on its own but the crux of the matter is this. Take a model case:

iu(x,t)=k\partial^2_{xx} u(x,t)+V(x,t)u(x,t)

where x\in \mathbb{T}, t\in [0,\infty) and the parameter k is real as well as the potential V(x,t). We define the Cauchy problem by taking u(x,0)=1. Then, assuming V is sufficiently regular, we can define the unique solution. Its L^2(\mathbb{T}) norm is preserved in time. However, how fast do the Sobolev norms can grow? Even for fixed k, this is an interesting problem but we want to know what happens for Lebesgue generic k. My conjecture is that for V\in L^2(\mathbb{T},[0,\infty)) the H^1(\mathbb{T}) Sobolev norm is bounded in time. This is true for a.e. k. The partial results were recently obtained here but more analysis is needed. For example, for the transport equation the solution can be explicitly written and the Carleson theorem on the Fourier series yields the sharp result. Here is Harmonic analysis in action. For the general case the combination of soft-analysis methods and the Christ-Kiselev type perturbation arguments is needed. The quantity most relevant to multidimensional case is

\displaystyle \int\limits_{-1}^1 \|u\|^2_{\dot{H}^1}dk

One wants to control it by L^2 norm of V. The small values of k correspond to the small gaps in the spectrum of the diagonal operator and this regime is the most difficult for analysis.

The difficulty of the problem is this– if one takes V complex-valued then the L^2 norm of the solution is not preserved in time anymore. So, whatever analytical machinery one applies, it must be sensitive enough to account for that fact. A good classical example is function \sin(x). Its Taylor expansion is hardly useful for the analysis of this function for large real x.

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June 2013. I recently posted the new preprint on my webpage in which, in particular, the following statement is proved:

Consider iu(x,t)=k\partial^2_{xx} u(x,t)+V(x,t)u(x,t), \, u(x,0,k)=1 in which V is only bounded in both x\in T and t\in [0,1]. Then, for a.e. k the solution u(x,t,k) has Sobolev norm H^\alpha(T), \alpha<1 uniformly bounded for all t\in [0,1].

This result is based on a recent generalization of Carleson theorem on Fourier maximal function (through the variational norm). The statement is rather striking as it gives more regularity for the solution than one can guess as long as the generic coupling constant is taken.

January 2012. Let \mathbb{T} denote the unit circle and let d\sigma be a probability measure on \mathbb{T}. Take the sequence \{1, z ,z^2, \ldots\} and orthonormalize in the Hilbert space L^2(d\sigma) to produce \{\phi_0(z),\phi_1(z),\phi_2(z),\ldots\}.
We have

\displaystyle \int_\mathbb{T} |\phi_n(z)|^2d\sigma=1

by definition. The question is what is the size of \|\phi_n\|_{L^p(d\sigma)} if p> 2 and we assume some additional information on the measure \sigma? This is a classical question in approximation theory.

The Steklov’s conjecture was: assume that d\sigma is purely a.c. and has the weight uniformly bounded away from zero, i.e.

d\sigma=w(\theta)d\theta, \, w(\theta)>\delta>0

Is it true that |\phi_n(z)|<C uniformly in n and z\in \mathbb{T}? The negative answer to this question was given by Rakhmanov. He proved that the possible growth is essentially \sqrt{n} up to an inverse logarithmic factor. Can one drop it? This is a nice problem. !!! SOLVED, see below.

Another important class of measures extensively studied in the literature is the so-called Szego class. It is defined as follows:
d\sigma=d\sigma_s+w(\theta)d\theta where the singular component is arbitrary and for w(\theta) one has

\displaystyle \int_\mathbb{T} \log w(\theta)d\theta>-\infty

How large the polynomials can be in this case? Well, for a while the conjecture was that the polynomial entropy is bounded, i.e.

\displaystyle \int_{\mathbb{T}} \log|\phi_n(z)| |\phi_n (z)|^2d\sigma(z)<C

uniformly in n. That, however, again tuned out to be wrong with the possible growth like \sqrt n. This result is sharp.

Then, what restriction on the size of \phi_n(z) do we get from the orthogonality? Not clear to me.

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August 2013. In the joint paper with A. Aptekarev and D. Tulyakov which you can find on my webpage, we proved the sharp estimates for the uniform norms of the orthogonal polynomials in the Steklov class. This class is defined by the condition that

\sigma'>\delta

for a.e. point on the circle where \delta is some small positive constant. The world record so far was made by Rakhmanov about 30 years ago. The method we used turned out to be powerful enough to also give the sharp bounds for the polynomial entropy in this class. We hope to iterate the construction to prove the lower bounds over the subsequence. If we succeed, this will give the full solution to the famous Steklov’s conjecture.

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February 2014. Building on the previous paper (discussed above), we were able to completely settle a problem by Steklov. That involved a lot of technical work. Here is the main result: suppose \sigma is a probability measure on the circle which satisfies

\sigma'(\theta)>\delta/(2\pi),\quad {\rm a.e.} \theta

and \delta\in (0,1). That condition defines the Steklov’s class. If \{\phi_n\} are the corresponding orthonormal polynomials, then the following upper bound is well-known and easy to prove:

\|\phi_n\|_\infty=o(\sqrt n)

In the paper, we prove

Theorem. For every sequence \{\beta_n\}: \beta_n\to 0, there is an absolutely continuous probability measure \sigma^* from the Steklov class such that

\|\phi_{k_n}(z,\sigma^*)\|_\infty>\beta_{k_n}\sqrt{k_n}

for some sequence k_n.

The proof is constructive and can be used when the measure satisfies different lower bounds. There are other interesting questions that naturally follow from the paper, some of them do not seem to be so hard anymore.

The project on getting the uniform bound on the polynomials is mainly finished. What is left wide open is the problem on controlling the \sup_{n}|\phi_n| for a.e. \theta provided the constructive information on measure is known. This is the so-called nonlinear Luzin’s conjecture and we know next to nothing about it.

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January 2012.  The 2D Euler equation can be written in the following form

\dot \theta=\nabla \theta\cdot \nabla^\perp \Delta^{-1}\theta \quad\quad (1)

where \theta(x,y,t) is 2\pi– periodic in both x and y and it denotes the vorticity of the velocity field. The symbol \nabla^\perp stands for (\partial_y, -\partial_x). To define the Cauchy problem, one needs to specify \theta(x,y,0)=\theta_0(x,y)

This is a transport equation with the divergence-free vector field and so all L^p norms of \theta are preserved in time. The global regularity is known for many functional spaces, e.g. C^\infty or H^s with large s will do. The quantitative version is a bound

\|\theta(t)\|_{H^s}\lesssim (1+\|\theta(0)\|_{H^s})^{e^{Ct}}, s\geq 2

Is this estimate sharp? I do not know but one can have double exponential growth for arbitrarily large but finite time. The lower bound valid for all time is only superlinear.

Another interesting problem is dynamics of patches. Assume that \theta_0(x,y) is the characteristic function of a domain \Omega(0) and the problem is considered on the whole plane, not on the 2d torus. Then, \theta(x,y,t)=\chi_{\Omega(t)} and the boundary of \Omega(t) is smooth provided that the boundary of \Omega(0) was smooth. However, one can study the growth of curvature and the rate of merging (if there are several patches).

The merging mechanism was studied recently and the INFINITE double exponential rate is possible at least if the regular strain is present. This result is sharp.

If one modifies equation (1) for vorticity by writing \Delta^{-1/2} instead of \Delta^{-1}, then the resulting equation is called SQG (surface quasi-geostrophic). The outstanding conjecture in the field is that the singularity forms in finite time. No proof so far.

 

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May 2013. I have finished one more revision of the paper on the corner formation and you can check either my webpage or arxiv. The estimates are now sharp and the regularity of the strain is as optimal as the method can give. I think I approach the stage when the paper can finally be read by the students.

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October 2013. In the recent preprint, Kiselev and Sverak considered the 2d Euler equation on the disc and proved the infinite in time double-exponential growth of the Lipschitz norm for the vorticity. The singularity forms on the boundary and happens as two identical vortices of different sign slide along the boundary towards each other. This interesting result might indicate that singularity formation is generic when the equation is considered on the domain with the boundary. The initial data is smooth but is not zero near the boundary (this is important for the construction). So, if one writes the equation on the Fourier side, the decay of

\hat \theta(n,t)

in n is very weak. The stable mechanism for the singularity formation in the bulk is still missing.

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October 2013. Another interesting result obtained for 2d Euler. My colleague Andrej Zlatos proved the possibility of infinite exponential growth of the Hessian of vorticity for smooth initial data. This was done for the 2d torus and no boundary effects were used. He also showed that the Lipschitz norm can grow exponentially for C^{1,\alpha} initial data. In this work the hyperbolic scenario is exploited. I think the interesting question is the possibility of the exponential growth of the Lipschitz norm for smooth initial data (e.g., trig polynomial). So far we have only superlinear estimate. Even better problem is to show that the sharp front can form with width decaying exponentially. This would prove that the results of Fefferman and D. Cordoba on the fronts are essentially sharp. There are plausible mechanisms but the justification for Euler equation is hard.

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