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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)| 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)

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. _______________________________________________________________________ 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. ______________________________________________________________________________ February 2014. Building on the previous paper (discussed above), we were able to 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 very little about it. ____________________________________________________________ November 6, 2014  Fix $\delta\in (0,1)$, large $n$ and $p$ such that $p. Consider the class of probability measures defined as $P_{p,\delta}=\{\sigma=\delta \mu/(2\pi)+\sum_{j=1}^p m_j\delta(\theta-\theta_j)\}$ where $d\mu=d\theta$$\{\theta_j\}$ are arbitrary points on $[-\pi,\pi]$, and $m_j\geq 0$. Let $M_{n,p}=\max_{\sigma\in P_{p,\delta}}\|\phi_n(z,\sigma)\|_{L^\infty(\mathbb{T})}$. In the recent paper, I proved the following estimate $M_{n,p}1.5$ It implies, for example, that the maximizer in the Steklov problem has $\sim n$ point masses. In the same paper, I also proved the following inequality improving the bound by Kos: if $\lambda_1<\ldots<\lambda_p$ and $T(x)=\sum_{j=1}^p x_j e^{i\lambda_j x}$, then $|T(0)|\leq p\pi/2\|T\|_{L^2[0,1]}$. That was done by using the duality between two extremal problems for one of which the bound is easy to obtain if the generalization of the Halasz’ result is used. ___________________________________________________________________________________________________________________ May 19, 2015 The solution to a problem by Steklov discussed above allows to study other variational problems. For example, $A_{n,\epsilon}=\sup_{|w-1|\leq \epsilon} \|\phi_n(z,w)\|_{L^\infty(\mathbb{T})}$ Then, the old idea by S.Bernstein and our method used for getting the lower bounds can be combined to get $\log A_{n,\epsilon}\sim \epsilon \log n$ when $\epsilon$ is fixed and $n\to \infty$. In the case when the deviation of the weight is large, we have two results: 1. For every large $T$, there is $\alpha(T)>0$ such that $\|\phi_n(z,w)\|_{L^\infty(\mathbb{T})}\lesssim n^{1/2-\alpha(T)}$ for every weight $w: \delta\leq w\leq T$. 2. For every small $\beta>0$, there is large $T(\beta)$ and a weight $w: \delta\leq w\leq T$ such that $\phi_n(1,w)>n^{1/2-\beta}$. One can see that the $\sqrt n$ growth can not be achieved by weights uniformly bounded in $L^\infty$.

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July 2015.  We significantly simplified several proofs in the paper “On a problem by Steklov” and submitted a new version to arXiv. We also fixed some minor mistakes and typos.

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September 2015. In his original paper on the Steklov’s conjecture, Rakhmanov used a formula for orthogonal polynomial obtained by adding mass points to the measure of orthogonality. Although its uniform norm can grow not faster than $\log n$ (where $n$ is a degree), it is interesting to deduce this polynomial by method used for getting the sharp lower bound in the problem by Steklov. This was carried out in my recent preprint. The interesting feature is a different kind of cancellation. It is less effective than the one giving the $\sqrt n$ bound but it still might be interesting.

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November 2015. The method used to solve the problem of Steklov proves the possibility of $\sqrt n$ growth of polynomials with weights $w$ that satisfy $w>\delta/2\pi$ and $\|w\|_{L^p[0,2\pi]} with arbitrary $p<\infty$. If $w, w^{-1}\in L^\infty[0,2\pi)$ then the upper bound gives an exponent smaller than $1/2$: $\|\phi_n\|_{L^\infty[0,2\pi)}. In a joint project with my student Keith Rush, we proved that $w,w^{-1}\in {\rm BMO}$ is sufficient to have $\|\phi_n\|_{L^\infty[0,2\pi)}. The proof gives more that that. Under these assumptions, we can show that the polynomial entropies are uniformly bounded. In terms of the regularity of weight, this results is rather sharp since the entropies are known to grow logarithmically for $w: w>\delta/(2\pi), \|w\|_{L^p[0,2\pi)} with any $p<\infty$.

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.