Small time global null controllability for a viscous Burgers' equation despite the presence of a boundary layer

In this work, we are interested in the small time global null controllability for the viscous Burgers' equation _yt−_yxx+y_yx=u(t)$y_t-y_{xx}+y{y}_x=u(t)$ on the line segment [0,1]$[0,1]$. The right-hand side is a scalar control playing a role similar to that of a pressure. We set y(t,1)=0$y(t,1)=0$ and restrict ourselves to using only two controls (namely the interior one u(t)$u(t)$ and the boundary one y(t,0)$y(t,0)$). In this setting, we show that small time global null controllability still holds by taking advantage of both hyperbolic and parabolic behaviors of our system. We use the Cole–Hopf transform and Fourier series to derive precise estimates for the creation and the dissipation of a boundary layer.     

New representation and factorizations of the higher-order ultraspherical-type differential equations

The paper deals with the class of linear differential equations of any even order 2α+4$2\alpha +4$, α∈_N0$\alpha \in {\mathbb{N}}_0$, which are associated with the so-called ultraspherical-type polynomials. These polynomials form an orthogonal system on the interval [−1,1]$[-1,1]$ with respect to the ultraspherical weight function (1−x²)^α${(1-x^2)}^{\alpha }$ and additional point masses of equal size at the two endpoints. The differential equations of “ultraspherical-type” were developed by R. Koekoek in 1994 by utilizing special function methods. In the present paper, a new and completely elementary representation of these higher-order differential equations is presented. This result is used to deduce the orthogonality relation of the ultraspherical-type polynomials directly from the differential equation property. Moreover, we introduce two types of factorizations of the corresponding differential operators of order 2α+4$2\alpha +4$ into a product of α+2$\alpha +2$ linear second-order operators.     

The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary

We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u   is a solution of (−Δ)^su=g${(-\mathrm{\Delta })}^{s}u=g$ in Ω  , u≡0$u\equiv 0$ in Rⁿ\Ω${\mathbb{R}}^n\backslash \Omega$, for some s∈(0,1)$s\in (0,1)$ and g∈L^∞(Ω)$g\in L^{\infty }(\Omega )$, then u   is C^s(Rⁿ)$C^s({\mathbb{R}}^n)$ and u/δ^s_|Ω$u/{\delta }^s{|}_{\Omega }$ is C^α$C^{\alpha }$ up to the boundary ∂Ω   for some α∈(0,1)$\alpha \in (0,1)$, where δ(x)=dist(x,∂Ω)$\delta (x)=\mathrm{dist}(x,\partial \Omega )$. For this, we develop a fractional analog of the Krylov boundary Harnack method.     

An improvement for the Trudinger–Moser inequality and applications

In line with the Concentration–Compactness Principle due to P.-L. Lions [19], we study the lack of compactness of Sobolev embedding of W^1,n(Rⁿ)$W^{1,n}({\mathbb{R}}^n)$, n?2$n?2$, into the Orlicz space _LΦα$L_{{\Phi }_{\alpha }}$ determined by the Young function _Φα(s)${\Phi }_{\alpha }(s)$ behaving like e^{α|s|n/(n−1)}−1$e^{\alpha {|s|}^{n/(n-1)}}-1$ as |s|→+∞$|s|\to +\infty$. In the light of this result we also study existence of ground state solutions for a class of quasilinear elliptic problems involving critical growth of the Trudinger–Moser type in the whole space Rⁿ${\mathbb{R}}^n$.     

Minimal time of controllability of two parabolic equations with disjoint control and coupling domains

We consider two parabolic equations coupled by a matrix A(x)=q(x)A₀$A(x)=q(x)A_0$, where A₀$A_0$ is a Jordan block of order 1, and controlled by a single localized function, or by a single boundary control. The support of the coupling coefficient, q  , and the control domain may be disjoint. We exhibit an explicit minimal time of null-controllability, T₀(q)∈[0,+∞]$T_0(q)\in [0,+\infty ]$.     

Characterizing graphic matroids by a system of linear equations

Given a rank-r   binary matroid we construct a system of O(r³)$O(r^3)$ linear equations in O(r²)$O(r^2)$ variables that has a solution over GF(2)$\mathrm{GF}(2)$ if and only if the matroid is graphic.     

Semilinear fractional elliptic equations involving measures

We study the existence of weak solutions to (E) (−Δ)^αu+g(u)=ν${(-\mathrm{\Delta })}^{\alpha }u+g(u)=\nu$ in a bounded regular domain Ω   in R^N(N≥2)${\mathbb{R}}^N(N\ge 2)$ which vanish in R^N?Ω${\mathbb{R}}^N?\Omega$, where (−Δ)^α${(-\mathrm{\Delta })}^{\alpha }$ denotes the fractional Laplacian with α∈(0,1)$\alpha \in (0,1)$, ν is a Radon measure and g is a nondecreasing function satisfying some extra hypotheses. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of weak solution for problem (E) for any measure. In the case where ν   is a Dirac measure, we characterize the asymptotic behavior of the solution. When g(r)=|r|^k−1r$g(r)={|r|}^{k-1}r$ with k supercritical, we show that a condition of absolute continuity of the measure with respect to some Bessel capacity is a necessary and sufficient condition in order (E) to be solved.     

Semilinear elliptic equations admitting similarity transformations

In this paper we study the equation −Δu+ρ^−(α+2)h(ρ^αu)=0$-\mathrm{\Delta }u+{\rho }^{-(\alpha +2)}h({\rho }^{\alpha }u)=0$ in a smooth bounded domain Ω   where ρ(x)=dist(x,∂Ω)$\rho (x)=\mathrm{dist}(x,\partial \Omega )$, α>0$\alpha >0$ and h is a nondecreasing function which satisfies Keller–Osserman condition. We introduce a condition on h which implies that the equation is subcritical, i.e., the corresponding boundary value problem is well posed with respect to data given by finite measures. Under additional assumptions on h we show that this condition is necessary as well as sufficient. We also discuss b.v. problems with data given by positive unbounded measures. Our results extend results of [13] treating equations of the form −Δu+ρ^βu^q=0$-\mathrm{\Delta }u+{\rho }^{\beta }{u}^q=0$ with q>1$q>1$, β>−2$\beta >-2$.     

Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation

We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical (α<1/2$\alpha <1/2$) dissipation α(−Δ)${(-\mathrm{\Delta })}^{\alpha }$: If a Leray–Hopf weak solution is Hölder continuous θ∈Cδ(R²)$\theta \in C^{\delta }({\mathbb{R}}^2)$ with δ>1−2α$\delta >1-2\alpha$ on the time interval [t₀,t]$[t_0,t]$, then it is actually a classical solution on (t₀,t]$(t_0,t]$.