This paper is concerned with the solvability of the reduced version of the Possio singular integral equation, which plays a fundamental role in theoretical aeroelasticity. This equation relates the pressure distribution over a typical section of a long slender wing in subsonic compressible air flow to the normal velocity of the points of a wing (down-wash). In spite of the importance of the Possio equation, the question of the existence of its solution has not yet been settled. In this paper, we provide a rigorous study of the reduced version of the Possio equation and prove its solvability. The reduced Possio equation is important in its own right for two reasons: (a) it can be used as an approximate equation for the numerical analysis of the coupled system describing a vibrating wing in a surrounding air flow; (b) its analysis is essential for identifying analytical difficulties associated with the general Possio equation.

In this paper, we deal with optimal boundary control for the Navier-Stokes problem. We establish the existence of such a control in appropriate functional spaces. Then we study a stabilization problem around a steady state. In view of numerical approximation, we derive rigorously Euler equations satisfied by the control.

We continue to study the degenerate parabolic equation derived from the kinetic theory using Rényi-Tsallis' entropy. The finiteness of type II blowup points and several structures of the blowup set are shown for the critical exponent case.

Let $p\in[2,N)$, ${\Omega}\subset{\mathbb{R}}^N$ an open set and let $\mu$ be a Borel measure on ${\partial\Omega}$. Under some assumptions on ${\Omega},\mu,f,g$ and $\beta$, we show that the quasi-linear elliptic equation with nonlinear inhomogeneous Robin-type boundary conditions \[ \begin{cases} -\Delta_pu+c(x)|u|^{p-2}u=f \;& \text{ in }{\Omega} \\ d{\operatorname{\mathsf N}}_p(u) + \beta(x,u)d\mu =gd\mu \;& \text{ on }{\partial\Omega} \end{cases} \] has a unique weak solution which is globally bounded on ${\overline{\Omega}};$ that is, the weak solution $u$ is in $L^\infty({\Omega})$ and its trace $u|_{{\partial\Omega}}$ belongs to $L^\infty({\partial\Omega},\mu)$. Here ${\operatorname{\mathsf N}}_p(u)$ is a generalization of the normal derivative for bad domains. When ${\Omega}$ and $u$ are smooth, then $d{\operatorname{\mathsf N}}_p(u)= | \nabla u | ^{p-2} (\partial u/\partial\nu) d\sigma$ where $\sigma$ is the surface measure and $\nu$ the outer normal to ${\partial\Omega}$. A priori estimates for solutions are also obtained.

We establish local well posedness for arbitrarily large initial data in the usual Sobolev spaces $H^{s}({\mathbb{R}}),$ $s>\frac{1}{2},$ for the Cauchy problem associated to the integro-differential equation $$ \partial_{t}u+i\alpha\partial^{2}_{x}u=\beta u\left(1+i\mathcal{T}_h\right) \partial_{x}(\left|u\right|^{2})+i\gamma|u|^{2}u, $$ where $u=u(x,t)\in{\mathbb{C}},$ $x, t\,\in{\mathbb{R}}$, and $\mathcal{T}_h$ denotes the singular operator defined by $$ \mathcal{T}_{h}f(x)=\frac{1}{2h}\,\mbox{p.v.} \int^{\infty}_{-\infty}\coth\left(\frac{\pi(x-y)}{2h}\right) f(y)dy, $$ when $0 < h\le \infty$. Note that $\mathcal{T}_{\infty}=\mathcal{H}$ is the Hilbert transform. Our method of proof relies on a gauge transformation localized in positive frequencies which allows us to weaken the high-low frequencies interaction in the nonlinearity.

In this paper, we provide a quantitative estimate of unique continuation (doubling property) for higher-order parabolic partial differential equations with non-analytic Gevrey coefficients. Also, a new upper bound is given on the number of zeros for the solutions with a polynomial dependence on the coefficients.

A general closed expression is derived in the frequency domain for the minimum free energy, related to a state of a linear electromagnetic conductor with memory effects in the constitutive equation of the current density, by evaluating the maximum recoverable work obtainable from the given state. The constitutive equation with memory is given by a linear relation between the current density and the electric field, expressed by a linear functional of the history of this field and a term proportional to its actual value. Another equivalent formulation is also given and used to derive explicit formulae for the particular case of a discrete spectrum model.