In this paper, the axisymmetric flow in an ideal fluid outside the infinite cylinder ($r\le d$) where $(r,\theta ,z)$ denotes the cylindrical co-ordinates in ${ℝ}^3$ is considered. The motion is with swirl (i.e. the $\theta$-component of the velocity of the flow is non constant). The (non-dimensional) equation governing the phenomenon is (Pd) displayed below. It is known from e.g. that for the problem without swirl ($f_q=0$ in (f)) in the whole space, as the flux constant $k$ tends to $\infty$, 1) $\mathrm{dist}(0z,\partial A)=O\left(k^{1/2}\right)$; $\mathrm{diam}A=O(exp(-c_0{k}^{3/2}))$; 2) ${\left(k^{1/2}\Psi \right)}_{k\in \mathbb{N}}$ converges to a vortex cylinder $U_m$ (see...

Steady-state system of equations for incompressible, possibly non-Newtonean of the $p$-power type, viscous flow coupled with the heat equation is considered in a smooth bounded domain $\Omega \subset {ℝ}^n$, $n=2$ or 3, with heat sources allowed to have a natural $L^1$-structure and even to be measures. The existence of a distributional solution is shown by a fixed-point technique for sufficiently small data if $p>3/2$ (for $n=2$) or if $p>9/5$ (for $n=3$).

The aim of this paper is to analyze the well posedness of the one-phase quasi-stationary Stefan problem with the Gibbs-Thomson correction in a two-dimensional domain which is a perturbation of the half plane. We show the existence of a unique regular solution for an arbitrary time interval, under suitable smallness assumptions on initial data. The existence is shown in the Besov-Slobodetskiĭ class with sharp regularity in the L₂-framework.

Vengono brevemente studiati i problemi di Stefan su «capacità concentrate»,seguendo l'approccio recentemente introdotto di G. Savaré e A. Visintin.

We develop a class of averaging lemmas for stochastic kinetic equations. The velocity is multiplied by a white noise which produces a remarkable change in time scale.Compared to the deterministic case and as far as we work in $L^2$, the nature of regularity on averages is not changed in this stochastic kinetic equation and stays in the range of fractional Sobolev spaces at the price of an additional expectation. However all the exponents are changed; either time decay rates are slower (when the right...

Consider the boundary value problem (L.P):$(h-A)u=f$ in $D$, $(v-\Gamma )u=g$ on $\partial D$ where $A$ is written as $A=1/2{\sum }_{i=1}^m{Y}_i^2+Y_0$, and $\Gamma$ is a general Venttsel’s condition (including the oblique derivative condition). We prove existence, uniqueness and smoothness of the solution of (L.P) under the Hörmander’s condition on the Lie brackets of the vector fields $Y_i$ ($0\le i\le m$), for regular open sets $D$ with a non-characteristic boundary.Our study lies on the stochastic representation of $u$ and uses the stochastic calculus of variations for the $(A,\Gamma )$-diffusion process...

Let H be a separable real Hilbert space and let E be a separable real Banach space. We develop a general theory of stochastic convolution of ℒ(H,E)-valued functions with respect to a cylindrical Wiener process ${W_t^H}_{t\in [0,T]}$ with Cameron-Martin space H. This theory is applied to obtain necessary and sufficient conditions for the existence of a weak solution of the stochastic abstract Cauchy problem (ACP) $d{X}_t=A{X}_{t}dt+Bd{W}_t^H$ (t∈ [0,T]), $X_0=0$ almost surely, where A is the generator of a $C_0$-semigroup ${S\left(t\right)}_{t\ge 0}$ of bounded linear operators on...

Let $H$ be a Hilbert space and $E$ a Banach space. We set up a theory of stochastic integration of $ℒ(H,E)$-valued functions with respect to $H$-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter $0<\beta <1$. For $0<\beta <\frac{1}{2}$ we show that a function $\Phi :(0,T)\to ℒ(H,E)$ is stochastically integrable with respect to an $H$-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an $H$-cylindrical fractional Brownian motion. We apply our results to stochastic evolution equations...

We consider non-degenerate SDEs with a β-Hölder continuous and bounded drift term and driven by a Lévy noise L which is of α-stable type. If β > 1 - α/2 and α ∈ [1,2), we show pathwise uniqueness and existence of a stochastic flow. We follow the approach of [Priola, Osaka J. Math. 2012] improving the assumptions on the noise L. In our previous paper L was assumed to be non-degenerate, α-stable and symmetric. Here we can also recover relativistic and truncated stable processes and some classes...

Stochastic homogenization (with multiple fine scales) is studied for a class of nonlinear monotone eigenvalue problems. More specifically, we are interested in the asymptotic behaviour of a sequence of realizations of the form $$-div\left(a\left(T_1\left(\frac{x}{{\epsilon }_1}\right){\omega }_1,T_2\left(\frac{x}{{\epsilon }_2}\right){\omega }_2,\nabla u_{\epsilon }^{\omega }\right)\right)={\lambda }_{\epsilon }^{\omega }𝒞\left(u_{\epsilon }^{\omega }\right).$$ It is shown, under certain structure assumptions on the random map $a({\omega }_1,{\omega }_2,\xi )$, that the sequence $\{{\lambda }_{\epsilon }^{\omega ,k},u_{\epsilon }^{\omega ,k}\}$ of $k$th eigenpairs converges to the $k$th eigenpair $\{{\lambda }^k,u^k\}$ of the homogenized eigenvalue problem $$-\mathrm{div}\left(b\left(\nabla u\right)\right)=\lambda \overline{𝒞}\left(u\right).$$ For the case of $p$-Laplacian type maps we characterize $b$ explicitly.

Let H be a separable real Hilbert space and let E be a real Banach space. In this paper we construct a stochastic integral for certain operator-valued functions Φ: (0,T) → ℒ(H,E) with respect to a cylindrical Wiener process ${W_H\left(t\right)}_{t\in [0,T]}$. The construction of the integral is given by a series expansion in terms of the stochastic integrals for certain E-valued functions. As a substitute for the Itô isometry we show that the square expectation of the integral equals the radonifying norm of an operator which is...