Related papers: A fast regularisation of a Newtonian vortex equati…

We consider the homogeneous Dirichlet problem for the parabolic equation \[ u_t- \operatorname{div} \left(|\nabla u|^{p(x,t)-2} \nabla u\right)= f(x,t) + F(x,t, u, \nabla u) \] in the cylinder $Q_T:=\Omega\times (0,T)$, where $\Omega\subset…

We consider a superfluid described by the Gross-Pitaevskii equation passing an obstacle \[\epsilon^2 \Delta u+ u(1-|u|^2)=0 \ \mbox{in} \ {\mathbb R}^d \backslash \Omega, \ \ \frac{\partial u}{\partial \nu}=0 \ \mbox{on}\ \partial \Omega \]…

Point vortex models are presented for the generalized Euler equations, which are characterized by a fractional Laplacian relation between the active scalar and the streamfunction. Special focus is given to the case of the surface…

We study the general nonlinear diffusion equation $u_t=\nabla\cdot (u^{m-1}\nabla (-\Delta)^{-s}u)$ that describes a flow through a porous medium which is driven by a nonlocal pressure. We consider constant parameters $m>1$ and $0<s<1$, we…

We consider non linear elliptic equations of the form $\Delta u = f(u,\nabla u)$ for suitable analytic nonlinearity $f$, in the vinicity of infinity in $\mathbb{R}^d$, that is on the complement of a compact set.We show that there is a…

The coupling between dilatation and vorticity, two coexisting and fundamental processes in fluid dynamics is investigated here, in the simplest cases of inviscid 2D isotropic Burgers and pressureless Euler-Coriolis fluids respectively…

We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation \[ \mathcal{L}u = \sigma u^{q} + \mu \quad \text{in} \;\; \Omega, \] in the sublinear case $0<q<1$, with…

We consider the reaction-diffusion problem $-\Delta_g u = f(u)$ in $\mathcal{B}_R$ with zero Dirichlet boundary condition, posed in a geodesic ball $\mathcal{B}_R$ with radius $R$ of a Riemannian model $(M,g)$. This class of Riemannian…

The lake equations $$\left\{\begin{aligned} \nabla \cdot \big( b \, \mathbf{u}\big) &= 0 & & \text{on}\ \mathbb{R}\times D,\\ \partial_t\mathbf{u} + (\mathbf{u}\cdot \nabla)\mathbf{u} &= -\nabla h & & \text{on}\ \mathbb{R}\times D ,\\…

An effort has been made to solve the Cauchy problem of the Navier-Stokes equations in the whole space by two methods. It is proved that the sum of the three vorticity components is a time-invariant in fluid motion. It has been proved that,…

In this article, we study an inverse problem for the following convective Brinkman-Forchheimer (CBF) equations: \begin{align*} \boldsymbol{u}_t-\mu…

We consider uniformly rotating incompressible Euler and Navier-Stokes equations. We study the suppression of vertical gradients of Lagrangian displacement ("vertical" refers to the direction of the rotation axis). We employ a formalism that…

We study the regularity of the extremal solution $u^*$ to the singular reaction-diffusion problem $-\Delta_p u = \lambda f(u)$ in $\Omega$, $u =0$ on $\partial \Omega$, where $1<p<2$, $0 < \lambda < \lambda^*$, $\Omega \subset \mathbb{R}^n$…

We consider solutions $u(t)$ to the 3d NLS equation $i\partial_t u + \Delta u + |u|^2u=0$ such that $\|xu(t)\|_{L^2} = \infty$ and $u(t)$ is nonradial. Denoting by $M[u]$ and $E[u]$, the mass and energy, respectively, of a solution $u$, and…

In this paper, we first consider a class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space $\mathbb{R}^{n+1}$ with speed $u^\alpha f^{-\beta}$, where $u$ is the support function of the hypersurface, $f$ is a…

In this paper we prove that if $u$ is a solution to second order hyperbolic equation $\partial^2_tu+a(x)\partial_tu-(div_x\left(A(x)\nabla_x u\right)+b(x)\cdot\nabla_x u+c(x)u)=0$ and $u$ is flat on a segment $\{x_0\}\times (-T,T)$ then $u$…

As a model for vortex-wall interactions, we consider the two-dimensional incompressible Navier--Stokes equations in the half-plane $R^2_+$ with no-slip boundary condition and point vortices as initial data. We focus on the paradigmatic…

In this paper, we deal with the following double phase problem $$ \left\{\begin{array}{ll} -\mbox{div}\left(|\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right)=…

We consider Burgers equation with transverse viscosity $$\partial_tu+u\partial_xu-\partial_{yy}u=0, \ \ (x,y)\in \mathbb R^2, \ \ u:[0,T)\times \mathbb R^2\rightarrow \mathbb R.$$ We construct and describe precisely a family of solutions…

The incompressible Euler equations on a compact Riemannian manifold $(M,g)$ take the form \begin{align*} \partial_t u + \nabla_u u &= - \mathrm{grad}_g p \\ \mathrm{div}_g u &= 0, \end{align*} where $u: [0,T] \to \Gamma(T M)$ is the…