Related papers: Sharp Sobolev estimates for concentration of solut…
We consider the drift-diffusion equation $$ u_t-\varepsilon \Delta u+\nabla\cdot(u\nabla K\star u)=0 $$ in the whole space with global-in-time bounded solutions. Mass concentration phenomena for radially symmetric solutions of this equation…
Our aim is to study the limit of the solution of reaction-diffusion porous medium equation with linear drift $\displaystyle\partial_t u -\Delta u^m +\nabla \cdot (u \: V)=g(t,x,u) $, as $m\to\infty.$ We study the problem in bounded domain…
We consider radial solutions of the slightly subcritical problem $-\Delta u_\varepsilon = |u_\varepsilon|^{\frac{4}{n-2}-\varepsilon}u_\varepsilon$ either on $\mathbb R^n$ ($n\geq 3$) or in a ball $B$ satisfying Dirichlet or Neumann…
We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth \[\left\{\begin{gathered} - \Bigl({\varepsilon ^2}a + \varepsilon b\int_{{\R^3}} {{{\left| {\nabla u} \right|}^2}}…
Existence and uniqueness of radially symmetric self-similar very singular solutions are proved for the singular diffusion equation with gradient absorption {equation*} \partial_t u -\Delta_{p}u+|\nabla u|^q=0, \ \hbox{in} \…
We study well-posedness and long-time behaviour of aggregation-diffusion equations of the form $\frac{\partial \rho}{\partial t} = \Delta \rho^m + \nabla \cdot( \rho (\nabla V + \nabla W \ast \rho))$ in the fast-diffusion range, $0<m<1$,…
Given a finite sequence of times $0<t_1<...<t_N$, we construct an example of a smooth solution of the free nonstationnary Navier--Stokes equations in $\R^d$, $d=2,3$, such that: (i) The velocity field $u(x,t)$ is spatially poorly localized…
This work establishes the existence and uniqueness of solutions to the fractional diffusion equation $$\frac{\partial^\alpha u}{\partial t^{\alpha}} + K(-\Delta)^{\beta} u - \nabla \cdot (\nabla V u) = f$$ on a $d$-dimensional torus,…
We determine the sharp mass threshold for Sobolev norm growth for the focusing continuum Calogero--Moser model. It is known that below the mass of $2\pi$, solutions to this completely integrable model enjoy uniform-in-time $H^s$ bounds for…
We study a priori estimates for a class of non-negative local weak solution to the weighted fast diffusion equation $u_t = |x|^{\gamma} \nabla\cdot (|x|^{-\beta} \nabla u^m)$, with $0 < m <1$ posed on cylinders of $(0,T)\times{\mathbb…
We establish quantitative blow-up criteria below the scaling threshold for radially symmetric solutions to the defocusing nonlinear Schr\"odinger equation with nonlinearity $|u|^6u$. This provides to our knowledge the first generic results…
We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ in the energy-supercritical regime p>4. For even values of the power p, we show that blowup (or failure to scatter) must be accompanied by blowup of the…
We study the large time behavior of non-negative solutions to the singular diffusion equation with gradient absorption $$ \partial_t u-\Delta_{p}u+|\nabla u|^q=0 \quad \hbox{in} \ (0,\infty)\times\real^N, $$ for $p_c:=2N/(N+1)
We establish -among other things- existence and multiplicity of solutions for the Dirichlet problem $\sum_i\partial_{ii}u+\frac{|u|^{\crit-2}u}{|x|^s}=0$ on smooth bounded domains $\Omega$ of $ \rn$ ($n\geq 3$) involving the critical…
This work is devoted to the analysis of the quantum drift-diffusion model derived by Degond et al. The model is obtained as the diffusive limit of the quantum Liouville-BGK equation, where the collision term is defined after a local quantum…
We investigate the homogeneous Dirichlet problem for the Fast Diffusion Equation $u_t=\Delta u^m$, posed in a smooth bounded domain $\Omega\subset \mathbb{R}^N$, in the exponent range $m_s=(N-2)_+/(N+2)<m<1$. It is known that bounded…
We study the quantitative stability of critical points of the fractional Sobolev inequality. We show that, for a non-negative function $u \in \dot H^s(\mathbb R^N)$ whose energy satisfies $$\tfrac{1}{2} S^\frac{N}{2s}_{N,s} \le \|u\|_{\dot…
We study the multidimensional aggregation equation $u_t+\Div(uv)=0$, $v=-\nabla K*u$ with initial data in $\cP_2(\bR^d)\cap L_{p}(\bR^d)$. We prove that with biological relevant potential $K(x)=|x|$, the equation is ill-posed in the…
We study the problem of estimating the coefficients of a diffusion (X_t,t\geq 0); the estimation is based on discrete data X_{n\Delta},n=0,1,...,N. The sampling frequency \Delta^{-1} is constant, and asymptotics are taken as the number N of…
We solve the problem of spatial distribution of inertial particles that sediment in turbulent flow with small ratio of acceleration of fluid particles to acceleration of gravity $g$. The particles are driven by linear drag and have…