Related papers: Stationary homogeneous solutions for the inviscid …
In this paper, we show the existence of a family of analytic stationary patch solutions of the SQG and gSQG equations. This answers an open problem in [F. de la Hoz, Z. Hassainia, T. Hmidi. Arch. Ration. Mech. Anal., 220(3):1209-1281,…
We consider homogeneous (stationary self-similar) solutions to the generalized surface quasi-geostrophic (gSQG) equations parametrized by the constant $0<s<1$, representing the 2D Euler equations ($s=1$), the SQG equations $(s=1/2)$, and…
In this paper we study classification of homogeneous solutions to the stationary Euler equation with locally finite energy. Written in the form $u = \nabla^\perp \Psi$, $\Psi(r,\theta) = r^{\lambda} \psi(\theta)$, for $\lambda >0$, we show…
We construct stationary statistical solutions of a deterministic unforced nonlinear Schr\"odinger equation, by perturbing it by a linear damping $\gamma u$ and a stochastic force whose intensity is proportional to $\sqrt \gamma$, and then…
We investigate stationary solutions of a non-local aggregation equation with degenerate power-law diffusion and bounded attractive potential in arbitrary dimensions. Compact stationary solutions are characterized and compactness…
We present general results on exponential decay of finite energy solutions to stationary nonlinear Schr\"odinger equations.
We consider a class of stationary viscous Hamilton--Jacobi equations as $$ \left\{\begin{array}{l} \la u-{\rm div}(A(x) \nabla u)=H(x,\nabla u)\mbox{in }\Omega, u=0{on}\partial\Omega\end{array} \right. $$ where $\la\geq 0$, $A(x)$ is a…
We consider ($-\alpha$)-homogeneous solutions (stationary self-similar solutions of degree $-\alpha$) to the two-dimensional inviscid Boussinesq equations in a half-plane. We show their non-existence and existence with both regular and…
In this paper, we prove the existence of locally non-radial solutions to the stationary 2D Euler equations with compact support but non-concentrated around one or several points. Our solutions are of patch type, have analytic boundary,…
In this paper we study the existence of stationary solutions for stochastic partial differential equations. We establish a new connection between $L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{1}}) \otimes…
We prove the existence and uniqueness of stationary spherically symmetric positive solutions for the Schr\"{o}dinger-Newton model in any space dimension $d$. Our result is based on an analysis of the corresponding system of second order…
We consider the inviscid generalized surface quasi-geostrophic equation (gSQG) in a patch setting, where the parameter $\alpha \in (1,2)$. The cases $\alpha = 0$ and $\alpha = 1$ correspond to 2d Euler and SQG respectively, and our choice…
In this paper we prove the existence and local uniqueness of stationary states for the nonlinear Dirac equation \[ i \sum_{j=0}^{3} \ga^j \pd_j \psi - m\psi + F(\bar{\psi}\psi)\psi =0 \] where $ m >0$ and $ F(s) = |s|^{\theta}$ for $ 1\leq…
We consider the regularity of stationary solutions to the linearized Boltzmann equations in bounded $C^1$ convex domains in $\mathbb{R}^3$ for gases with cutoff hard potential and cutoff Maxwellian gases. We prove that the stationary…
We show the existence of non-unique stationary weak solutions for forced surface quasi-geostrophic (SQG) equation via a convex integration scheme. The scheme is implemented for the sum-difference system of two distinct solutions. Through…
We prove a general theorem that the $L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solution of an infinite horizon backward doubly stochastic differential equation, if exists,…
Analytical analysis of spatially extended autocatalytic and hypercyclic systems is presented. It is shown that spatially explicit systems in the form of reaction-diffusion equations with global regulation possess the same major qualitative…
We consider the existence of radially symmetric stationary solutions of the compressible viscous and heat-conductive polytropic ideal fluid on the unbounded exterior domain of a sphere where the boundary and far-field conditions are…
We consider the focusing wave equation outside a ball of $\R^3$, with Dirichlet boundary condition and a superquintic power nonlinearity. We classify all radial stationary solutions, and prove that all radial global solutions are…
In this paper we prove the existence of non-stationary periodic solutions of delay Lotka-Volterra equations. In the proofs we use the degree for $S^1$-equivariant maps.