Related papers: Multiple Vortices for the Shallow Water Equation
By using a shooting technique, we prove that the quasilinear boundary value problem $$ \textrm{div} \, \left( \frac{\nabla u}{\sqrt{1-| \nabla u |^2}}\right) + \lambda q(| x |) | u |^{p-1} u = 0, \qquad u|_{\partial \mathcal{B}} = 0,$$…
In this article, we study the existence of non-trivial weak solutions for the following boundary-value problem \begin{gather*} -\frac{\partial^2 u}{\partial x^2} -\left|x\right|^{2k}\frac{\partial^2 u}{\partial y^2}=f(x,y,u) \quad\text{ in…
We study an initial-boundary value problem of three-dimensional (3D) compressible isentropic magneto-micropolar fluid equations with Coulomb force and slip boundary conditions in a bounded simply connected domain, whose boundary has a…
We study existence of nontrivial solutions to problem \begin{equation*} \left\lbrace \begin{array}{rcll} -\Delta u &=& \lambda u+f(u)&\text{ in }\Omega,\\ u&=&0&\text{ on }\partial \Omega, \end{array}\right. \end{equation*} where $\Omega…
Interesting analogies between shallow water dynamics and astrophysical phenomena have offered valuable insight from both the theoretical and experimental point of view. To help organize these efforts, here we analyze systematically the…
We establish the first finite-time blow-up results for generalized 3D stochastic fractional Navier-Stokes equations \[ \Caputo \mathbf{u} = -(\mathbf{u} \cdot \nabla)\mathbf{u} - \nabla p + \nu \fLaplacian \mathbf{u} +…
Let $u$ be a nonnegative, local, weak solution to the porous medium equation for $m\ge2$ in a space-time cylinder $\Omega_T$. Fix a point $(x_o,t_o)\in\Omega_T$: if the average \[…
This article investigates the existence, nonexistence, and multiplicity of positive solutions to the sublinear fractional elliptic problem $(P_{\lambda}^s)$. We begin by establishing several a priori estimates that provide regularity…
We consider initial boundary-value problems for nonlinear systems of conservation laws in one space variable. It is known that in general different viscous mechanisms yield different solutions in the zero-viscosity limit. Here we focus on…
We are concerned with the nodal set of solutions to equations of the form \begin{equation*} -\Delta u = \lambda_+ \left(u^+\right)^{q-1} - \lambda_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $\lambda_+,\lambda_- >…
In this paper, we study the existence and multiplicity of weak solutions for a general class of elliptic equations (\mathscr{P}_{\lambda}) in a smooth bounded domain, driven by a nonlocal integrodifferential operator…
We apply iteration schemes and perturbation methods to provide a complete solution of the boundary Yamabe problem with minimal boundary scenario, or equivalently, the existence of a real, positive, smooth solution of $ -\frac{4(n -1)}{n -…
For the stationary nonlinear Schr\"odinger equation $-\Delta u+ V(x)u- f(u) = \lambda u$ with periodic potential $V$ we study the existence and stability properties of multibump solutions with prescribed $L^2$-norm. To this end we introduce…
We consider an infinite vortex line in a fluid which interacts with a boundary surface as a simplified model for tornadoes. We study self-similar solutions for stationary axisymmetric Navier-Stokes equations and investigate the types of…
We investigate the existence of time-periodic vortex patch solutions, in both simply and doubly-connected cases, for the two-dimensional lake equation where the depth function of the lake is assumed to be non-degenerate and radial. The…
In this manuscript we study geometric regularity estimates for quasi-linear elliptic equations of $p$-Laplace type ($1 < p< \infty$) with strong absorption condition: $$ -\text{div}\,(\Phi(x, u, \nabla u)) + \lambda_0(x) u_{+}^q(x) = 0…
By introducing a suitable setting, we study the behavior of finite Morse index solutions of the equation \[ -\{div} (|x|^\theta \nabla v)=|x|^l |v|^{p-1}v \;\;\; \{in $\Omega \subset \R^N \; (N \geq 2)$}, \leqno(1) \] where $p>1$, $\theta,…
Inspired by some experimental (numerical) works on fractional diffusion PDEs, we develop a rigorous framework to prove that solutions to the fractional Navier-Stokes equations, which involve the fractional Laplacian operator…
In this paper we apply the approach of formal asymptotic expansions and perturbation theory to derive a new highly nonlinear shallow-water model from the full governing equations for two dimensional incompressible fluid with constant…
In this paper, we investigate the asymptotic behavior of solutions to the Navier-Stokes equations in the half-plane under high Reynolds number conditions, where the initial vorticity belongs to the Yudovich class and is supported away from…