The aim of this paper is to show the existence of $\mathcal{R}$-bounded solution operator families for two-phase Stokes resolvent equations in $\dot\Omega =\Omega _+\cup\Omega _-$, where $\Omega _\pm$ are uniform $W_r^{2-1/r}$ domains of $N$-dimensional Euclidean space ${\mathbf{R}^N}$ ($N\geq 2$, $N < r < \infty$). More precisely, given a uniform $W_r^{2-1/r}$ domain $\Omega $ with two boundaries $ \Gamma _\pm$ satisfying $ \Gamma _+\cap \Gamma _-=\emptyset$, we suppose that some hypersurface $ \Gamma $ divides $\Omega $ into two sub-domains, that is, there exist domains $\Omega _\pm\subset\Omega $ such that $ \Omega _+\cap\Omega _-=\emptyset$ and $\Omega \setminus \Gamma =\Omega _+\cup\Omega _-, $ where $ \Gamma \cap \Gamma _+=\emptyset$, $ \Gamma \cap \Gamma _-=\emptyset$, and the boundaries of $\Omega _\pm$ consist of two parts $ \Gamma $ and $ \Gamma _\pm$, respectively. The domains $\Omega _\pm$ are filled with viscous, incompressible, and immiscible fluids with density $\rho_\pm$ and viscosity $\mu_\pm$, respectively. Here, $\rho_\pm$ are positive constants, while $\mu_\pm=\mu_\pm(x)$ are functions of $x\in{\mathbf{R}^N}$. On the boundaries $ \Gamma $, $ \Gamma _+$, and $ \Gamma _-$, we consider an interface condition, a free boundary condition, and the Dirichlet boundary condition, respectively. We also show, by using the $\mathcal{R}$-bounded solution operator families, some maximal $L_p{\theta}xt{-}L_q$ regularity as well as generation of analytic semigroup for a time-dependent problem associated with the two-phase Stokes resolvent equations. This kind of problems arises in the mathematical study of the motion of two viscous, incompressible, and immiscible fluids with free surfaces. The essential assumption of this paper is the unique solvability of a weak elliptic transmission problem for $\mathbf{f}\in L_q(\Omega )^N$, that is, it is assumed that the unique existence of solutions ${\theta}\in\mathcal{W}_q^1(\Omega )$ to the variational problem: $ (\rho^{-1}\nabla{\theta},\nabla{\varphi})_{\dot\Omega }=(\mathbf{f},\nabla{\varphi})_{\Omega } $ for any ${\varphi}\in\mathcal{W}_{q'}^1(\Omega )$ with $1 < q < \infty$ and $q'=q/(q-1)$, where $\rho$ is defined by $\rho=\rho_+$ ($x\in\Omega _+$), $\rho=\rho_-$ ($x\in\Omega _-$) and $\mathcal{W}_q^1(\Omega )$ is a suitable Banach space endowed with norm $\|\cdot\|_{\mathcal{W}_q^1(\Omega )}:=\|\nabla\cdot\|_{L_q(\Omega )}$. Our assumption covers e.g. the following domains as $\Omega $: ${\mathbf{R}^N}$, $\mathbf{R}_\pm^N$, perturbed $\mathbf{R}_\pm^N$, layers, perturbed layers, and bounded domains, where $\mathbf{R}_+^N$ and $\mathbf{R}_-^N$ are the open upper and lower half spaces, respectively.

Our aim, in this paper, is to study a generalization of the Caginalp phase-field system based on the Gurtin--Pipkin law with two temperatures for heat conduction with memory. In particular, we obtain well-posedness results and study the dissipativity, in terms of the global attractor with optimal regularity, of the associated solution operators. We also study the stability of the system as the memory kernel collapses to a Dirac mass.

In this paper, we will use a gauge transform to prove the local existence and uniqueness of the derivative nonlinear Schrödinger equation with additive noise, showing that for the initial data $u_0\in H^\frac{1}{2}(\mathbb{R})$, there is a local and unique solution almost surely.

We show some regularity criteria for Navier-Stokes equations, the harmonic heat flow, two liquid crystals models, and a model for magneto-elastic materials. The method of proof depends on a systematic use of interpolation inequalities in Besov spaces and is independent on logarithmic inequalities.

We prove the existence of a ground state solution for the following fractional scalar field equation \begin{align*} (-\Delta)^{s} u= g(u) \mbox{ in } \mathbb R^{N} \end{align*} where $s\in (0,1)$, $N> 2s$, $(-\Delta)^{s}$ is the fractional Laplacian, and $g\in C^{1, \beta}( \mathbb R, \mathbb R)$ is an odd function satisfying the critical growth assumption.

We consider the following nonlinear elliptic equation \begin{equation} \tag{0.1} A_\frac{1}{2} u = K(x) |u|^{p-1}u \hbox{ in } \Omega, \;\; u=0 \hbox{ on } \partial\Omega, \end{equation} where $\Omega$ is a bounded domain of $\mathbb{R}^n, n\geq 1, K(x)$ is a given function, $A_\frac{1}{2}$ represents the square root of $-\Delta$ in $\Omega$ with zero Dirichlet boundary condition and $1 < p < \frac{n+1}{n-1}$, $(p > 1$ if $n=1$). We apply the Brouwer's fixed point theorem to prove that (0.1) has infinitely many distinct solutions.

In this paper, we consider the quasilinear elliptic equation $-\Delta_m u=\lambda f(u)$, in a bounded, smooth and convex domain. When the nonnegative nonlinearity $f$ has multiple positive zeros, we prove the existence of at least two positive solutions for each of these zeros, for $\lambda$ large, without any hypothesis on the behavior at infinity of $f$. We also prove a result concerning the behavior of the solutions as $\lambda\to\infty$.