We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the initial data $u_0\in X$, where $X\in \{M_{2,q}^s\left(ℝ\right),H^{\sigma }\left(𝕋\right),H^{s_1}\left(ℝ\right)+H^{s_2}\left(𝕋\right)\}$ and $q\in [1,2]$, $s\ge 0$, or $\sigma \ge 0$, or $s_2\ge s_1\ge 0$. Moreover, if $M_{2,q}^s\left(ℝ\right)\hookrightarrow L^3\left(ℝ\right)$, or if $\sigma \ge \frac{1}{6}$, or if $s_1\ge \frac{1}{6}$ and $s_2>\frac{1}{2}$ we show that the Cauchy problem is unconditionally wellposed in $X$. Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ the normal...

We investigate the existence and uniqueness of solutions to the Dirichlet problem for a degenerate nonlinear elliptic equation $-{\sum }_{i,j=1}^n{D}_j\left(a_{ij}\left(x\right)D_{i}u\left(x\right)\right)+b\left(x\right)u\left(x\right)+div\left(\Phi \left(u\left(x\right)\right)\right)=g\left(x\right)-{\sum }_{j=1}^n{f}_j\left(x\right)$ on Ω in the setting of the space H₀(Ω).

Consider the Navier-Stokes equation with the initial data $a\in L_{\sigma }^2\left({ℝ}^d\right)$. Let $u$ and $v$ be two weak solutions with the same initial value $a$. If $u$ satisfies the usual energy inequality and if $\nabla v\in L^2((0,T);{\dot{X}}_1{\left({ℝ}^d\right)}^d)$ where ${\dot{X}}_1\left({ℝ}^d\right)$ is the multiplier space, then we have $u=v$.