Considered here is a model system derived by Gear and Grimshaw to describe the strong interaction of weakly nonlinear, long waves. It has the structure of a pair of Korteweg-de Vries equations coupled through both dispersive and nonlinear effects. Our aim is to investigate the controllability properties of the model, posed on a bounded interval, by means of distributed controls. When the control region is a neighborhood of the right end point of the interval, we prove the local exact controllability of the nonlinear problem in a well chosen weighted L2-spaces. The results are first established for the linearized system through a classical duality approach and then extended for the full system via a fixed point argument.

In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian systems \begin{eqnarray*} \begin{cases} _tD_\infty^\alpha(_{-\infty}D_t^\alpha u(t))+L(t)u(t) =\nabla F(t, u(t)),\\ u\in H^\alpha(\mathbb R,\mathbb R^n), \end{cases} \end{eqnarray*} where $\alpha \in (1/2, 1), u\in {\mathbb{R}}^{N}$, $L\in C({\mathbb{R}}, {\mathbb{R}}^{N\times N})$ and $F\in C^1({\mathbb{R}}\times {\mathbb{R}}^{N}, {\mathbb{R}})$. The novelty of this paper is that, under the relaxed assumptions on $F(t, x)$ and $L(t)$, we obtain infinitely many solutions via genus properties in critical point theory. Recent results are generalized and significantly improved.

Let $(M,g)$ be a compact Riemann surface with no boundary and $u^*=(u_1^*,u_2^*)$ be a solution of the following singular Liouville system: $$ \Delta_g u_i^*+\sum_{j=1}^2 a_{ij} \rho_j \Big (\frac{h_j^*e^{u_j^*}}{\int_M h_j^* e^{u_j^*}dV_g}-1 \Big )=\sum_{l=1}^{N}4\pi \gamma_l(\delta_{p_l}-1), $$ where $h_1^*,h_2^*$ are positive smooth functions, $p_1,\cdots,p_N$ are distinct points on $M$, $\delta_{p_l}$ are Dirac masses, $\rho=(\rho_1,\rho_2)$ $(\rho_i\geq 0)$ and $(\gamma_1,\cdots,\gamma_N)$ $(\gamma_l > -1)$ are constant vectors. In the previous work, we derived a degree counting formula for the singular Liouville system when $A$ satisfies standard assumptions. In this article, we establish a more general degree counting formula for 2$\times$2 singular Liouville system when the coefficient matrix $A$ is non-symmetric and non-invertible. Finally, the existence of solution can be proved by the degree counting formula which depends only on the topology of the domain and the location of $\rho$.

We consider a sufficient condition for the regularity of a weak solution for the incompressible Navier--Stokes equations on $\mathbb R^d$. We prove a smoothing property of the weak solution near the initial time under the condition that a scale invariant norm written by $BMO$ is finite.

We consider the well-posedness of the Cauchy problem for convection-diffusion equations in uniformly local Lebesgue spaces $L^r_{\text{uloc}}(\mathbb R^n)$. In our setting, an initial function that is spatially periodic or converges to a nonzero constant at infinity is admitted. Our result is applicable to the one dimensional viscous Burgers equation. For the proof, we use the $L^p_{\text{uloc}}- L^q_{\text{uloc}}$ estimate for the heat semigroup obtained by Maekawa--Terasawa [20], the Banach fixed point theorem, and the comparison principle.