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We address the global existence of solutions for the 2D Kuramoto-Sivashinsky equations in a periodic domain $$[0,L_1]\times [0,L_2]$$ with initial data satisfying $$\Vert u_0\Vert _{L^2}\le C^{-1}L_2^{-2}$$ , where C is a constant. We prove that the global solution exists under the condition $$L_2\le 1/C L_1^{3/5}$$ , improving earlier results. The solutions are smooth and decrease energy until they are dominated by $$C L_1^{3/2}L_2^{1/2}$$ , implying the existence of an absorbing ball in $$L^2$$ .

We establish the strong unique continuation property for differences of solutions to the Navier–Stokes system with Gevrey forcing. For this purpose, we provide Carleman-type inequalities with the same singular weight for the Laplacian and the heat operator.

We address the problem of local uniqueness of weak solutions to the Navier–Stokes system, with the initial datum in a subspace of $${BMO^{-1}(\mathbb{R}^{n})}$$ . The existence and uniqueness of local mild solutions has been proven by Koch and Tataru (Adv Math 157:22–35, 2001). We present a necessary and sufficient condition for two weak solutions to evolve from the same initial datum, and for weak solutions to be mild.

We prove that a weak solution $$u \in L_{\rm loc}^{\infty} ([0,T),L^{2}({\mathbb R}^{3})) \cap L_{\rm loc}^{2} ([0,T),H^{1}({\mathbb R}^{3}))$$ of the Navier–Stokes system satisfies the energy equality if the associated pressure is locally square integrable. A similar statement is shown to hold for the Euler system.

For dissipative equations of the form $$u_t + ( - 1)^m u^{(2m)} + f(x,u,u_x ,...,u^{(2m - 1)} ) = 0$$ we show that the global attractor is a Lipschitz graph over a finite dimensional Fourier eigenspace. In particular, the statement applies to the Burgers equation u t −u xx +uu x =f and the modified Burgers equation u t −u xx +uu x −u=0.

We show that solutions of the 1D Kuramoto-Sivashinsky equation with periodic boundary conditions are asymptotically determined by their values at four points. That is, there existx1,x2,x3, andx4 in the (periodic) domainΩ such that if $$\mathop {\lim }\limits_{t \to \infty } \left| {u_1 (x_j ,t) - u_2 (x_j ,t)} \right| = 0, j = 1,2,3,4$$ for two solutionsu1 andu2, then $$\mathop {\lim }\limits_{t \to \infty } \left\| {u_1 ( \cdot ,t) - u_2 ( \cdot ,t)} \right\|_{L^2 (\Omega )} = 0$$

In the case of the 3D Navier-Stokes equations, it is proved that there exists a constantɛ>0 with the following property: Every time-periodic solution with a period smaller thanɛ is necessarily a stationary solution. An explicit formula for ɛ is also provided.

In the case of solutions of the two-diensional Navier-Stokes equations, the following analyticity property is established. If the initial datum lies on the global attractor and is close enough to a stationary solution, then the analyticity radius att = 0 of the solution can be made arbitrarily large.