Springer Science and Business Media LLC

This paper is devoted to present some results on the controllability of the hydrostatic Stokes equations. The first main result of this paper states that the approximate null controllability of this system holds. This is proved whatever the boundary conditions are. Then, we extend this result to an exact null controllability result when the boundary conditions are ∂ z u = 0 (the vertical derivative of the horizontal velocity) at the bottom.

We prove the global well-posedness in the critical Besov spaces for the incompressible Oldroyd-B model without damping mechanism on the stress tensor in $${\mathbb {R}}^d$$ for the small initial data. Our proof is based on the observation that the behaviors of Green’s matrix to the system of $$\big (u,(-\Delta )^{-\frac{1}{2}}{\mathbb {P}}\nabla \cdot \tau \big )$$ as well as the effects of $$\tau $$ change from the low frequencies to the high frequencies and the construction of the appropriate energies in different frequencies.

A two-dimensional water wave system is examined consisting of two discrete incompressible fluid domains separated by a free common interface. In a geophysical context this is a model of an internal wave, formed at a pycnocline or thermocline in the ocean. The system is considered as being bounded at the bottom and top by a flatbed and wave-free surface respectively. A current profile with depth-dependent currents in each domain is considered. The Hamiltonian of the system is determined and expressed in terms of canonical wave-related variables. Limiting behaviour is examined and compared to that of other known models. The linearised equations as well as long-wave approximations are presented.

We study the so-called damped Navier–Stokes equations in the whole 2D space. The global well-posedness, dissipativity and further regularity of weak solutions of this problem in the uniformly-local spaces are verified based on the further development of the weighted energy theory for the Navier–Stokes type problems. Note that any divergent free vector field $${u_0 \in L^\infty(\mathbb{R}^2)}$$ is allowed and no assumptions on the spatial decay of solutions as $${|x| \to \infty}$$ are posed. In addition, applying the developed theory to the case of the classical Navier–Stokes problem in $${\mathbb{R}^2}$$ , we show that the properly defined weak solution can grow at most polynomially (as a quintic polynomial) as time goes to infinity.

It is shown both locally and globally that $${L_t^{\infty}(L_x^{3,q})}$$ solutions to the three-dimensional Navier–Stokes equations are regular provided $${q\neq\infty}$$ . Here $${L_x^{3,q}}$$ , $${0 < q \leq\infty}$$ , is an increasing scale of Lorentz spaces containing $${L^3_x}$$ . Thus the result provides an improvement of a result by Escauriaza et al. (Uspekhi Mat Nauk 58:3–44, 2003; translation in Russ Math Surv 58, 211–250, 2003), which treated the case q = 3. A new local energy bound and a new $${\epsilon}$$ -regularity criterion are combined with the backward uniqueness theory of parabolic equations to obtain the result. A weak-strong uniqueness of Leray–Hopf weak solutions in $${L_t^{\infty}(L_x^{3,q})}$$ , $${q\neq\infty}$$ , is also obtained as a consequence.

Bài báo này nhằm nghiên cứu sự tồn tại và tính độc nhất của giải yếu bảo tồn chấp nhận toàn cầu cho phương trình xung đơn vòng định kỳ mà không có giả thiết bổ sung nào. Đầu tiên, bằng cách giới thiệu một tập hợp biến mới, chúng tôi biến đổi phương trình xung đơn vòng thành một hệ thống bán tuyến tính tương đương. Sử dụng lý thuyết phương trình vi phân thông thường tiêu chuẩn, giải pháp toàn cầu của hệ thống bán tuyến tính được nghiên cứu. Thứ hai, khi trở về hệ tọa độ ban đầu, chúng tôi nhận được một giải pháp yếu bảo tồn chấp nhận toàn cầu cho phương trình xung đơn vòng định kỳ. Cuối cùng, bằng cách chọn một số hàm kiểm tra thiết yếu khác với [Bressan (Discrete Contin. Dyn. Syst 35:25-42, 2015), Brunelli (Phys. Lett. A 353:475-478, 2006)], chúng tôi tìm thấy một phương trình để phân biệt một đường cong đặc trưng duy nhất đi qua mỗi điểm khởi đầu. Hơn nữa, tính độc nhất của giải pháp yếu bảo tồn chấp nhận toàn cầu được xác nhận.

In this paper we consider the Cauchy problem for incompressible flows governed by the Navier-Stokes or MHD equations. We give a new proof for the time decay of the spatial $ L_2 $ norm of the solution, under the assumption that the solution of the heat equation with the same initial data decays. By first showing decay of the first derivatives of the solution, we avoid some technical difficulties of earlier proofs based on Fourier splitting.

In this paper, we investigate the dependence on initial data of solutions to the Novikov equation. We show that the solution map is not uniformly continuous dependence on the initial data in Besov spaces $$B^s_{p,r}({\mathbb {R}}),\ s>\max \{1+\frac{1}{p},\frac{3}{2}\}$$ .