In the present paper we give a new proof of the Caffarelli-Kohn-Nirenberg theorem based on a direct approach. Given a pair (u,p) of suitable weak solutions to the Navier-Stokes equations in ℝ³ × ]0,∞[ the velocity field u satisfies the following property of partial regularity: The velocity u is Lipschitz continuous in a neighbourhood of a point (x₀,t₀) ∈ Ω × ]0,∞ [ if $limsu{p}_{R\to 0⁺}1/R{\int }_{Q_R(x₀,t₀)}|curlu\times u/|u\left|\right|²dxdt\le {\epsilon }_{*}$ for a sufficiently small ${\epsilon }_{*}>0$.

We deal with a suitable weak solution $(𝐯,p)$ to the Navier-Stokes equations in a domain $\Omega \subset {ℝ}^3$. We refine the criterion for the local regularity of this solution at the point $(𝐟x_0,t_0)$, which uses the $L^3$-norm of $𝐯$ and the $L^{3/2}$-norm of $p$ in a shrinking backward parabolic neighbourhood of $({𝐱}_0,t_0)$. The refinement consists in the fact that only the values of $𝐯$, respectively $p$, in the exterior of a space-time paraboloid with vertex at $({𝐱}_0,t_0)$, respectively in a ”small” subset of this exterior, are considered. The consequence is that...

A regularity criterion for strong solutions of the Ericksen-Leslie equations is established in terms of both the pressure and orientation field in homogeneous multiplier spaces.

We prove that there exists a suitable weak solution of the Navier-Stokes equation, which satisfies the generalized energy inequality for every nonnegative test function. This improves the famous result on existence of a suitable weak solution which satisfies this inequality for smooth nonnegative test functions with compact support in the space-time.

We propose a modification of the classical Boussinesq approximation for buoyancy-driven flows of viscous, incompressible fluids in situations where viscous heating cannot be neglected. This modification is motivated by unresolved issues regarding the global solvability of the original system. A very simple model problem leads to a coupled system of two parabolic equations with a source term involving the square of the gradient of one of the unknowns. Based on adequate notions of weak and strong...

The incompressible three-dimensional Navier-Stokes equations are considered. A new regularity criterion for weak solutions is established in terms of the pressure gradient.

We study higher local integrability of a weak solution to the steady Stokes problem. We consider the case of a pressure- and shear-rate-dependent viscosity, i.e., the elliptic part of the Stokes problem is assumed to be nonlinear and it depends on $p$ and on the symmetric part of a gradient of $u$, namely, it is represented by a stress tensor $T(Du,p):=\nu (p,|D{|}^2)D$ which satisfies $r$-growth condition with $r\in (1,2]$. In order to get the main result, we use Calderón-Zygmund theory and the method which was presented for example in...

We review several regularity criteria for the Navier-Stokes equations and prove some new ones, containing different components of the velocity gradient.

We consider a continuum model describing steady flows of a miscible mixture of two fluids. The densities ${\rho }_i$ of the fluids and their velocity fields $u^{\left(i\right)}$ are prescribed at infinity: ${\rho }_i{|}_{\infty }={\rho }_{i\infty }>0$, $u^{\left(i\right)}{|}_{\infty }=0$. Neglecting the convective terms, we have proved earlier that weak solutions to such a reduced system exist. Here we establish a uniqueness type result: in the absence of the external forces and interaction terms, there is only one such solution, namely ${\rho }_i\equiv {\rho }_{i\infty }$, $u^{\left(i\right)}\equiv 0$, $i=1,2$.

We present a simplified proof of a theorem proved recently concerning the number of singular points of weak solutions to the Navier-Stokes equations. If a weak solution $𝐮$ belongs to $L^{\infty }(0,T,L^3{\left(\Omega \right)}^3)$, then the set of all possible singular points of $𝐮$ in $\Omega$ is at most finite at every time $t_0\in (0,T)$.

Almost global in time existence of solutions for equations describing the motion of a magnetohydrodynamic incompressible fluid in a domain bounded by a free surfaced is proved. In the exterior domain we have an electromagnetic field which is generated by some currents which are located on a fixed boundary. We prove that a solution exists for t ∈ (0,T), where T > 0 is large if the data are small.

As observed by Yamazaki, the third component $b_3$ of the magnetic field can be estimated by the corresponding component $u_3$ of the velocity field in $L^{\lambda }$$(2\le \lambda \le 6)...\; <p></p>\; <article>\; <h3>\; <a\; href="\; /doc/42294">\; An\; optimal\; control\; problem\; for\; a\; generalized\; Boussinesq\; model:\; The\; time\; dependent\; case.\; </a>\; </h3>\; <div\; class="meta\; meta-detail">\; <p\; class="author">\; Jose\; Luis\; Boldrini,\; Enrique\; Fernández-Cara,\; Marko\; Antonio\; Rojas-Medar\; (2007)\; </p>\; <p\; class="source">\; Revista\; Matemática\; Complutense\; </p>\; </div>\; <div\; class="excerpt">\; <p>\; </p>\; </div>\; </article>\; <section\; class="unit\; unit-results-currently-displaying-subject">\; <h2>Currently\; displaying\; 1\; –\; 20\; of\; 223</h2>\; <div>\; <p\; class="pagination"\; aria-label="Pages">\; <strong>Page\; 1</strong>\; <a\; href="?pageNumber=2\&prefix=">\; <span\; class="next">Next</span>\; </a>\; </p>\; </div>\; </section>\; <!--\; /\; end\; \#tabs-1\; -->\; <!--\; /\; end\; \#tabs\; -->\; <!--\; /\#primary-content\; -->\; <!--\; /\#page-content\; -->\; <!--\; \#container\; -->$