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We study a nonlinear parabolic system for a time dependent solenoidal vector field on ℝ3. The nonlinear term of these new model equations is obtained slightly modifying that of the Navier–Stokes equations. The system has the same scaling property and Galilean invariance as the Navier–Stokes equations. For such a system we prove the global regularity for smooth initial data.

In this paper we extend the results of Foias-Guillope-Temam on the regularity and a priori estimates for the weak solutions of the Navier-Stokes equations. More specifically, we obtain upperbounds for the temporal averages of the Gevrey class norm for the weak solutions of the equations. The estimates are obtained first by getting integrated version of Foias-Temam's local in time estimate for Gevrey class norms of strong solutions and next by an induction procedure. We also strengthen slightly the local in time Gevrey class regularization of strong solutions.

We study a system of elliptic equations from the Abelian Born-Infeld system coupled with the Einstein equations under the boundary condition of symmetric vacuum (nontopological type). When the total string number satisfies 1 ? N< 1/4? G, where Gis the gravitational constant, we construct a family of solutions to the system. The qualitative properties of the solutions are quite different from the solutions with the boundary condition of broken vacuum symmetry.

In this paper we prove that the finite time blow-up of classical solutions of the three-dimensional homogeneous incompressible Euler equations are controlled by the Besov space, , norm of the two componentsof the vorticity. For the axisymmetric flows with swirl we deduce that the blow-up of solution is controlled by the same Besov space norm of the angular componentof the vorticity. For a proof of these results we use the Beale-Kato-Majda criterion, and the special structure of the vortex stretching term in the vorticity formulation of the Euler equations.

In this paper we consider two model systems of equations obtained by taking the Hodge duals of the Euler and the Navier–Stokes systems, respectively. The ‘vorticity’ formulation of the new system becomes a nonlinear, non-local scalar equation having similarities to the quasi-geostrophic equations. We prove the local existence of classical solutions, and derive the Beale-Kato-Majda type of blow-up criterion for the dual Euler system. Moreover, we actually show that the finite-time blow-up occurs for generic non-trivial initial data for this system. For the dual Navier-Stokes equations we construct an explicitly global-in-time smooth solution for general initial data, using the Hopf-Cole transform. In the appendix we formulate the Euler system and the dual Euler system in terms of the Clifford algebra.

Abstract.: In this paper we construct a non-topological multivortex solution of a generalized version of the relativistic self-dual Chern-Simons-Higgs system in that makes the energy functional finite. Our method of proof is an extension of the previous argument used by the authors to prove the existence of general type of non-topological multivortex solutions of the relativistic Chern-Simons-Higgs system, using an implicit function theorem argument with features similar to the Liapunov-Schmidt decomposition.

We prove local-in-time unique existence and a blowup criterion for solutions in the Triebel-Lizorkin space for the Euler equations of inviscid incompressible fluid flows in ℝⁿ, n ≥ 2. As a corollary we obtain global persistence of the initial regularity characterized by the Triebel-Lizorkin spaces for the solutions of two-dimensional Euler equations. To prove the results, we establish the logarithmic inequality of the Beale-Kato-Majda type, the Moser type of inequality, as well as the commutator estimate in the Triebel-Lizorkin spaces. The key methods of proof used are the Littlewood-Paley decomposition and the paradifferential calculus by J. M. Bony. © 2002 John Wiley & Sons, Inc.

Abstract.: We obtain improved regularity criteria for the axisymmetric weak solutions of the three dimensional Navier-Stokes equations with nonzero swirl. In particular we prove that the integrability of single component of vorticity or velocity fields, in terms of norms with zero scaling dimension give sufficient conditions for the regularity of weak solutions. To obtain these criteria we derive new a priori estimates for the axisymmetric smooth solutions of the Navier-Stokes equations.

We prove the global unique existence of classical solutions to the Einstein equations coupled with the nonlinear Klein-Gordon equations for small initial data under spherical symmetry. We also obtain the decay estimates of the solutions. For its proof we reduce the system to a single first-order integro-differential equation, and use the contraction mapping theorem in the appropriate function spaces.

Abstract:: We construct a general type of multivortex solutions of the self-duality equations (the Bogomol'nyi equations) of (2+1) dimensional relativistic Chern–Simons model with the non-topological boundary condition near infinity. For such construction we use a perturbation argument around the explicit solutions of the Liouville equation.

In this paper we first prove a new criterion of global solvability for smooth axisymmetric solutions of the 3-D Euler equations for a cylindrical type of domain, which can be whole R3. Then, using this result, we obtain generic solvability of the equations in the cylindrical domain which does not include the axis of symmetry. By a similar method we also obtain generic solvability of the 2-D Boussinesq system.

In this Letter, we investigate multiple existence of the multivortex solutions of the self-dual Chern–Simons CP(1) model on a flat torus for each range of the symmetry breaking parameter. We also study asymptotics of solutions as the Chern–Simons coupling constant goes to zero.

We present a new family of travelling wave-like solutions to the Navier–Stokes equations of incompressible fluid flows, and other regularized equations of the Euler equations, obtain their trend to the solutions of the Euler equations as the viscosity tends to zero, and estimate the rate of convergence. We also find a “singularizing effect” of the viscosity term in the Navier–Stokes equations, i.e., we have a local moving blow-up of unbounded solutions with the blow-up's speed depending on viscosity. We demonstrate that if the initial function is the Beltrami flow then the solution of the Navier–Stokes equations conserves the Beltrami flow property for all time.

We study the existence and various behaviors of topological multivortex solutions of the relativistic self-dual Maxwell–Chern–Simons–Higgs system. We first prove the existence of general topological solutions by applying variational methods to the newly discovered minimizing functional. Then, by an iteration method, we prove the existence of topological solutions satisfying some extra conditions, which we call admissible solutions. We establish asymptotic exponential decay estimates for these topological solutions. We also investigate the limiting behavior of the admissible solutions as parameters in our system go to some limits. For the Abelian Higgs limit we obtain strong convergence result, while for the Chern–Simons limit we only obtained that our admissible solutions weakly approximate one of the Chern–Simons solutions.

We prove the local existence theorem for general Smoluchovsky's coagulation equation with coagulation kernels which allow the multiplicative growth. If the system concerned has absorption, then the local existence theorem converts into the global existence theorem provided that initial data and sources are sufficiently small. We prove uniqueness, mass conservation and continuous dependence on initial data in the domain of its existence. We show that the solution “in large” asymptotically tends to zero as time goes to infinity and demonstrate that, in general, the sequence of approximated solutions does not converge to the exact solution of the original problem with the multiplicative kernel. This fact reveals the limits of numerical simulation of the coagulation equation.

We consider the notion of a functional solution of the Euler equations for incompressible fluid flows. We show that a functional solution can be constructed under “very weak” a priori estimates on approximate solution sequences of the equation; an estimate uniform in Lloc1 together with weak consistency with the equation is sufficient to construct a solution. We prove that if we have an estimate uniform in Lloc2 available for the approximate solution sequence, then the structured functional solution just described becomes a measure-valued solution in the sense of DiPerna & Majda. We also show that a functional solution can be obtained from a measure-valued solution. We give an example showing that a much higher concentration of energy than in the case of measure-valued solutions is allowed by the approximation procedure of a functional solution.

We refine the Beale-Kato-Majda criterion for the breakdown of smooth solutions of the 3-D incompressible Euler equations in the case of axisymmetry. In this case the angular component of vorticity in the cylindrical coordinates alone controls blow-up of the higher Sobolev norms of the velocity.

This paper studies the problem of exact boundary controllability of second-order semilinear parabolic equations when the control is under Neumann's boundary conditions. For a nonlinear term with a sublinear growth we prove the global null-controllability and for the superlinear growth case we prove the local exact controllability of the equations.