In the paper we introduce the concept of the Kuratowski distance between nonempty and bounded subsets of a Banach space. We show that the Kuratowski measure of noncompactness of a nonempty and bounded subset of a Banach space is equal to the Kuratowski distance of that subset to the family of all nonempty and relatively compact subsets of the mentioned Banach space.
In this paper, we are concerned with the validity of energy equality for weak solutions of the incompressible magnetohydrodynamic equations in the framework of anisotropic regularities. We present some new energy equality criteria which allow the velocity field to lie within the classical Shinbrot or Taniuchi-Beir & atilde;o da Veiga-Yang energy equality classes, while permitting the magnetic field to extend beyond these classes. Furthermore, we also establish energy equality criteria in terms of the gradient of the velocity only.
In this paper, we study a two-phase flow system consisting of the compressible pressureless Euler equations and incompressible Navier-Stokes equations coupled through the drag force rho(u - v). Due to the absence of pressure term in the Euler equations, the classical existence theory of the hyperbolic-parabolic system developed by Kawashima in [Doctoral Dissertation, Kyoto University, 1984] is not applicable to this system. Therefore, a new approach has been introduced here to address this issue. Firstly, we employ the pure energy method to derive a faster time-decay rate for the velocity of Euler flow parallel to & nabla;(2)u parallel to(2)(H) <= C (1 + t)(-2+sigma/2) under the assumption that initial velocity of Navier-Stokes flow v(0) is an element of H-center dot(-sigma) (& Ropf;(3)) for some sigma is an element of (0,3/2) and initial density rho(0) > 0. Then, combining the above decay estimate and characteristic method yields the uniform boundedness of the density rho. Finally, utilizing the uniform estimates of velocities and the continuity argument, we establish the global well-posedness of this two-phase flow system. As a by-product, it is also shown that the velocities (u, v) decay to the motionless state at the optimal algebraic time-decay rates in L-2-norm. Our results reveal that the smoothing effect of the Navier-Stokes equations can be propagated to Euler equations through the drag force.
In this paper we study existence, uniqueness, regularity and asymptotic behavior of the solutions to a class of anisotropic parabolic equations of p-Laplacian type when the initial datum is a summable function. We prove the existence of a unique solution that immediately becomes regular under very weak limitations on the diffusion exponents which may also be very sparse.
This paper addresses the quasilinear Schr & ouml;dinger equation with critical growth in the zero-mass case: -Delta u - u Delta(u(2)) = lambda g(x)|u|(p-2)u + |u|(2 & sdot;2 & lowast;-2)u, x is an element of & Ropf;(N) , where N >= 3, lambda > 0, p is an element of (4, 2 & sdot;2(& lowast;)) or p is an element of (1, 2) with 2(& lowast;) = 2N/N-2 the critical Sobolev exponent, and g is an element of L-p0 (& Ropf;(N)), where p(0) = 2.2 & lowast; 2 & sdot; 2 & lowast; - p if p is an element of (4, 2 & sdot;2(& lowast;)), p0 = 2(& lowast;)/2 & lowast;- p if p is an element of (1, 2). To address the inherent lack of compactness in the zero-mass case, we develop a perturbative variational framework inspired by established techniques. For the case p is an element of (4, 2 & sdot; 2(& lowast;)), we establish the existence of a positive-energy ground state solution when lambda is sufficiently large. For the case p is an element of (1, 2), we prove that for small lambda > 0, the equation admits a ground state solution with negative energy. Additionally, for p is an element of (1, 2), we show the multiplicity of solutions to the above equation for small lambda
