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A blow-up criterion for the strong solutions to the nonhomogeneous Navier-Stokes-Korteweg equations in dimension three

Huanyuan Li (2021)

Applications of Mathematics

This paper proves a Serrin’s type blow-up criterion for the 3D density-dependent Navier-Stokes-Korteweg equations with vacuum. It is shown that if the density ρ and velocity field u satisfy ρ L ( 0 , T ; W 1 , q ) + u L s ( 0 , T ; L ω r ) < for some q > 3 and any ( r , s ) satisfying 2 / s + 3 / r 1 , 3 < r , then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally over [ 0 , T ] . Here L ω r denotes the weak L r space.

A global existence result for the compressible Navier-Stokes-Poisson equations in three and higher dimensions

Zhensheng Gao, Zhong Tan (2012)

Annales Polonici Mathematici

The paper is dedicated to the global well-posedness of the barotropic compressible Navier-Stokes-Poisson system in the whole space N with N ≥ 3. The global existence and uniqueness of the strong solution is shown in the framework of hybrid Besov spaces. The initial velocity has the same critical regularity index as for the incompressible homogeneous Navier-Stokes equations. The proof relies on a uniform estimate for a mixed hyperbolic/parabolic linear system with a convection term.

Global strong solutions of a 2-D new magnetohydrodynamic system

Ruikuan Liu, Jiayan Yang (2020)

Applications of Mathematics

The main objective of this paper is to study the global strong solution of the parabolic-hyperbolic incompressible magnetohydrodynamic model in the two dimensional space. Based on Agmon, Douglis, and Nirenberg’s estimates for the stationary Stokes equation and Solonnikov’s theorem on L p - L q -estimates for the evolution Stokes equation, it is shown that this coupled magnetohydrodynamic equations possesses a global strong solution. In addition, the uniqueness of the global strong solution is obtained.

Growing Sobolev norms for the cubic defocusing Schrödinger equation

Zaher Hani, Benoit Pausader, Nikolay Tzvetkov, Nicola Visciglia (2013/2014)

Séminaire Laurent Schwartz — EDP et applications

This text aims to describe results of the authors on the long time behavior of NLS on product spaces with a particular emphasis on the existence of solutions with growing higher Sobolev norms.

On the global existence for the Muskat problem

Peter Constantin, Diego Córdoba, Francisco Gancedo, Robert M. Strain (2013)

Journal of the European Mathematical Society

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an L 2 ( ) maximum principle, in the form of a new “log” conservation law which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance f 1 1 / 5 . Previous results of this...

On the solvability of the equation div u = f in L 1 and in C 0

Bernard Dacorogna, Nicola Fusco, Luc Tartar (2003)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We show that the equation div u = f has, in general, no Lipschitz (respectively W 1 , 1 ) solution if f is C 0 (respectively L 1 ).

Parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces

Vagif S. Guliyev, Mehriban N. Omarova (2016)

Open Mathematics

We obtain the global weighted Morrey-type regularity of the solution of the regular oblique derivative problem for linear uniformly parabolic operators with VMO coefficients. We show that if the right-hand side of the parabolic equation belongs to certain generalized weighted Morrey space Mp,ϕ(Q, w), than the strong solution belongs to the generalized weighted Sobolev- Morrey space [...] W˙2,1p,φ(Q,ω) W ˙ 2 , 1 p , ϕ Q , ω .

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