We prove the existence of infinitely many non-zero time-periodic solutions (breathers) to the dispersive wave equation of the form which are localized in the spatial variable, that is The main tool employed is the concentration compactness principle of P. L. Lions. 相似文献
We show the weak–strong uniqueness property for the compressible Navier–Stokes system with general non-monotone pressure law. A weak solution coincides with the strong solution emanating from the same initial data as long as the latter solution exists. 相似文献
We consider the zero-electron-mass limit for the Navier?CStokes?CPoisson system in unbounded spatial domains. Assuming smallness of the viscosity coefficient and ill-prepared initial data, we show that the asymptotic limit is represented by the incompressible Navier?CStokes system, with a Brinkman damping, in the case when viscosity is proportional to the electron-mass, and by the incompressible Euler system provided the viscosity is dominated by the electron mass. The proof is based on the RAGE theorem and dispersive estimates for acoustic waves, and on the concept of suitable weak solutions for the compressible Navier?CStokes system. 相似文献
We consider a general Euler-Korteweg-Poisson system in R3, supplemented with the space periodic boundary conditions, where the quantum hydrodynamics equations and the classical fluid dynamics equations with capillarity are recovered as particular examples. We show that the system admits infinitely many global-in-time weak solutions for any sufficiently smooth initial data including the case of a vanishing initial density - the vacuum zones. Moreover, there is a vast family of initial data, for which the Cauchy problem possesses infinitely many dissipative weak solutions, i.e. the weak solutions satisfying the energy inequality. Finally, we establish the weak-strong uniqueness property in a class of solutions without vacuum. In this paper we show that, even in presence of a dispersive tensor, we have the same phenomena found by De Lellis and Székelyhidi. 相似文献
We prove that the solution semigroup $$S_t \left[ {u_0 ,v_0 } \right] = \left[ {u(t),u_t (t)} \right]$$ generated by the evolutionary problem $$\left\{ P \right\}\left\{ \begin{gathered} u_{tt} + g(u_t ) + Lu + f(u) = 0, t \geqslant 0 \hfill \\ u(0) = u_0 , u_t (0) = \upsilon _0 \hfill \\ \end{gathered} \right.$$ possesses a global attractorA in the energy spaceEo=V×L2(Ω). Moreover,A is contained in a finite-dimensional inertial setA attracting bounded subsets ofE1=D(L)×V exponentially with growing time. 相似文献
We establish the asymptotic limit of the compressible Navier–Stokes system in the regime of low Mach and high Reynolds number on unbounded spatial domains with slip boundary condition. The result holds in the class of suitable weak solutions satisfying a relative entropy inequality. 相似文献