Authors:Willie Wai Yeung Wong Pages: 1 - 13 Abstract: Journal of Hyperbolic Differential Equations, Volume 15, Issue 01, Page 1-13, March 2018. We prove that there does not exist global-in-time axisymmetric solutions to the time-like minimal submanifold system in Minkowski space. We further analyze the limiting geometry as the maximal time of existence is approached. Citation: Journal of Hyperbolic Differential Equations PubDate: 2018-03-28T09:06:52Z DOI: 10.1142/S0219891618500017
Authors:Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu Pages: 15 - 35 Abstract: Journal of Hyperbolic Differential Equations, Volume 15, Issue 01, Page 15-35, March 2018. We consider a model of liquid crystals, based on a nonlinear hyperbolic system of differential equations, that represents an inviscid version of the model proposed by Qian and Sheng. A new concept of dissipative solution is proposed, for which a global-in-time existence theorem is shown. The dissipative solutions enjoy the following properties: (i)they exist globally in time for any finite energy initial data; (ii)dissipative solutions enjoying certain smoothness are classical solutions; (iii)a dissipative solution coincides with a strong solution originating from the same initial data as long as the latter exists. Citation: Journal of Hyperbolic Differential Equations PubDate: 2018-03-28T09:06:59Z DOI: 10.1142/S0219891618500029
Authors:Tarek Elgindi, Donghyun Lee Pages: 37 - 118 Abstract: Journal of Hyperbolic Differential Equations, Volume 15, Issue 01, Page 37-118, March 2018. We study the zero-viscosity limit of free boundary Navier–Stokes equations with surface tension in unbounded domain thus extending the work of Masmoudi and Rousset [Uniform regularity and vanishing viscosity limit for the free surface Navier–Stokes equations, Arch. Ration. Mech. Anal. (2016), doi:10.1007/s00205-016-1036-5] to take surface tension into account. Due to the presence of boundary layers, we are unable to pass to the zero-viscosity limit in the usual Sobolev spaces. Indeed, as viscosity tends to zero, normal derivatives at the boundary should blow-up. To deal with this problem, we solve the free boundary problem in the so-called Sobolev co-normal spaces (after fixing the boundary via a coordinate transformation). We prove estimates which are uniform in the viscosity. And after inviscid limit process, we get the local existence of free-boundary Euler equation with surface tension. Our main idea is to use Dirichlet–Neumann operator and time-derivatives. Citation: Journal of Hyperbolic Differential Equations PubDate: 2018-03-28T09:06:47Z DOI: 10.1142/S0219891618500030
Authors:Darko Mitrović, Andrej Novak Pages: 119 - 132 Abstract: Journal of Hyperbolic Differential Equations, Volume 15, Issue 01, Page 119-132, March 2018. We extend Brenier’s transport collapse scheme on the Cauchy problem for heterogeneous scalar conservation laws i.e. for the conservation laws with spacetime-dependent coefficients. The method is based on averaging out the solution to the corresponding kinetic equation, and it necessarily converges toward the entropy admissible solution. We also provide numerical examples. Citation: Journal of Hyperbolic Differential Equations PubDate: 2018-03-28T09:06:54Z DOI: 10.1142/S0219891618500042
Authors:Paolo Antonelli, Stefano Spirito Pages: 133 - 147 Abstract: Journal of Hyperbolic Differential Equations, Volume 15, Issue 01, Page 133-147, March 2018. We consider the quantum Navier–Stokes (QNS) system in three space dimensions. We prove compactness of finite energy weak solutions for large initial data. The main novelties are that vacuum regions are included in the weak formulation and no extra terms, like damping or cold pressure, are considered in the equations in order to define the velocity field. Our argument uses an equivalent formulation of the system in terms of an effective velocity, in order to eliminate the third-order terms in the new system. This will allow to obtain the same compactness properties as for the Navier–Stokes equations with degenerate viscosity. Citation: Journal of Hyperbolic Differential Equations PubDate: 2018-03-28T09:06:50Z DOI: 10.1142/S0219891618500054
Authors:Yoshihiro Ueda, Renjun Duan, Shuichi Kawashima Pages: 149 - 174 Abstract: Journal of Hyperbolic Differential Equations, Volume 15, Issue 01, Page 149-174, March 2018. This paper is concerned with the weak dissipative structure for linear symmetric hyperbolic systems with relaxation. The authors of this paper had already analyzed the new dissipative structure called the regularity-loss type in [Y. Ueda, R. Duan and S. Kawashima, Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal. 205 (2012) 239–266]. Compared with the dissipative structure of the standard type in [T. Umeda, S. Kawashima and Y. Shizuta, On the devay of solutions to the linearized equations of electro-magneto-fluid dynamics, Japan J. Appl. Math. 1 (1984) 435–457; Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985) 249–275], the regularity-loss type possesses a weaker structure in the high-frequency region in the Fourier space. Furthermore, there are some physical models which have more complicated structure, which we discussed in [Y. Ueda, R. Duan and S. Kawashima, Decay structure of two hyperbolic relaxation models with regularity loss, Kyoto J. Math. 57(2) (2017) 235–292]. Under this situation, we introduce new concepts and extend our previous results developed in [Y. Ueda, R. Duan and S. Kawashima, Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal. 205 (2012) 239–266] to cover those complicated models. Citation: Journal of Hyperbolic Differential Equations PubDate: 2018-03-28T09:06:56Z DOI: 10.1142/S0219891618500066
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