Related papers: Smooth solutions for a ${p}$-system of mixed type
The stability of stationary solutions of first-order systems of PDE's are considered. They may include some singular geometric terms, leading to discontinuous flux and non-conservative products. Based on several examples in Fluid Mechanics,…
In this article we address some issues related to the initial value problems for a rotating shallow water hyperbolic system of equations and the diffusive regularization of this system. For initial data close to the solution at rest, we…
The stability of the system is an important part of the research on differential dynamical systems. This paper considers a pointwise hyperbolic system defined on a connected open subset N of a compact smooth Riemannian manifold M. The…
This article examines the smoothness of the solution to the Navier-Stokes equation from a novel perspective. Here, the existence of the smoother solution relative to x and to the time t was shown only for a finite time. Moreover, for each…
We show that any polyhomogeneous asymptotically hyperbolic constant-mean-curvature solution to the vacuum Einstein constraint equations can be approximated, arbitrarily closely in H\"older norms determined by the physical metric, by…
We establish the existence of smooth, finite-energy solutions to the 2D incompressible porous media equation (IPM), with a compactly supported uniformly smooth source, which develop singularities in finite time.
Under the genuinely nonlinear assumption for 1-D $n\times n$ strictly hyperbolic conservation laws, we investigate the geometric blowup of smooth solutions and the development of singularities when the small initial data fulfill the generic…
The idea of using fast sweeping methods for solving stationary systems of conservation laws has previously been proposed for efficiently computing solutions with sharp shocks. We further develop these methods to allow for a more challenging…
We consider a characteristic initial value problem for a class of symmetric hyperbolic systems with initial data given on two smooth null intersecting characteristic surfaces. We prove existence of solutions on a future neighborhood of the…
We show that in one space dimension, a linearly degenerate hyperbolic system of rich type admits exact traveling wave solutions if the initial data are Riemann type outside of a space interval. In a particular case of the system including…
Here a mixed problem for a nonlinear hyperbolic equation with Neumann boundary value condition is investigated, and a priori estimations for the possible solutions of the considered problem are obtained. These results demonstrate that any…
In this paper, we will show the blow-up of smooth solutions to the Cauchy problem for the full compressible Navier-Stokes equations and isentropic compressible Navier-Stokes equations with constant and degenerate viscosities in arbitrary…
Compositional simulation is challenging, because of highly nonlinear couplings between multi-component flow in porous media with thermodynamic phase behavior. The coupled nonlinear system is commonly solved by the fully-implicit scheme.…
We consider equations of M\"uller-Israel-Stewart type describing a relativistic viscous fluid with bulk viscosity in four-dimensional Minkowski space. We show that there exists a class of smooth initial data that are localized perturbations…
In this paper we describe a method to derive solutions of the incompressible Navier- Stokes system of equations for non-stationary initial value problems in $\mathbb{R}^n$. We show that for a given smooth solenoidal initial velocity vector…
The existence of singular solutions of the incompressible Navier-Stokes system with singular external forces, the existence of regular solutions for more regular forces as well as the asymptotic stability of small solutions (including…
In this paper we prove that for a certain class of initial data, smooth solutions of the hydrostatic Euler equations blow up in finite time.
In a joint work with Palmer we have formulated sufficient conditions under which there exist continuous and invertible transformations of the form $H_n(x,y)$ taking solutions of a coupled system \begin{equation*} x_{n+1} =A_nx_n+f_n(x_n,…
For the first order 1D $n\times n$ quasilinear strictly hyperbolic system $\partial_tu+F(u)\partial_xu=0$ with $u(x, 0)=\varepsilon u_0(x)$, where $\varepsilon>0$ is small, $u_0(x)\not\equiv 0$ and $u_0(x)\in C_0^2(\mathbb R)$, when at…
The incompressible smoothed particle hydrodynamics method (ISPH) is a numerical method widely used for accurately and efficiently solving flow problems with free surface effects. However, to date there has been little mathematical…