Related papers: Cauchy Problem for Cylinder-like Capillary Jets
When a water jet impinges upon a solid surface it produces a so called hydraulic jump that everyone can observe in the sink of its kitchen. It is characterized by a thin liquid sheet bounded by a circular rise of the surface due to…
This work focuses on the mathematical analysis of the Cauchy problem associated with a two-dimensional equation describing the dynamics of a thin fluid film flowing down an inclined flat plate under the influence of gravity and an electric…
We prove that the Cauchy problem associated to the Zakharov-Schulman system $iu_t+L_1u=uv$, $L_2v=L_3(|u|^2)$ is locally well-posed for given initial data in Sobolev spaces $H^s(R^n)$, $s\geq n/4$, for n =2,3. Here, L_j denote second order…
We consider the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) equation \begin{align*} \partial_{x}\left(u_{t}-\beta\partial_{x}^{3}u +\partial_{x}(u^{2})\right)+\partial_{y}^{2}u-\gamma u=0 \end{align*} in the…
Experiments and global linear stability analysis are used to obtain the critical flow rate below which the highly stretched capillary jet generated when a Newtonian liquid issues from a vertically oriented tube, is no longer steady. The…
In this paper, we consider the Cauchy problem for the fifth-order KP-I equation \begin{align*} u_t + \partial_x^5u+\partial_x^{-1}\partial_y^2u + \frac{1}{2}\partial_x(u^2)=0. \end{align*} Firstly, we establish the local well-posedness of…
We introduce a new family of p-adic non-linear evolution equations. We establish the local well-posedness of the Cauchy problem for these equations in Sobolev-type spaces. For a certain subfamily, we show that the blow-up phenomenon occurs…
We consider the free boundary problem for a 3-dimensional, incompressible, irrotational liquid drop of nearly spherical shape with capillarity. We study the problem from the beginning, extending some classical results from the flat case…
Capillary jetting of a fluid dispersed into another immiscible phase is usually limited by a critical Capillary number, a function of the Reynolds number and the fluid properties ratios. Critical conditions are set when the minimum…
We prove that the Cauchy problem for the two-dimensional Zakharov system is locally well-posed for initial data which are localized perturbations of a line solitary wave. Furthermore, for this Zakharov system, we prove weak convergence to a…
The Cauchy problem of the Cahn-Hilliard equations is studied in three-dimensional space. Firstly, we construct its approximate fourth-order parabolic equation, obtaining the existence of solutions by the Aubin-Lions's compactness lemma.…
New general results of non-existence and rigidity of spacelike submanifolds immersed in a spacetime, whose mean curvature is a time-oriented causal vector field, are given. These results hold for a wide class of spacetimes which includes…
This paper investigates the nonlinear Schr\"{o}dinger equation with a singular convolution potential. It demonstrates the local well-posedness of this equation in a modified Sobolev space linked to the energy. Additionally, we derive…
In the paper, we consider the Cauchy problem for a fifth order pseudoparabolic equation that appears in studying the issues of fluid filtration in fissured media, the moisture transfer in soils and etc. The Cauchy problem with non-classic…
We study the Cauchy problem for a generalized derivative nonlinear Schr\"odinger equation with the Dirichlet boundary condition. We establish the local well-posedness results in the Sobolev spaces $H^1$ and $H^2$. Solutions are constructed…
The uniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard \cite{Ha} these kind of problems are known to be ill-posed and even severely ill-posed. Until now there are only few partial results concerning…
Physical experiments show that a capillary water jet is exponentially unstable under long wave perturbations, while remaining stable under short wave perturbations. Measurements indicate that the exponential growth rate in the long wave…
In this paper, we systematically investigate the wetting behavior of a liquid ring in a cylindrical capillary tube. We obtain analytical solutions of the axisymmetric Young-Laplace equation for arbitrary contact angles. We find that, for…
We consider the Cacuhy problem for a viscous compressible rotating shallow water system with a third-order surface-tension term involved, derived recently in the modelling of motions for shallow water with free surface in a rotating…
We study singular jets from the collapse of drop-impact craters, when the drop and pool are of different immiscible liquids. These jets emerge from a dimple at the bottom of the rebounding crater, when no bubble is pinched off. The…