arXiv subjects
Publications and source records attributed to Manwai Yuen.
In this article, we study the blowup phenomena of compressible Euler equations with non-vacuum initial data. Our new results, which cover a general class of testing functions, present new initial value blowup conditions. The corresponding blowup results of the 1-dimensional case in non-radial symmetry are also included.
In this paper, we study the blowup phenomena for the regular solutions of the isentropic relativistic Euler-Poisson equations with a vacuum state in spherical symmetry. Using a general family of testing functions, we obtain new blowup conditions for the relativistic Euler-Poisson equations. We also show that the proposed blowup conditions are valid regardless of the speed requirement, which was one of the key constraints stated in "Y. Geng, Singularity Formation for Relativistic Euler and Euler-Poisson Equations with Repulsive Force, Commun. Pure Appl. Anal., 14 (2015), 549--564.".
In this paper, we present rotational and self-similar solutions for the compressible Euler equations in R^3 using the separation method. These solutions partly complement Yuen's irrotational and elliptic solutions in R^3 [Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 4524-4528] as well as rotational and radial solutions in R^2 [Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 2172-2180]. A newly deduced Emden dynamical system is obtained. Some blowup phenomena and global existences of the responding solutions can be determined. The 3D rotational solutions provide concrete reference examples for vortices in computational fluid dynamics.
Compressible Euler-Poisson equations are the standard self-gravitating models for stellar dynamics in classical astrophysics. In this article, we construct periodic solutions to the isothermal ($\gamma=1$) Euler-Poisson equations in $R^{2}$ with possible applications to the formation of plate, spiral galaxies and the evolution of gas-rich, disk-like galaxies. The results complement Yuen's solutions without rotation (M.W. Yuen, Analytical Blowup Solutions to the 2-dimensional Isothermal Euler-Poisson Equations of Gaseous Stars, J. Math. Anal. Appl. 341(2008), 445--456.). Here, the periodic rotation prevents the blowup phenomena that occur in solutions without rotation. Based on our results, the corresponding $3$D rotational results for Goldreich and Weber's solutions are conjectured.
Pressureless Euler-Poisson equations with attractive forces are standard models in Newtonian cosmology. In this article, we further develop the spectral dynamics method and apply a novel spectral-dynamics-integration method to study the blowup conditions for $C^{2}$ solutions with a bounded domain, $\left\Vert X(t)\right\Vert \leq X_{0}$, where $\left\Vert\cdot\right\Vert $ denotes the volume and $X_{0}$ is a positive constant. In particular, we show that if the cosmological constant $\Lambda<M/X_{0}$, with the total mass $M$, then the non-trivial $C^{2}$ solutions in $R^{N}$ with the irrotational initial condition blow up at a finite time.
For the 2D and 3D Euler equations, their existing exact solutions are often in linear form with respect to variables x,y,z. In this paper, the Clarkson-Kruskal reduction method is applied to reduce the 2D incompressible Euler equations to a system of completely solvable ordinary equations, from which several novel nonlinear exact solutions with respect to the variables x and y are found.
This paper presents the vortical and self-similar solutions for 2D compressible Euler equations using the separation method. These solutions complement Makino's solutions in radial symmetry without rotation. The rotational solutions provide new information that furthers our understanding of ocean vortices and reference examples for numerical methods. In addition, the corresponding blowup, time-periodic or global existence conditions are classified through an analysis of the new Emden equation. A conjecture regarding rotational solutions in 3D is also made.
In this paper, we construct a new class of blowup solutions with elementary functions to the 3-dimensional compressible or incompressible Euler and Navier-Stokes equations. In detail, we obtain a class of global rotational exact solutions for the compressible fluids with $\gamma>1$:%} [c]{c}% \rho=\max\{\frac{\gamma-1}{K\gamma}[ C^{2}[ x^{2}% +y^{2}+z^{2}-(xy+yz+xz)] -\dot{a}(t)(x+y+z)+b(t)], 0\} ^{\frac{1}{\gamma-1}} u_{1}=a(t)+C(y-z) u_{2}=a(t)+C(-x+z) u_{3}=a(t)+C(x-y). where a(t)=c_{0}+c_{1}t and b(t)=3c_{0}c_{1}t+{3/2}c_{1}^{2}t^{2}+c_{2}% with $C$, $c_{0}$, $c_{1}$ and $c_{2}$ are arbitrary constants; And the corresponding blowup or global solutions for the incompressible Euler equations are also given. Our constructed solutions are similar to the famous Arnold-Beltrami-Childress (ABC) flow. The solutions with infinite energy can exhibit the interesting behaviors locally. Besides, the corresponding global solutions are also given for the compressible Euler equations. Furthermore, due to $\operatorname{div}\vec{u}=0$ for the solutions, the solutions also work for the 3-dimnsional incompressible Euler and Navier-Stokes equations.
In this paper, we investigate the analytical solutions of the compressible Navier-Stokes equations with dependent-density viscosity. By using the characteristic method, we successfully obtain a class of drifting solutions with elliptic symmetry for the Navier-Stokes model wherein the velocity components are governed by a generalized Emden dynamical system. In particular, when the viscosity variables are taken the same as Yuen in [Yuen M.W. (2008), Analytical Solutions to the Navier-Stokes Equations, J. Math. Phys. 49, 113102], our solutions constitute a generalization of that obtained by Yuen.
Based on Makino's solutions with radially symmetry, we extend the corresponding ones with elliptic symmetry for the compressible Euler and Navier-Stokes equations in R^{N} (N\geq2). By the separation method, we reduce the Euler and Navier-Stokes equations into 1+N differential functional equations. In detail, the velocity is constructed by the novel Emden dynamical system: { | a_{i}(t)=({\xi}/(a_{i}(t)({\Pi}a_{k}(t))^{{\gamma}-1})), for i=1,2,....,N a_{i}(0)=a_{i0}>0, a_{i}(0)=a_{i1} with arbitrary constants {\xi}, a_{i0} and a_{i1}. Some blowup phenomena or global existences of the solutions obtained could be shown.
The Newtonian Euler-Poisson equations with attractive forces are the classical models for the evolution of gaseous stars and galaxies in astrophysics. In this paper, we use the integration method to study the blowup problem of the $N$-dimensional system with adiabatic exponent $\gamma>1$, in radial symmetry. We could show that the $C^{1}$ non-trivial classical solutions $(\rho,V)$, with compact support in $[0,R]$, where $R>0$ is a positive constant with $\rho(t,r)=0$ and $V(t,r)=0$ for $r\geq R$, under the initial condition \begin{equation} H_{0}=\int_{0}^{R}r^{n}V_{0}dr>\sqrt{\frac{2R^{2n-N+4}M}{n(n+1)(n-N+2)}}% \end{equation} with an arbitrary constant $n>\max(N-2,0),$\newline blow up before a finite time $T$ for pressureless fluids or $\gamma>1.$ Our results could fill some gaps about the blowup phenomena to the classical $C^{1}$ solutions of that attractive system with pressure under the first boundary condition.\newline In addition, the corresponding result for the repulsive systems is also provided. Here our result fully covers the previous case for $n=1$ in "M.W. Yuen, \textit{Blowup for the Euler and Euler-Poisson Equations with Repulsive Forces}, Nonlinear Analysis Series A: Theory, Methods & Applications 74 (2011), 1465--1470".
In this paper, we continue to study the blowup problem of the $N$-dimensional compressible Euler or Euler-Poisson equations with repulsive forces, in radial symmetry. In details, we extend the recent result of "M.W. Yuen, \textit{Blowup for the Euler and Euler-Poisson Equations with Repulsive Forces}, Nonlinear Analysis Series A: Theory, Methods & Applications \textbf{74} (2011), 1465--1470.". We could further apply the integration method to obtain the more general results which the non-trivial classical solutions $(\rho,V)$, with compact support in $[0,R]$, where $R>0$ is a positive constant with $\rho(t,r)=0$ and $V(t,r)=0$ for $r\geq R$, under the initial condition% \begin{equation} H_{0}=\int_{0}^{R}r^{n}V_{0}dr>0 \end{equation} where an arbitrary constant $n>0$, blow up on or before the finite time $T=2R^{n+2}/(n(n+1)H_{0})$ for pressureless fluids or $\gamma>1.$ The results obtained here fully cover the previous known case for $n=1$.
We study the construction of analytical non-radially solutions for the 1-dimensional compressible adiabatic Euler equations in this article. We could design the perturbational method to construct a new class of analytical solutions. In details, we perturb the linear velocity:% \begin{equation} u=c(t)x+b(t) \end{equation} and substitute it into the compressible Euler equations. By comparing the coefficients of the polynomial, we could deduce the corresponding functional differential system of $(c(t),b(t),\rho^{\gamma-1}(0,t)).$ Then by skillfully applying the Hubble's transformation: \begin{equation} c(t)=\frac{\dot{a}(t)}{a(t)}, \end{equation} the functional differential equations can be simplified to be the system of $(a(t),b(t),\rho^{\gamma-1}(0,t))$. After proving the existence of the corresponding ordinary differential equations, a new class of blowup or global solutions can be shown. Here, our results fully cover the previous known ones by choosing $b(t)=0$.
In this article, we study the perturbational method to construct the non-radially symmetric solutions of the compressible 2-component Camassa-Holm equations. In detail, we first combine the substitutional method and the separation method to construct a new class of analytical solutions for that system. In fact, we perturb the linear velocity: u=c(t)x+b(t), and substitute it into the system. Then, by comparing the coefficients of the polynomial, we can deduce the functional differential equations involving $(c(t),b(t),\rho^{2}(0,t)).$ Additionally, we could apply the Hubble's transformation c(t)={\dot{a}(3t)}/{a(3t)}, to simplify the ordinary differential system involving $(a(3t),b(t),\rho ^{2}(0,t))$. After proving the global or local existences of the corresponding dynamical system, a new class of analytical solutions is shown. And the corresponding solutions in radial symmetry are also given. To determine that the solutions exist globally or blow up, we just use the qualitative properties about the well-known Emden equation: {array} [c]{c} {d^{2}/{dt^{2}}}a(3t)= {\xi}{a^{1/3}(3t)}, a(0)=a_{0}>0 ,\dot{a}(0)=a_{1} {array} . Our solutions obtained by the perturbational method, fully cover the previous known results in "M.W. Yuen, \textit{Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations,}J. Math. Phys., \textbf{51} (2010) 093524, 14pp." by the separation method.
In this article, we study the self-similar solutions of the 2-component Degasperis-Procesi water system:% [c]{c}% \rho_{t}+k_{2}u\rho_{x}+(k_{1}+k_{2})\rho u_{x}=0 u_{t}-u_{xxt}+4uu_{x}-3u_{x}u_{xx}-uu_{xxx}+k_{3}\rho\rho_{x}=0. By the separation method, we can obtain a class of self-similar solutions,% [c]{c}% \rho(t,x)=\max(\frac{f(\eta)}{a(4t)^{(k_{1}+k_{2})/4}},\text{}0),\text{}u(t,x)=\frac{\overset{\cdot}{a}(4t)}{a(4t)}x \overset{\cdot\cdot}{a}(s)-\frac{\xi}{4a(s)^{\kappa}}=0,\text{}a(0)=a_{0}% \neq0,\text{}\overset{\cdot}{a}(0)=a_{1} f(\eta)=\frac{k_{3}}{\xi}\sqrt{-\frac{\xi}{k_{3}}\eta^{2}+(\frac{\xi}{k_{3}}\alpha) ^{2}}% where $\eta=\frac{x}{a(s)^{1/4}}$ with $s=4t;$ $\kappa=\frac{k_{1}}{2}% +k_{2}-1,$ $\alpha\geq0,$ $\xi<0$, $a_{0}$ and $a_{1}$ are constants. which the local or global behavior can be determined by the corresponding Emden equation. The results are very similar to the one obtained for the 2-component Camassa-Holm equations. Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems. With the characteristic line method, blowup phenomenon for $k_{3}\geq0$ is also studied.
In this article, we study the self-similar solutions of the 2-component Camassa-Holm equations% \begin{equation} \left\{ \begin{array} [c]{c}% \rho_{t}+u\rho_{x}+\rho u_{x}=0 m_{t}+2u_{x}m+um_{x}+\sigma\rho\rho_{x}=0 \end{array} \right. \end{equation} with \begin{equation} m=u-\alpha^{2}u_{xx}. \end{equation} By the separation method, we can obtain a class of blowup or global solutions for $\sigma=1$ or $-1$. In particular, for the integrable system with $\sigma=1$, we have the global solutions:% \begin{equation} \left\{ \begin{array} [c]{c}% \rho(t,x)=\left\{ \begin{array} [c]{c}% \frac{f\left( \eta\right) }{a(3t)^{1/3}},\text{ for }\eta^{2}<\frac {\alpha^{2}}{\xi} 0,\text{ for }\eta^{2}\geq\frac{\alpha^{2}}{\xi}% \end{array} \right. ,u(t,x)=\frac{\overset{\cdot}{a}(3t)}{a(3t)}x \overset{\cdot\cdot}{a}(s)-\frac{\xi}{3a(s)^{1/3}}=0,\text{ }a(0)=a_{0}% >0,\text{ }\overset{\cdot}{a}(0)=a_{1} f(\eta)=\xi\sqrt{-\frac{1}{\xi}\eta^{2}+\left( \frac{\alpha}{\xi}\right) ^{2}}% \end{array} \right. \end{equation} where $\eta=\frac{x}{a(s)^{1/3}}$ with $s=3t;$ $\xi>0$ and $\alpha\geq0$ are arbitrary constants.\newline Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems.
In this paper, we study the Euler and Euler-Poisson equations in $R^{N}$, with multiple $\gamma$-law for pressure function: \begin{equation} P(\rho)=e^{s}\sum_{j=1}^{m}\rho^{\gamma_{j}}, \end{equation} where all $\gamma_{i+1}>\gamma_{i}\geq1$, is the constants. The analytical line solutions are constructed for the systems. It is novel to discover the analytical solutions to handle the systems with mixed pressure function. And our solutions can be extended to the systems with the generalized multiple damping and pressure function.
This article extends the previous paper in "M.W. Yuen, \textit{Stabilities for Euler-Poisson Equations in Some Special Dimensions}, J. Math. Anal. Appl. \textbf{344} (2008), no. 1, 145--156.", from the Euler-Poisson equations for attractive forces to the repulsive ones in $R^{N}$ $(N\geq2)$. The similar stabilities of the system are studied. Additionally, we explain that it is impossible to have the density collapsing solutions with compact support to the system with repulsive forces for $γ>1$.