Manwai Yuen

Analytical blowup solutions to the 2-dimensional isothermal Euler-Poisson equations of gaseous stars II

In this article, we construct analytical blowup solutions with non-radial symmetry for the 2-dimensional Euler-Poisson equations. Based on the previous solutions with radial symmetry for the 2-dimensional isothermal Euler-Poisson equations, some special blowup solutions with non-radial symmetry are constructed by the separation method.

Self-Similar Solutions with Elliptic Symmetry for the Compressible Euler and Navier-Stokes Equations in R N

Abstract 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 ⩾ 2 ). 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: (1) a ¨ i ( t ) = ξ a i ( t ) ∏ N a k ( t ) γ - 1 , for i = 1 , 2 , … , N a i ( 0 ) = a i 0 > 0 , a ˙ i ( 0 ) = a i 1 with arbitrary constants ξ , a i 0 and a i 1 . Some blowup phenomena or global existences of the solutions obtained can be shown. Computing simulation or rigorous mathematical proofs for the Emden dynamical system (1) , are expected to be followed in the future research.

Self-similar blowup solutions to the 2-component Camassa–Holm equations

In this article, we study the self-similar solutions of the 2-component Camassa–Holm equations ρt+uρx+ρux=0, mt+2uxm+umx+σρρx=0, with m=u−α2uxx. By the separation method, we can obtain a class of blowup or global solutions for σ=1 or −1. In particular, for the integrable system with σ=1, we have the global solutions, ρ(t,x)=f(η)/a(3t)1/3 for η2 0,a(0)=a1, f(η)=ξ−1/ξη2+(α/ξ)2, where η=xa(s)1/3 with s=3t; ξ>0 and α≥0 are arbitrary constants. Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems.

Perturbational blowup solutions to the compressible 1-dimensional Euler equations

Abstract We construct non-radially symmetry solutions for the compressible 1-dimensional adiabatic Euler equations in this Letter. In detail, we perturb the linear velocity with a drifting term: (1) u = c ( t ) x + b ( t ) , to seek new solutions. Then, we transform the problem into the analysis of ordinary differential equations. By investigating the corresponding ordinary differential equations, a new class of blowup or global solutions can be given. Here, our constructed solutions can provide the mathematical explanations for the drifting phenomena of some propagation wave like Tsunamis. And when we adopt the Galilean-like transformation to a drifting frame, the constructed solutions are self-similar.

Analytical blowup solutions to the pressureless Navier–Stokes–Poisson equations with density-dependent viscosity in RN

We study the pressureless Navier–Stokes–Poisson equations with density-dependent viscosity. With the extension of the blowup solutions for the Euler–Poisson equations, the analytical blowup solutions, in radial symmetry, in RN (N ≥ 2) are constructed.

Exact, rotational, infinite energy, blowup solutions to the 3-dimensional Euler equations

Abstract In this Letter, we construct a new class of blowup or global solutions with elementary functions to the 3-dimensional compressible or incompressible Euler and Navier–Stokes equations. And the corresponding blowup or global solutions for the incompressible Euler and Naiver–Stokes equations are also given. Our constructed solutions are similar to the famous Arnold–Beltrami–Childress (ABC) flow. The obtained solutions with infinite energy can exhibit the interesting behaviors locally. Furthermore, due to div u → = 0 for the solutions, the solutions also work for the 3-dimensional incompressible Euler and Navier–Stokes equations.

Blowup for the C1 solutions of the Euler–Poisson equations of gaseous stars in RN

Abstract 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 γ > 1 , in radial symmetry. We could show that the C 1 non-trivial classical solutions ( ρ , V ) , with compact support in [ 0 , R ] , where R > 0 is a positive constant with ρ ( t , r ) = 0 and V ( t , r ) = 0 for r ⩾ R , under the initial condition (1) H 0 = ∫ 0 R r n V 0 d r > 2 R 2 n − N + 4 M n ( n + 1 ) ( n − N + 2 ) with an arbitrary constant n > max ( N − 2 , 0 ) and the total mass M, blow up before a finite time T for pressureless fluids or γ > 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. 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, Blowup for the Euler and Euler–Poisson equations with repulsive forces, Nonlinear Anal. 74 (2011) 1465–1470].

Analytically periodic solutions to the three-dimensional Euler–Poisson equations of gaseous stars with a negative cosmological constant

By extension of the three-dimensional analytical solutions of Goldreich and Weber (1980 Homologously collapsing stellar cores Astrophys. J. 238 991) with an adiabatic exponent γ = 4/3, to the (classical) Euler–Poisson equations without a cosmological constant, the self-similar (almost re-collapsing) time-periodic solutions with a negative cosmological constant (Λ < 0) are constructed. The solutions with time periodicity are novel. Based on these solutions, the time-periodic and almost re-collapsing model is conjectured in this paper, for some gaseous stars.

Perturbational blowup solutions to the 2-component Camassa–Holm equations

Abstract 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: (1) 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 ) , ρ 2 ( 0 , t ) ) . Additionally, we could apply Hubbleʼs transformation c ( t ) = a ˙ ( 3 t ) a ( 3 t ) , to simplify the ordinary differential system involving ( a ( 3 t ) , b ( t ) , ρ 2 ( 0 , t ) ) . After proving the global or local existences of the corresponding dynamical system, a new class of analytical solutions is shown. To determine that the solutions exist globally or blow up, we just use the qualitative properties about the well-known Emden equation. Our solutions obtained by the perturbational method, fully cover Yuenʼs solutions by the separation method.

Analytical solutions to the Navier-Stokes-Poisson equations with density-dependent viscosity and with pressure

We study some particular solutions to the Navier-Stokes-Poisson equations with densitydependent viscosity and with pressure, in radial symmetry. With extension of the previous known blowup solutions for the Euler-Poisson equations / pressureless Navier-Stokes-Poisson with density-dependent viscosity, we constructed the corresponding analytical blowup solutions for the Navier-Stokes-Poisson Equations with density-dependent viscosity and with pressure.