N. Lai
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View article: Long-Time Existence of Quasilinear Wave Equations Exterior to Star-shaped Obstacle in $2\mathbf{D}$
Long-Time Existence of Quasilinear Wave Equations Exterior to Star-shaped Obstacle in $2\mathbf{D}$ Open
In this paper, we study the long-time existence result for small data solutions of quasilinear wave equations exterior to star-shaped regions in two space dimensions. The key novelty is that we establish a Morawetz type energy estimate for…
View article: Morawetz type estimate for damped wave equation in $\mathbb{R}^n (n\geq 4)$ and its application
Morawetz type estimate for damped wave equation in $\mathbb{R}^n (n\geq 4)$ and its application Open
In this paper we establish a Morawetz type etimate for the linear inhomogeneous wave equation with time-dependent scale invariant damping in $\mathbb{R}^n (n\geq 4)$. The novelty is that we view the differential operator $\Box+\fracμ{t}\pa…
View article: A blow-up result for the semilinear Euler-Poisson-Darboux-Tricomi equation with critical power nonlinearity
A blow-up result for the semilinear Euler-Poisson-Darboux-Tricomi equation with critical power nonlinearity Open
In this paper, we prove a blow-up result for a generalized semilinear Euler-Poisson-Darboux equation with polynomially growing speed of propagation, when the power of the semilinear term is a shift of the Strauss' exponent for the classica…
View article: Self-dual solution of 3D incompressible Navier-Stokes equations
Self-dual solution of 3D incompressible Navier-Stokes equations Open
Whether the 3D incompressible Navier-Stokes equations will have a global smooth solution for all smooth, finite energy initial data is a Millennium Prize problem. One of the main difficulties of this problem is that the Navier-Stokes equat…
View article: Sharp lifespan estimate for the compressible Euler system with critical time-dependent damping in $\R^2$
Sharp lifespan estimate for the compressible Euler system with critical time-dependent damping in $\R^2$ Open
This paper concerns the long time existence to the smooth solutions of the compressible Euler system with critical time dependent damping in $\R^2$. We establish the sharp lifespan estimate from below, with respect to the small parameter o…
View article: On the global well-posedness for the compressible Hall-MHD system
On the global well-posedness for the compressible Hall-MHD system Open
In this paper, we prove the global well-posedness of the Cauchy problem to the 3-D compressible Hall-magnetohydrodynamic (MHD) system supplemented with a class of large initial data in Besov spaces. By exploiting the intrinsic structure of…
View article: Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions Open
Compressible Euler equations with space-dependent damping in high dimensions R n ( n = 2 , 3 ) {{\bf{R}}}^{n}\hspace{0.33em}\hspace{0.33em}\left(n=2,3) are considered in this article. Assuming that the small initial velocity …
View article: Formation of Finite Time Singularity for Axially Symmetric Magnetohydrodynamic Waves in 3-D
Formation of Finite Time Singularity for Axially Symmetric Magnetohydrodynamic Waves in 3-D Open
In this paper we study the compressible magnetohydrodynamics equations in three dimensions, which offer a good model for plasmas. Formation of singularity for C1-solution in finite time is proved with axisymmetric initial data. The key obs…
View article: Blow-up for compressible Euler system with space-dependent damping in 1-D
Blow-up for compressible Euler system with space-dependent damping in 1-D Open
This article considers the Cauchy problem for compressible Euler system in R {\bf{R}} with damping, in which the coefficient depends on the space variable. Assuming the initial density has a small perturbation around a constant state and…
View article: Blow-up and lifespan estimate to a nonlinear wave equation in Schwarzschild spacetime
Blow-up and lifespan estimate to a nonlinear wave equation in Schwarzschild spacetime Open
We study the semilinear wave equation with power type nonlinearity and small initial data in Schwarzschild spacetime. If the nonlinear exponent $p$ satisfies $2\le p<1+\sqrt 2$, we establish the sharp upper bound of lifespan estimate, whil…
View article: Lifespan estimates for the compressible Euler equations with damping via Orlicz spaces techniques
Lifespan estimates for the compressible Euler equations with damping via Orlicz spaces techniques Open
In this paper we are interested in the upper bound of the lifespan estimate for the compressible Euler system with time dependent damping and small initial perturbations. We employ some techniques from the blow-up study of nonlinear wave e…
View article: Global existence of strong solution to non-isothermal ideal gas system
Global existence of strong solution to non-isothermal ideal gas system Open
This paper aims to establish the global existence of strong solutions to a non-isothermal ideal gas model. We first show global well-posedness in the Sobolev space $H^2(\mathbb{R}^3)$ by using energy estimates. We then prove the global wel…
View article: Lifespan estimates for semilinear wave equations with space dependent damping and potential
Lifespan estimates for semilinear wave equations with space dependent damping and potential Open
In this work, we investigate the influence of general damping and potential terms on the blow-up and lifespan estimates for energy solutions to power-type semilinear wave equations. The space-dependent damping and potential functions are a…
View article: Global existence for semilinear wave equations with scaling invariant damping in 3-D
Global existence for semilinear wave equations with scaling invariant damping in 3-D Open
Global existence for small data Cauchy problem of semilinear wave equations with scaling invariant damping in 3-D is established in this work, assuming that the data are radial and the constant in front of the damping belongs to $[1.5, 2)$…
View article: Short time blow-up by negative mass term for semilinear wave equations with small data and scattering damping
Short time blow-up by negative mass term for semilinear wave equations with small data and scattering damping Open
In this paper we study blow-up and lifespan estimate for solutions to the Cauchy problem with small data for semilinear wave equations with scattering damping and negative mass term. We show that the negative mass term will play a dominant…
View article: Positivity of temperature for some non-isothermal fluid models
Positivity of temperature for some non-isothermal fluid models Open
We establish three partial differential equation models describing the thermodynamics of the fluid, by combining the energetic variational approach, appropriate constitutive relations, and classical thermodynamics laws. What is more, by us…
View article: Lifespan estimates for $2$-dimensional semilinear wave equations in asymptotically Euclidean exterior domains
Lifespan estimates for $2$-dimensional semilinear wave equations in asymptotically Euclidean exterior domains Open
In this paper we study the initial boundary value problem for two-dimensional semilinear wave equations with small data, in asymptotically Euclidean exterior domains. We prove that if $1
View article: Nonexistence results for a higher-order evolution equation with an inhomogeneous term depending on time and space
Nonexistence results for a higher-order evolution equation with an inhomogeneous term depending on time and space Open
We consider a higher-order evolution equation with an inhomogeneous term depending on time and space. We first derive a general criterion for the nonexistence of weak solutions. Next, we study the particular case when the inhomogeneity dep…
View article: Strauss exponent for semilinear wave equations with scattering space dependent damping
Strauss exponent for semilinear wave equations with scattering space dependent damping Open
It is believed or conjectured that the semilinear wave equations with scattering space dependent damping admit the Strauss critical exponent, see Ikehata-Todorova-Yordanov \cite{ITY}(the bottom in page 2) and Nishihara-Sobajima-Wakasugi \c…
View article: Blow-up and lifespan estimate to a nonlinear wave equation in Schwarzschild spacetime
Blow-up and lifespan estimate to a nonlinear wave equation in Schwarzschild spacetime Open
In \cite{Luk}, Luk proved global existence for semilinear wave equations in Kerr spacetime with small angular momentum($a\ll M$) \[ \Box_{g_K}ϕ=F(\partial ϕ), \] when the quadratic nonlinear term satisfies the null condition. In this work,…
View article: Global Instability of the Multi-dimensional Plane Shocks for the isothermal flow
Global Instability of the Multi-dimensional Plane Shocks for the isothermal flow Open
In this paper, we are concerned with the long time behavior of the piecewise smooth solutions to the generalized Riemann problem governed by the compressible isothermal Euler equations in two and three dimensions. Non-existence result is e…
View article: Weighted $L^2-L^2$ estimate for wave equation and its applications
Weighted $L^2-L^2$ estimate for wave equation and its applications Open
In this work we establish a weighted $L^2-L^2$ estimate for inhomogeneous wave equation in 3-D, by introducing a Morawetz multiplier with weight of power $s(1
View article: Nonexistence of global solutions of wave equations with weak time-dependent damping and combined nonlinearity
Nonexistence of global solutions of wave equations with weak time-dependent damping and combined nonlinearity Open
In our previous two works, we studied the blow-up and lifespan estimates for\ndamped wave equations with a power nonlinearity of the solution or its\nderivative, with scattering damping independently. In this work, we are devoted\nto estab…
View article: Blow up for initial boundary value problem of critical semilinear wave equation in two space dimensions
Blow up for initial boundary value problem of critical semilinear wave equation in two space dimensions Open
This paper focuses on the initial boundary value problem of semilinear wave equation in exterior domain in two space dimensions with critical power. Based on the contradiction argument, we prove that the solution will blow up in a finite t…
View article: Nonexistence of global solutions of nonlinear wave equations with weak time-dependent damping related to Glassey conjecture
Nonexistence of global solutions of nonlinear wave equations with weak time-dependent damping related to Glassey conjecture Open
This work is devoted to the nonexistence of global-in-time energy solutions of nonlinear wave equation of derivative type with weak time-dependent damping in the scattering and scale invariant range. By introducing some multipliers to abso…