The problem we are concerned with is whether singularities form in finite time in incompressible fluid flows. It is well known that the answer is ``no'' in the case of Euler and Navier-Stokes equations in dimension two. In dimension three it is still an open problem for these equations.
In this paper we focus on a two-dimensional active scalar model for the 3D Euler vorticity equation. Constantin, Majda and Tabak suggested, by studying rigorous theorems and detailed numerical experiments, a general principle: ``If the level set topology in the temperature field for the 2D quasi-geostrophic active scalar in the region of strong scalar gradients does not contain a hyperbolic saddle, then no finite time singularity is possible.''
Numerical simulations showed evidence of singular behavior when the geometry of the level sets of the active scalar contain a hyperbolic saddle. There is a naturally associated notion of simple hyperbolic saddle breakdown. The main theorem we present in this paper shows that such breakdown cannot occur in finite time. We also show that the angle of the saddle cannot close in finite time and it cannot be faster than a double exponential in time. Using the same techniques, we see that analogous results hold for incompressible 2D and 3D Euler.
In this paper, we study the super-critical Quasi-Geostrophic equation in Gevrey-Sobolev space. We prove the local existence of $(QG)$ for any large initial data and we give an exponential type of Blow-up to the solution. Moreover, we establish the existence global for a small initial data and we show that $\|θ\|_{H^s_{a, α^{-1}}}$ decays to zero as time goes to infinity. Fourier analysis and standard techniques are used.
In the present paper, we investigate the blow-up dynamics for local solutions to the semilinear generalized Tricomi equation with combined nonlinearity. As a result, we enlarge the blow-up region in comparison to the ones for the corresponding semilinear models with either power nonlinearity or nonlinearity of derivative type. Our approach is based on an iteration argument to establish lower bound estimates for the space average of local solutions. Finally, we obtain upper bound estimates for the lifespan of local solutions as byproduct of our iteration argument.
We consider the energy critical semilinear heat equation $$ \left\{\begin{aligned} &\partial_t u-Δu =|u|^{\frac{4}{n-2}}u &\mbox{in } {\mathbb R}^n\times(0,T),\\ &u(x,0)=u_0(x), \end{aligned}\right. $$ where $ n\geq 3$, $u_0\in L^\infty({\mathbb R}^n)$, and $T\in {\mathbb R}^+$ is the first blow up time. We prove that if $ n \geq 7$ and $ u_0 \geq 0$, then any blowup must be of Type I, i.e., \[\|u(\cdot, t)\|_{L^\infty({\mathbb R}^n)}\leq C(T-t)^{-\frac{1}{p-1}}.\] A similar result holds for bounded convex domains. The proof relies on a reverse inner-outer gluing mechanism and delicate analysi…
Nonexistence of blow-up solutions with smooth radiation for energy-critical equivariant wave maps
Institut de Mathématiques de Jussieu, Sorbonne Université, Université Paris Cité（巴黎数学研究所，索邦大学，巴黎西岱大学）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
We address a general system of nonlinear Dirac equations in (1+1) dimensions and prove nonexistence of classical self-similar blowup solutions in the space of bounded functions. While this argument does not exclude the possibility of finite-time blowup, it still suggests that smooth solutions to the nonlinear Dirac equations in (1+1) dimensions do not develop self-similar singularities in a finite time. In the particular case of the cubic Dirac equations, we characterize (unbounded) self-similar solutions in the closed analytical form.
This paper studies of a variation of the hyperbolic blow up scenario suggested by Hou and Luo's recent numerical simulation [12]. In particular, we propose a "hyperbolic" surface quasi-geostrophic equation characterized by a incompressible velocity field with a modified Biot-Savart law. For this model, we will show Finite time blow up for a wide class of initial data.
The one-dimensional quasi-geostrophic equation is the one-dimensional Fourier-space analogue of the famous Navier-Stokes equations. In their work Li and Sinai have proposed a renormalization approach to the problem of existence of finite-time blow up solutions of this equation. In this setting, existence of finite time blow ups is a consequence of existence of a fixed point for a certain renormalization operator on an appropriate functional space. They have provided a proof of existence of complex-valued finite time blow up solutions of the quasi-geostrophic equation. In this paper we revisit …
We introduce a determining wavenumber for the surface quasi-geostrophic (SQG) equation defined for each individual trajectory and then study its dependence on the force. While in the subcritical and critical cases this wavenumber has a uniform upper bound, it may blow up when the equation is supercritical. A bound on the determining wavenumber provides determining modes, which in some sense measure the number of degrees of freedom of the flow, or resolution needed to describe a solution to the SQG equation.
In this paper we consider the blow-up of solutions to a weakly coupled system of semilinear damped wave equations in the scattering case with nonlinearities of mixed type, namely, in one equation a power nonlinearity and in the other a semilinear term of derivative type. The proof of the blow-up results is based on an iteration argument. As expected, due to the assumptions on the coefficients of the damping terms, we find as critical curve in the p-q plane for the pair of exponents (p,q) in the nonlinear terms the same one found by Hidano-Yokoyama and, recently, by Ikeda-Sobajima-Wakasa for th…
We exclude Type I blow-up, which occurs in the form of atomic concentrations of the $L^2$ norm for the solution of the 3D incompressible Euler equations. As a corollary we prove nonexistence of discretely self-similar blow-up in the energy conserving scale.
In this paper we consider the blow-up for solutions to a weakly coupled system of semilinear damped wave equations of derivative type in the scattering case. After introducing suitable functionals proposed by Lai-Takamura for the corresponding single semilinear equation, we employ Kato's lemma to derive the blow-up result in the subcritical case. On the other hand, in the critical case an iteration procedure based on the slicing method is employed. Let us point out that we find as critical curve in the p-q plane for the pair of exponents (p, q) in the nonlinear terms the same one as for the we…
In this article, we consider the damped wave equation in the \textit{scale-invariant case} with time-dependent speed of propagation, mass term and time derivative nonlinearity. More precisely, we study the blow-up of the solutions to the following equation: $$ (E) \quad u_{tt}-t^{2m}Δu+\fracμ{t}u_t+\frac{ν^2}{t^2}u=|u_t|^p, \quad \mbox{in}\ \mathbb{R}^N\times[1,\infty), $$ that we associate with small initial data. Assuming some assumptions on the mass and damping coefficients, $ν$ and $μ>0$, respectively, that the blow-up region and the lifespan bound of the solution of $(E)$ remain the same …
This work involves theoretical and numerical analysis of the Thermal Quasi-Geostrophic (TQG) model of submesoscale geophysical fluid dynamics (GFD). Physically, the TQG model involves thermal geostrophic balance, in which the Rossby number, the Froude number and the stratification parameter are all of the same asymptotic order. The main analytical contribution of this paper is to construct local-in-time unique strong solutions for the TQG model. For this, we show that solutions of its regularized version $α$-TQG converge to solutions of TQG as its smoothing parameter $α\rightarrow 0$ and we ob…
In our previous two works, we studied the blow-up and lifespan estimates for damped wave equations with a power nonlinearity of the solution or its derivative, with scattering damping independently. In this work, we are devoted to establishing a similar result for a combined nonlinearity. Comparing to the result of wave equation without damping, one can say that the scattering damping has no influence.
The present article is devoted to the 3D dissipative quasi-geostrophic system (QG). This system can be obtained as limit model of the Primitive Equations in the asymptotics of strong rotation and stratification, and involves a non-radial, non-local, homogeneous pseudo-differential operator of order 2 denoted by $Γ$ (and whose semigroup kernel reaches negative values). After a refined study of the non-local part of $Γ$, we prove apriori estimates (in the general L^p setting) for the 3D QG-model. The main difficulty of this article is to study the commutator of $Γ$ with a Lagrangian change of va…
We consider large time behavior of solutions to the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. We treat the case in which the nonlinearity contains non-oscillating factor $|u|^{1+2/d}$. The case is excluded in our previous studies. It turns out that there are no solutions that behave like a free solution with or without logarithmic phase corrections. We also prove nonexistence of an asymptotic free solution in the case that the gauge invariant nonlinearity is dominant, and give a finite time blow-up result.
We study fractional parabolic equations with indefinite nonlinearities $$ \frac{\partial u} {\partial t}(x,t) +(-Δ)^s u(x,t)= x_1 u^p(x, t),\,\, (x, t) \in \mathbb{R}^n \times \mathbb{R}, $$ where $0<s<1$ and $1<p<\infty$. We first prove that all positive bounded solutions are monotone increasing along the $x_1$ direction. Based on this we derive a contradiction and hence obtain non-existence of solutions. These monotonicity and nonexistence results are crucial tools in a priori estimates and complete blow-up for fractional parabolic equations in bounded domains. To this end, we introduce seve…
In this paper, we establish blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime with nonlinearity of derivative type. Our approach is based on the integral representation formula for the solution to the corresponding linear problem in the one-dimensional case, that we will determine through Yagdjian's Integral Transform approach. As upper bound for the exponent of the nonlinear term, we discover a Glassey-type exponent which depends both on the space dimension and on the Lorentzian metric in the generalized Einstein-de Sitter spacetime.
This article is to study the nonexistence of global solutions to semi-linear structurally damped wave equation with nonlinear memory in $\R^n$ for any space dimensions $n\ge 1$ and for the initial arbitrarily small data being subject to the positivity assumption. We intend to apply the method of a modified test function to establish blow-up results and to overcome some difficulties as well caused by the well-known fractional Laplacian $(-Δ)^{σ/2}$ in structural damping terms.
We study patch solutions of a family of transport equations given by a parameter $α$, $0< α<2$, with the cases $α=0$ and $α=1$ corresponding to the Euler and the surface quasi-geostrophic equations respectively. In this paper, using several new cancellations, we provide the following new results. First, we prove local well-posedness for $H^{2}$ patches in the half-space setting for $0<α< 1/3$, allowing self-intersection with the fixed boundary. Furthermore, we are able to extend the range of $α$ for which finite time singularities have been shown in \cite{KYZ} and \cite{KRYZ}. Second, we estab…
We construct examples of solutions to the conservative surface quasi-geostrophic (SQG) equation that must either exhibit infinite in time growth of derivatives or blow up in finite time.
In the present paper, we study an inhomogeneous pseudo-parabolic equation with nonlocal nonlinearity $$u_t-kΔu_t-Δu=I^γ_{0+}(|u|^{p})+ω(x),\,\ (t,x)\in(0,\infty)\times\mathbb{R}^N,$$ where $p>1,\,k\geq 0$, $ω(x)\neq0$ and $I^γ_{0+}$ is the left Riemann-Liouville fractional integral of order $γ\in(0,1).$ Based on the test function method, we have proved the blow-up result for the critical case $γ=0,\,p=p_c$ for $N\geq3$, which answers an {\bf open question} posed in \cite{Zhou}, and in particular when $k=0$ it improves the result obtained in \cite{Bandle}. An interesting fact is that in the cas…
In this work, the Cauchy problem for the semilinear Moore-Gibson-Thompson (MGT) equation with power nonlinearity $|u|^p$ on the right-hand side is studied. Applying $L^2-L^2$ estimates and a fixed point theorem, we obtain local (in time) existence of solutions to the semilinear MGT equation. Then, the blow-up of local in time solutions is proved by using an iteration method, under certain sign assumption for initial data, and providing that the exponent of the power of the nonlinearity fulfills $1 < p \leqslant p_{\mathrm{Str}}(n)$ for $n \geqslant2$ and $p>1$ for $n=1$. Here the Strauss expon…
We investigate existence and nonexistence of global in time nonnegative solutions to the semilinear heat equation, with a reaction term of the type $e^{μt}u^p$ ($μ\in\mathbb{R}, p>1$), posed on cones of the hyperbolic space. Under a certain assumption on $μ$ and $p$, related to the bottom of the spectrum of $-Δ$ in $\mathbb H^n$, we prove that any solution blows up in finite time, for any nontrivial nonnegative initial datum. Instead, if the parameters $μ$ and $p$ satisfy the opposite condition we have: (a) blow-up when the initial datum is large enough, (b) existence of global solutions when …
We prove existence and nonexistence results concerning elliptic problems whose basic model is \begin{equation*} \begin{cases} \displaystyle-Δu+μ(x)\frac{|\nabla u|^2}{(u+δ)^γ}= λu^p, &x\in Ω, \\ u> 0, &x\in Ω, \\ u=0, &x\in\partialΩ, \end{cases} \end{equation*} where $Ω\subset\mathbb{R}^N (N\geq 3)$ is a bounded smooth domain, $λ>0$, $p>1$, $δ\geq 0$, $γ>0$ and $μ\in L^\infty(Ω)$. The main achievement resides in handling a possibly singular ($δ=0$) first order term having a nonconstant coefficient $μ$ in the presence of a superlinear zero order term. Our approach for the existence results is b…
