相关论文: Analytic solutions and Singularity formation for t…
We analyze a system of reacting elements harmonically coupled to nearest neighbors in the continuum limit. An analytic solution is found for traveling waves. The procedure is used to find oscillatory as well as solitary waves. A comparison…
In this paper, we define the rarefaction and compression characters for the supersonic expanding wave of the compressible Euler equations with radial symmetry. Under this new definition, we show that solutions with rarefaction initial data…
Given that the terminal condition is of at most linear growth, it is well known that a Cauchy problem admits a unique classical solution when the coefficient multiplying the second derivative (i.e., the volatility) is also a function of at…
In this paper, we develop the blow-up analysis and establish the energy quantization for solutions to super-Liouville type equations on Riemann surfaces with conical singularities at the boundary. In other problems in geometric analysis,…
We prove well-posedness of global solutions for a class of coagulation equations which exhibit the gelation phase transition. To this end, we solve an associated partial differential equation involving the generating functions before and…
We analyze the Vlasov equation coupled with the compressible Navier--Stokes equations with degenerate viscosities and vacuum. These two equations are coupled through the drag force which depends on the fluid density and the relative…
In this paper we prove the existence of a family of self-similar solutions for a class of coagulation equations with a constant flux of particles from the origin. These solutions are expected to describe the longtime asymptotics of…
We present a family of time-dependent solutions to 2+1 gravity with negative cosmological constant and a massless scalar field as source. These solutions are continuously self-similar near the central singularity. We analyze linear…
It is studied the Cauchy problem for the equations of Burgers' type but with bounded dissipation flux. Such equations degenerate to hyperbolic ones as the velocity gradient tends to infinity. Thus the discontinuous solutions are permitted.…
When a solution to the Cauchy problem for nonlinear dispersive equations is obtained by a fixed point argument using auxiliary function spaces, it is non-trivial to ensure uniqueness of solutions in a natural space such as the class of…
This paper studies global solvability of the Cauchy problem for a generalized time-fractional Kuramoto-Sivashinsky equation in the Shwartz space, which is a complete topological space generated by a family of semi-norms. The main approach…
We construct an explicit family of smooth finite-time blow-up solutions for the focusing Calogero--Sutherland derivative NLS given by $$ i \partial_t u = -\partial_x^2 u - 2 D \Pi(|u|^2) u \quad \mbox{with} \quad (t,x) \in \mathbb{R} \times…
We study the formation of singularities for the Euler-Alignment system with influence function $\psi=\frac{k_\alpha}{|x|^\alpha}$ in 1D. As in [20] the problem is reduced to the analysis of a nonlocal 1D equation. We show the existence of…
We show unconditional uniqueness of solutions to the Cauchy problem associated with the Benjamin-Ono equation under the periodic boundary condition with initial data given in $H^s$ for $s>1/6$. This improves the previous unconditional…
We show that the solution (in the sense of distribution) to the Cauchy problem with the periodic boundary condition associated with the modified Benjamin-Ono equation is unique in $L^\infty_t(H^s(\mathbb{T}))$ for $s>1/2$. The proof is…
We study the classical dynamics of the universe undergoing a transition from contraction to expansion through a big bang singularity. The dynamics is described by a system of differential equations for a set of physical quantities, such as…
A unique analytic continuation result is proved for solutions of a relatively general class of difference equations, using techniques of generalized Borel summability. This continuation allows for Painlev\'e property methods to be extended…
We study a 1D transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data. By adding a diffusion term the finite time singularity is prevented and the solutions exist…
We study multi-parameters deformations of isolated singularity function-germs on either a subanalytic set or a complex analytic spaces. We prove that if such a deformation has no coalescing of singular points, then it has constant…
The spatial dimensions 1 and 4 play an exceptional role for radial solutions of the pressureless repulsive Euler-Poisson equations. Namely, for any spatial dimension except 1 and 4, any nontrivial solution of the Cauchy problem blows up in…