Journal of Differential Equations最新文献

We obtain the global well-posedness to the Cauchy problem of the Fokas-Lenells (FL) equation on the line without the small-norm assumption on initial data $u_0\in H^3\left(R\right)\cap H^{2,1}\left(R\right)$. Our main technical tool is the inverse scattering transform method based on the representation of a Riemann-Hilbert (RH) problem associated with the above Cauchy problem. The existence and the uniqueness of the RH problem is shown via a general vanishing lemma. By representing the solutions of the RH problem via the Cauchy integral protection and the reflection coefficients, the reconstruction formula is used to obtain a unique local solution of the FL equation. Further, the eigenfunctions and the reflection coefficients are shown Lipschitz continuous with respect to initial data, which provides a prior estimate of the solution to the FL equation. Based on the local solution and the uniformly prior estimate, we construct a unique global solution in $H^3\left(R\right)\cap H^{2,1}\left(R\right)$ to the FL equation.

This paper focuses on inverse problems arising in studying multi-population aggregations. The goal is to reconstruct the diffusion coefficient, advection coefficient, and interaction kernels of the aggregation system, which characterize the dynamics of different populations. In the theoretical analysis of the physical setup, it is crucial to ensure non-negativity of solutions. To address this, we employ the high-order variation method and introduce modifications to the systems. Additionally, we propose a novel approach called transformative asymptotic technique that enables the recovery of the diffusion coefficient preceding the Laplace operator, presenting a pioneering method for this type of problems. Through these techniques, we offer comprehensive insights into the unique identifiability aspect of inverse problems associated with multi-population aggregation models.

We study the existence and the properties of ground states at fixed mass for a focusing nonlinear Schrödinger equation in dimension two with a point interaction, an attractive Coulomb potential and a nonlinearity of power type. We prove that for any negative value of the Coulomb charge, for any positive value of the mass and for any $L^2$-subcritical power nonlinearity, such ground states exist and exhibit a logarithmic singularity where the interaction is placed. Moreover, up to multiplication by a phase factor, they are positive, radially symmetric and decreasing. An analogous result is obtained also for minimizers of the action restricted to the Nehari manifold, getting the existence also in the $L^2$-critical and supercritical cases.

We obtain the local existence and uniqueness for a system describing interaction of an incompressible inviscid fluid, modeled by the Euler equations, and an elastic plate, represented by the fourth-order hyperbolic PDE. We provide a priori estimates for the existence with the optimal regularity $H^r$, for $r>2.5$, on the fluid initial data and construct a unique solution of the system for initial data $u_0\in H^r$ for $r\ge 3$. An important feature of the existence theorem is that the Taylor-Rayleigh instability does not occur. This is in contrast to the free-boundary Euler problem, where the stability condition on the initial pressure needs to be imposed.

In this paper, the existence of non-trivial time periodic solutions of first order mean field games is proved. It is assumed that there is a non-trivial periodic orbit contained in the Mather set. The whole system is autonomous with a monotonic coupling term. Moreover, the large time convergence of solutions of first order mean field games to time periodic solutions is also considered.

In this paper, we consider the Cauchy problem of the hyperbolic Keller-Segel equations in $H^s\left(T^d\right)$ on torus with $d\ge 1$. Firstly, developing the dissipative mechanism through translation, we establish the global well-posedness in $H^s\left(T^d\right)$ ($s>1+\frac{d}{2}$) with initial data near some equilibrium state. Secondly, by capturing the feature of the preservation of zero directional derivative, we give a class of initial date that lead to finite time blow-up. It's worth noting that our method of proving blow-up phenomenon does not require any conservation law. Finally, the characterization of this blow-up motivates us to show the ill-posedness of this system in $H^{\frac{3}{2}}\left(T^d\right)$ in the sense of “norm inflation”, which implies that our ill-posedness result for this system is sharp on one dimensional torus.

We consider a general class of non-diffusive active scalar equations with constitutive laws obtained via an operator T that is singular of order $r_0\in [0,2]$. For $r_0\in (0,1]$ we prove well-posedness in Gevrey spaces $G^s$ with $s\in [1,\frac{1}{r_0})$, while for $r_0\in [1,2]$ and further conditions on T we prove ill-posedness in $G^s$ for suitable s. We then apply the ill/well-posedness results to several specific non-diffusive active scalar equations including the magnetogeostrophic equation, the incompressible porous media equation and the singular incompressible porous media equation.

The well-posedness and exponential ergodicity are proved for stochastic Hamiltonian systems containing a singular drift term which is locally integrable in the component with noise. As an application, the well-posedness and uniform exponential ergodicity are derived for a class of singular degenerated McKean-Vlasov SDEs.

In this paper, we examine the effect of a new free boundary condition on the propagation dynamics of the nonlocal diffusion model considered in [9], which describes the spreading of a species with density $u(t,x)$ and population range $\left[g\right(t),h(t\left)\right]\subset R$. The existing free boundary condition can be written as$\{\begin{array}{c}{h}^{\prime }\left(t\right)=\mu \underset{g\left(t\right)}{\overset{h\left(t\right)}{\int }}u(t,x)W_J\left(h\right(t)-x)dx,\\ g^{\prime }\left(t\right)=-\mu \underset{g\left(t\right)}{\overset{h\left(t\right)}{\int }}u(t,x)W_J(x-g(t\left)\right)dx,\end{array}$ where $W_J\left(x\right)={\int }_x^{+\infty }J\left(y\right)dy$, and J is the kernel function of the nonlocal diffusion operator in the model. In the new free boundary condition, we replace $W_J$ by a general nonnegative locally Lipschitz continuous function W with $W\left(0\right)>0$, independent of J. This represents a very different assumption that the movement of the range boundary of the species is independent of its dispersal strategy, as in [20]. Our analysis shows that the dynamics of the model with the new free boundary condition resembles that of the old model except in the case that J is thin-tailed and ${\int }_0^{\infty }W\left(x\right)dx=\infty$, where new propagation phenomena appear.

We describe regularizing effects in the linearization of a kinetic equation for nonlinear waves satisfying the Schrödinger equation in terms of weak turbulence and condensate. The problem is first considered in spaces of bounded functions with weights, where existence of solutions and some first regularity properties are proved. After a suitable change of variables the equation is written in terms of a pseudo differential operator. Homogeneity of the equation and classical arguments of freezing of coefficients may then be used to prove regularizing effect in local Sobolev type spaces.