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Register a new userOn the local and global existence of solutions to 1D transport equations with nonlocal velocity
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We consider the 1D transport equation with nonlocal velocity field: begin{equation*}label{intro eq} begin{split} &theta_t+utheta_x+ u Lambda^{gamma}theta=0, & u=mathcal{N}(theta), end{split} end{equation*} where $mathcal{N}$ is a nonlocal operator. In this paper, we show the existence of solutions of this model locally and globally in time for various types of nonlocal operators.
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We consider the porous medium equation with a power-like reaction term, posed on Riemannian manifolds. Under certain assumptions on $p$ and $m$ in (1.1), and for small enough nonnegative initial data, we prove existence of global in time solutions, provided that the Sobolev inequality holds on the manifold. Furthermore, when both the Sobolev and the Poincare inequality hold, similar results hold under weaker assumptions on the forcing term. By the same functional analytic methods, we investigate global existence for solutions to the porous medium equation with source term and variable density in ${mathbb R}^n$.
We prove the equivalence between the notion of Wasserstein gradient flow for a one-dimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and the notion of entropy solution of a Burgers-type scalar conservation law on the other. The solution of the former is obtained by spatially differentiating the solution of the latter. The proof uses an intermediate step, namely the $L^2$ gradient flow of the pseudo-inverse distribution function of the gradient flow solution. We use this equivalence to provide a rigorous particle-system approximation to the Wasserstein gradient flow, avoiding the regularization effect due to the singularity in the repulsive kernel. The abstract particle method relies on the so-called wave-front-tracking algorithm for scalar conservation laws. Finally, we provide a characterization of the sub-differential of the functional involved in the Wasserstein gradient flow.
In this paper, we are concerned with the global existence and blowup of smooth solutions of the 3-D compressible Euler equation with time-depending damping $$ partial_trho+operatorname{div}(rho u)=0, quad partial_t(rho u)+operatorname{div}left(rho uotimes u+p,I_{3}right)=-,frac{mu}{(1+t)^{lambda}},rho u, quad rho(0,x)=bar rho+varepsilonrho_0(x),quad u(0,x)=varepsilon u_0(x), $$ where $xinmathbb R^3$, $mu>0$, $lambdageq 0$, and $barrho>0$ are constants, $rho_0,, u_0in C_0^{infty}(mathbb R^3)$, $(rho_0, u_0) otequiv 0$, $rho(0,cdot)>0$, and $varepsilon>0$ is sufficiently small. For $0leqlambdaleq1$, we show that there exists a global smooth solution $(rho, u)$ when $operatorname{curl} u_0equiv 0$, while for $lambda>1$, in general, the solution $(rho, u)$ will blow up in finite time. Therefore, $lambda=1$ appears to be the critical value for the global existence of small amplitude smooth solutions.
We study a multi-dimensional nonlocal active scalar equation of the form $u_t+vcdot abla u=0$ in $mathbb R^+times mathbb R^d$, where $v=Lambda^{-2+alpha} abla u$ with $Lambda=(-Delta)^{1/2}$. We show that when $alphain (0,2]$ certain radial solutions develop gradient blowup in finite time. In the case when $alpha=0$, the equations are globally well-posed with arbitrary initial data in suitable Sobolev spaces.
In this paper, we investigate the nonhomogeneous boundary value problem for the steady Navier-Stokes equations in a helically symmetric spatial domain. When data is assumed to be helical invariant and satisfies the compatibility condition, we prove this problem has at least one helical invariant solution.
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