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Register a new userSelf-similar solutions for the Muskat equation
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We show the existence of self-similar solutions for the Muskat equation. These solutions are parameterized by $0<s ll 1$; they are exact corners of slope $s$ at $t=0$ and become smooth in $x$ for $t>0$.
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We consider the nonlinear heat equation $u_t = Delta u + |u|^alpha u$ with $alpha >0$, either on ${mathbb R}^N $, $Nge 1$, or on a bounded domain with Dirichlet boundary conditions. We prove that in the Sobolev subcritical case $(N-2) alpha <4$, for every $mu in {mathbb R}$, if the initial value $u_0$ satisfies $u_0 (x) = mu |x-x_0|^{-frac {2} {alpha }}$ in a neighborhood of some $x_0in Omega $ and is bounded outside that neighborhood, then there exist infinitely many solutions of the heat equation with the initial condition $u(0)= u_0$. The proof uses a fixed-point argument to construct perturbations of self-similar solutions with initial value $mu |x-x_0|^{-frac {2} {alpha }}$ on ${mathbb R}^N $. Moreover, if $mu ge mu _0$ for a certain $ mu _0( N, alpha )ge 0$, and $u_0 Ige 0$, then there is no nonnegative local solution of the heat equation with the initial condition $u(0)= u_0$, but there are infinitely many sign-changing solutions.
We construct forward self-similar solutions (expanders) for the compressible Navier-Stokes equations. Some of these self-similar solutions are smooth, while others exhibit a singularity do to cavitation at the origin.
We investigate the existence and the boundary regularity of source-type self-similar solutions to the thin-film equation $h_t=-(h^nh_{zzz})_z+(h^{n+3})_{zz},$ $ t>0,, zin R;, h(0,z)= M delta$ where $nin (3/2,3),, M > 0$ and $delta$ is the Dirac mass at the origin. It is known that the leading order expansion near the edge of the support coincides with that of a travelling-wave solution for the standard thin-film equation: $h_t=-(h^nh_{zzz})_z$. In this paper we sharpen this result, proving that the higher order corrections are analytic with respect to three variables: the first one is just the spacial variable, whereas the second and third (except for $n = 2$) are irrational powers of it. It is known that this third order term does not appear for the thin-film equation without gravity.
We consider the nonlinear heat equation $u_t - Delta u = |u|^alpha u$ on ${mathbb R}^N$, where $alpha >0$ and $Nge 1$. We prove that in the range $0 < alpha <frac {4} {N-2}$, for every $mu >0$, there exist infinitely many sign-changing, self-similar solutions to the Cauchy problem with initial value $u_0 (x)= mu |x|^{-frac {2} {alpha }}$. The construction is based on the analysis of the related inverted profile equation. In particular, we construct (sign-changing) self-similar solutions for positive initial values for which it is known that there does not exist any local, nonnegative solution.
In the supercritical range of the polytropic indices $gammain(1,frac43)$ we show the existence of smooth radially symmetric self-similar solutions to the gravitational Euler-Poisson system. These solutions exhibit gravitational collapse in the sense that the density blows-up in finite time. Some of these solutions were numerically found by Yahil in 1983 and they can be thought of as polytropic analogues of the Larson-Penston collapsing solutions in the isothermal case $gamma=1$. They each contain a sonic point, which leads to numerous mathematical difficulties in the existence proof.
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