We consider traveling wave phenomena for a viscoelastic generalization of Burgers’ equation. For asymptotically constant velocity profiles we find three classes of solutions corresponding to smooth traveling waves, piecewise smooth waves, and piecewise constant (shock) solutions. Each solution type is possible for a given pair of asymptotic limits, and we characterize the dynamics in terms of t...

We consider surface quasi-geostrophic equation with dispersive forcing and critical dissipation. We prove global existence of smooth solutions given sufficiently smooth initial data. This is done using a maximum principle for the solutions involving conservation of a certain family of moduli of continuity.

We consider the aggregation equation ut + ∇ · (u∇K ∗ u) = 0 in R n, n ≥ 2, where K is a rotationally symmetric, nonnegative decaying kernel with a Lipschitz point at the origin, e.g. K(x) = e−|x|. We prove finite-time blow-up of solutions from specific smooth initial data, for which the problem is known to have short time existence of smooth solutions.

In this paper, we study the partial convexity of smooth solutions to the heat equation on a compact or complete noncompact Riemannian manifold M or on Kaehler-Ricci flow. We show that under a natural assumption, a new partial convexity property for smooth solutions to the heat equation is preserved.

We explore local existence and properties of classical weak solutions to the initial-boundary value problem of a one-dimensional quasilinear equation of elastodynamics with non-convex stored-energy function, a model of phase transitions in elastic bars proposed by Ericksen [19]. The instantaneous formation of microstructures of local weak solutions is observed for all smooth initial data with i...

In this paper, we study stochastic functional differential equations (sfde’s) whose solutions are constrained to live on a smooth compact Riemannian manifold. We prove the existence and uniqueness of solutions to such sfde’s. We consider examples of geometrical sfde’s and establish the smooth dependence of the solution on finite-dimensional parameters.

We prove that there are solutions to the Euler equation on the torus with C vorticity and smooth except at one point such that the vorticity gradient grows in L∞ at least exponentially as t → ∞. The same result is shown to hold for the vorticity Hessian and smooth solutions. Our proofs use a version of a recent result by Kiselev and Šverák [5].

We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the generalized KP-BBM equation. A number of explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain periodic wave solutions, kink wave solutions, unbounded wave solutions, blow-up wave sol...

In this paper we establish a number of implications between various qualitative and quantitative versions of the global regularity problem for the Navier-Stokes equations, in the periodic, smooth finite energy, smooth H, Schwartz, or mild H categories, and with or without a forcing term. In particular, we show that if one has global well-posedness in H for the periodic Navier-Stokes problem wit...