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Register a new userFull blow-up range for co-rotaional wave maps to surfaces of revolution
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Authors
Can Gao
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We construct blow-up solutions of the energy critical wave map equation on $mathbb{R}^{2+1}to mathcal N$ with polynomial blow-up rate ($t^{-1- u}$ for blow-up at $t=0$) in the case when $mathcal{N}$ is a surface of revolution. Here we extend the blow-up range found by C^arstea ($ u>frac 12$) based on the work by Krieger, Schlag and Tataru to $ u>0$. This work relies on and generalizes the recent result of Krieger and the author where the target manifold is chosen as the standard sphere.
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We prove that the critical Wave Maps equation with target $S^2$ and origin $mathbb{R}^{2+1}$ admits energy class blow up solutions of the form $$u(t,r)=Q(lambda(t)r)+epsilon(t,r)$$where $Q: mathbb{R}^2 to S^2$ is the ground state harmonic map and $lambda(t) = t^{-1- u}$ for any $ u > 0$. This extends the work [13], where such solutions were constructed under the assumption $ u > 1/2$. In light of a result of Struwe [22], our result is optimal for polynomial blow up rates.
Consider the energy critical focusing wave equation on the Euclidian space. A blow-up type II solution of this equation is a solution which has finite time of existence but stays bounded in the energy space. The aim of this work is to exhibit universal properties of such solutions. Let W be the unique radial positive stationary solution of the equation. Our main result is that in dimension 3, under an appropriate smallness assumption, any type II blow-up radial solution is essentially the sum of a rescaled W concentrating at the origin and a small remainder which is continuous with respect to the time variable in the energy space. This is coherent with the solutions constructed by Krieger, Schlag and Tataru. One ingredient of our proof is that the unique radial solution which is compact up to scaling is equal to W up to symmetries.
In this paper, we investigate the problem of blow up and sharp upper bound estimates of the lifespan for the solutions to the semilinear wave equations, posed on asymptotically Euclidean manifolds. Here the metric is assumed to be exponential perturbation of the spherical symmetric, long range asymptotically Euclidean metric. One of the main ingredients in our proof is the construction of (unbounded) positive entire solutions for $Delta_{g}phi_lambda=lambda^{2}phi_lambda$, with certain estimates which are uniform for small parameter $lambdain (0,lambda_0)$. In addition, our argument works equally well for semilinear damped wave equations, when the coefficient of the dissipation term is integrable (without sign condition) and space-independent.
Following our previous paper in the radial case, we consider blow-up type II solutions to the energy-critical focusing wave equation. Let W be the unique radial positive stationary solution of the equation. Up to the symmetries of the equation, under an appropriate smallness assumption, any type II blow-up solution is asymptotically a regular solution plus a rescaled Lorentz transform of W concentrating at the origin.
In this paper, inspired by the study of the energy flux in local energy inequality of the 3D incompressible Navier-Stokes equations, we improve almost all the blow up criteria involving temperature to allow the temperature in its scaling invariant space for the 3D full compressible Navier-Stokes equations. Enlightening regular criteria via pressure $Pi=frac{text {divdiv}}{-Delta}(u_{i}u_{j})$ of the 3D incompressible Navier-Stokes equations on bounded domain, we generalize Beirao da Veigas result in [1] from the incompressible Navier-Stokes equations to the isentropic compressible Navier-Stokes system in the case away from vacuum.
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