Recent works have demonstrated that self-similar radial Euler flows of an ideal polytropic gas in 2 and 3 dimensions can drive primary (non-differentiated) flow variables to infinity at the center of motion. Unsurprisingly, this type of behavior can generate an outgoing shock wave from the point of collapse. In this work we prove that there is an alternative scenario possible in 3-d flow: an incoming, radial C^1 flow suffers blowup at a single point, including in density and pressure, and yet remains C^1 beyond collapse. No shock is generated despite the fact that the density gradient ∇ _xρ , as well as the density of characteristics, becomes unbounded at the center of motion at time of blowup. Furthermore, we verify that this behavior is possible even in cases where the fluid is everywhere moving toward the center of motion at time of collapse, in which case also div _xu→ -∞ . These phenomena underscore the subtlety of shock formation in multi-d flow.
This corrigendum corrects an error in the proof of Theorem 3.2 in [Gal, Grasselli and Wu, Global weak solutions to a diffuse interface model for incompressible two-phase flows with moving contact lines and different densities, Arch. Ration. Mech. Anal., 234 (2019), 1–56], which was caused by the unjustified strong convergence asserted in (3.90). We provide an alternative proof of the required nonlinear convergence. The statement and conclusions of Theorem 3.2 remain unchanged.
In this paper, and in the companion ([152]), we study the gauge-invariant components of linear perturbations of Kerr black hole spacetimes in the full subextremal range of parameters | a | < M . We show that, for all | a | < M , solutions of the Teukolsky equation of spin ± 1 and spin ± 2 arising from regular initial data remain bounded and decay in time. Our result makes completely explicit the dependence of these bounds on the Kerr black hole parameters, while being entirely self-contained. For completeness, we also include the case of the scalar wave equation, s = 0 . The present paper, which is an abridged version of our preprint ([151]), considers fixed frequency solutions of the transformed system of equations introduced by Dafermos et al. (Acta Math 222(1):1-214, 2019). We obtain estimates which are uniform in the separation parameters for subextremal | a | < M , as well as estimates away from the superradiant frequency threshold up to and including the extremal | a | = M . In the subextremal | a | < M case, these estimates are summed in our companion paper ([152]) to yield the conclusion of the proof of our main result.
Abstract We establish that locally bounded relaxed minimizers of degenerate elliptic symmetric gradient functionals on $$\textrm{BD}(\Omega )$$ BD ( Ω ) have weak gradients in $$\textrm{L}_{\textrm{loc}}^{1}(\Omega ;\mathbb {R}^{n\times n})$$ L loc 1 ( Ω ; R n × n ) . This is achieved for the sharp ellipticity range that is presently known to yield $$\textrm{W}_{\textrm{loc}}^{1,1}$$ W loc 1 , 1 -regularity in the full gradient case on $$\textrm{BV}(\Omega ;\mathbb {R}^{n})$$ BV ( Ω ; R n ) . As a consequence, we also obtain the first Sobolev regularity results for minimizers of the area-type functional on $$\textrm{BD}(\Omega )$$ BD ( Ω ) .
Abstract In this work we establish the formation of singularities of classical solutions with finite energy of the forced fractional Navier Stokes equations where the dissipative term is given by $$|\nabla |^{\alpha }$$ | ∇ | α for any $$\alpha \in [0, \alpha _0)$$ α ∈ [ 0 , α 0 ) ( $$\alpha _0 = \frac{22-8\sqrt{7}}{9} > 0$$ α 0 = 22 - 8 7 9 > 0 ). We construct solutions in $$\mathbb {R}^3\times [0,T]$$ R 3 × [ 0 , T ] with a finite $$T>0$$ T > 0 and with an external forcing which is in $$L^1_t([0, T]) C_x^{1,\epsilon }\cap L^{\infty }_{t} L^{2}_{x}$$ L t 1 ( [ 0 , T ] ) C x 1 , ϵ ∩ L t ∞ L x 2 , such that for each time $$t \in [0, T)$$ t ∈ [ 0 , T ) , the velocity u is in the space $$C^\infty \cap L^2$$ C ∞ ∩ L 2 and such that as the time t approaches the blow-up moment T , the integral $$\int _0^t |\nabla u| \text {d}s$$ ∫ 0 t | ∇ u | d s tends to infinity. Since this result lays in a well-posedness class, the blow-up is generated by the dynamics of the equation and not by the force itself. This is the first blow-up result for hypodissipative Navier–Stokes in a well-posedness class.
