h-Principles for the Incompressible Euler Equations

Recently, De Lellis and Sz\'ekelyhidi constructed H\"older continuous, dissipative (weak) solutions to the incompressible Euler equations in the torus $\mathbb T^3$. The construction consists in adding fast oscillations to the trivial solution. We extend this result by establishing optimal h-principles in two and three space dimensions. Specifically, we identify all subsolutions (defined in a suitable sense) which can be approximated in the $H^{-1}$-norm by exact solutions. Furthermore, we prove that the flows thus constructed on $\mathbb T^3$ are genuinely three-dimensional and are not trivially obtained from solutions on $\mathbb T^2$.Comment: 29 pages, no figure

Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line

In this paper, we study ill-posedness of cubic fractional nonlinear Schr\"odinger equations. First, we consider the cubic nonlinear half-wave equation (NHW) on $\mathbb R$. In particular, we prove the following ill-posedness results: (i) failure of local uniform continuity of the solution map in $H^s(\mathbb R)$ for $s\in (0,\frac 12)$, and also for $s=0$ in the focusing case; (ii) failure of $C^3$-smoothness of the solution map in $L^2(\mathbb R)$; (iii) norm inflation and, in particular, failure of continuity of the solution map in $H^s(\mathbb R)$, $s<0$. By a similar argument, we also prove norm inflation in negative Sobolev spaces for the cubic fractional NLS. Surprisingly, we obtain norm inflation above the scaling critical regularity in the case of dispersion $|D|^\beta$ with $\beta>2$.Comment: Introduction expanded, references updated. We would like to thank Nobu Kishimoto for his comments on the previous version and for pointing out the related article of Iwabuchi and Uriy

Local structure of the set of steady-state solutions to the 2D incompressible Euler equations

It is well known that the incompressible Euler equations can be formulated in a very geometric language. The geometric structures provide very valuable insights into the properties of the solutions. Analogies with the finite-dimensional model of geodesics on a Lie group with left-invariant metric can be very instructive, but it is often difficult to prove analogues of finite-dimensional results in the infinite-dimensional setting of Euler's equations. In this paper we establish a result in this direction in the simple case of steady-state solutions in two dimensions, under some non-degeneracy assumptions. In particular, we establish, in a non-degenerate situation, a local one-to-one correspondence between steady-states and co-adjoint orbits.Comment: 81 page

On the probabilistic Cauchy theory for nonlinear dispersive PDEs

In this note, we review some of the recent developments in the well-posedness theory of nonlinear dispersive partial differential equations with random initial data.Comment: 26 pages. To appear in Landscapes of Time-Frequency Analysis, Appl. Numer. Harmon. Ana

Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line

In this paper, we study ill-posedness of cubic fractional nonlinear Schrödinger equations. First, we consider the cubic nonlinear half-wave equation (NHW) on R. In particular, we prove the following ill-posedness results: (i) failure of local uniform continuity of the solution map in Hs(R) for s∈(0,1/2), and also for s=0 in the focusing case; (ii) failure of C3- smoothness of the solution map in L2(R); (iii) norm inflation and, in particular, failure of continuity of the solution map in Hs(R), s2

Traveling wave tube-based LINC transmitters

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