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Register a new userRemark on the semilinear ill-posedness for a periodic higher order KP-I equation
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Authors
Tristan Robert
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We prove that, for some irrational torus, the flow map of the periodic fifth-order KP-I equation is not locally uniformly continuous on the energy space, even on the hyperplanes of fixed x-mean value.
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Norm inflation implies certain discontinuous dependence of the solution on the initial value. The well-posedness of the mild solution means the existence and uniqueness of the fixed points of the corresponding integral equation. For ${rm BMO}^{-1}$, Auscher-Dubois-Tchamitchian proved that Koch-Tatarus solution is stable. In this paper, we construct a non-Gauss flow function to show that, for classic Navier-Stokes equations, wellposedness and norm inflation may have no conflict and stability may have meaning different to $L^{infty}(({rm BMO}^{-1})^{n})$.
The aim of this paper is to investigate the Cauchy problem for the periodic fifth order KP-I equation [partial_t u - partial_x^5 u -partial_x^{-1}partial_y^2u + upartial_x u = 0,~(t,x,y)inmathbb{R}timesmathbb{T}^2] We prove global well-posedness for constant $x$ mean value initial data in the space $mathbb{E} = {uin L^2,~partial_x^2 u in L^2,~partial_x^{-1}partial_y u in L^2}$ which is the natural energy space associated with this equation.
We prove the discontinuity for the weak $ L^2(T) $-topology of the flow-map associated with the periodic Benjamin-Ono equation. This ensures that this equation is ill-posed in $ H^s(T) $ as soon as $ s<0 $ and thus completes exactly the well-posedness result obtained by the author.
In this paper, we investigate the problem of optimal regularity for derivative semilinear wave equations to be locally well-posed in $H^{s}$ with spatial dimension $n leq 5$. We show this equation, with power $2le ple 1+4/(n-1)$, is (strongly) ill-posed in $H^{s}$ with $s = (n+5)/4$ in general. Moreover, when the nonlinearity is quadratic we establish a characterization of the structure of nonlinear terms in terms of the regularity. As a byproduct, we give an alternative proof of the failure of the local in time endpoint scale-invariant $L_{t}^{4/(n-1)}L_{x}^{infty}$ Strichartz estimates. Finally, as an application, we also prove ill-posed results for some semilinear half wave equations.
In this article, we address the Cauchy problem for the KP-I equation [partial_t u + partial_x^3 u -partial_x^{-1}partial_y^2u + upartial_x u = 0] for functions periodic in $y$. We prove global well-posedness of this problem for any data in the energy space $mathbb{E} = left{uin L^2left(mathbb{R}timesmathbb{T}right),~partial_x u in L^2left(mathbb{R}timesmathbb{T}right),~partial_x^{-1}partial_y u in L^2left(mathbb{R}timesmathbb{T}right)right}$. We then prove that the KdV line soliton, seen as a special solution of KP-I equation, is orbitally stable under this flow, as long as its speed is small enough.
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