We demonstrate norm inflation for nonlinear nonlocal equations, which extend the Korteweg-de Vries equation to permit fractional dispersion, in the periodic and non-periodic settings. That is, an initial datum is smooth and arbitrarily small in a Sobolev space, but the solution becomes arbitrarily large in the Sobolev space after an arbitrarily short time.
Fractional calculus of variation plays an important role to formulate the non-conservative physical problems. In this paper we use semi-inverse method and fractional variational principle to formulate the fractional order generalized Korteweg-deVries (KdV) equation with Jumarie type fractional derivative and proposed a new method to solve the non-linear fractional differential equation named as expansion method. Using this method we obtained the solutions of fractional order generalized KdV. The obtained solutions are more general compare to other method and the solutions are expressed in term…
The linear operator $c + (-Δ)^{α/2}$, where $c > 0$ and $(-Δ)^{α/2}$ is the fractional Laplacian on the periodic domain, arises in the existence of periodic travelling waves in the fractional Korteweg--de Vries equation. We establish a relation of the Green's function of this linear operator with the Mittag--Leffler function, which was previously used in the context of Riemann--Liouville's and Caputo's fractional derivatives. By using this relation, we prove that Green's function is strictly positive and single-lobe (monotonically decreasing away from the maximum point) for every $c > 0$ and e…
We investigated the analytical solution of fractional order K(m,n) type equation with variable coefficient which is an extended type of KdV equations into a genuinely nonlinear dispersion regime. By using the Lie symmetry analysis, we obtain the Lie point symmetries for this type of time-fractional partial differential equations (PDE). Also we present the corresponding reduced fractional differential equations (FDEs) corresponding to the time-fractional K(m,n) type equation.
We explain how the invariant subspace method can be extended to a scalar and coupled system of time-space fractional partial differential equations. The effectiveness and applicability of the method have been illustrated through time-space (i) fractional diffusion-convection equation, (ii) fractional reaction-diffusion equation, (iii) fractional diffusion equation with source term, (iv) two-coupled system of the fractional diffusion equation, (v) two-coupled system of fractional stationary transonic plane-parallel gas flow equation and (vi) three-coupled system of fractional Hirota-Satsuma KdV…
For $0<α<1$ and with initial data $\vert\vert u_0\vert\vert_{H^{N+\fracα{2}}}=\varepsilon$, sufficently small, we show that the existence time for solutions of the fractional BBM equation $\partial_tu+\partial_xu+u\partial_xu+\vert\mathrm{D}\vert^α\partial_tu=0$, can be extended beyond the hyperbolic existence time $\frac{1}{\varepsilon}$, to $\frac{1}{\varepsilon^2}$. For the proof we use a modified energy, based on a normal form transformation as in [Hunter, Ifrim, Tataru, Wong, 2015]. In addition we employ ideas and techniques from [Ehrnström, Wang, 2018], in which the authors obtain an enh…
In this paper we prove local well-posedness of a space-time fractional generalization of the nonlinear Schrödinger equation with a power-type nonlinearity. The linear equation coincides with a model proposed by Naber, and displays a nonlocal behavior both in space and time which accounts for long-range interactions and a so-called memory effect. Because of a loss of derivatives produced by the latter and the lack of semigroup structure of the solution operator, we employ a strategy of proof based on exploiting some smoothing effect similar to that used by Kenig, Ponce and Vega for the KdV equa…
We construct $N$-soliton solutions for the fractional Korteweg-de Vries (fKdV) equation $$ \partial_t u - \partial_x\left(|D|^αu - u^2 \right)=0, $$ in the whole sub-critical range $α\in]\frac12,2[$. More precisely, if $Q_c$ denotes the ground state solution associated to fKdV evolving with velocity $c$, then given $0<c_1< \cdots < c_N$, we prove the existence of a solution $U$ of (fKdV) satisfying $$ \lim_{t\to\infty} \| U(t,\cdot) - \sum_{j=1}^NQ_{c_j}(x-ρ_j(t)) \|_{H^{\fracα2}}=0, $$ where $ρ'_j(t) \sim c_j$ as $t \to +\infty$. The proof adapts the construction of Martel in the generalized …
We consider two types of the generalized Korteweg - de Vries equation, where the nonlinearity is given with or without absolute values, and, in particular, including the low powers of nonlinearity, an example of which is the Schamel equation. We first prove the local well-posedness of both equations in a weighted subspace of $H^1$ that includes functions with polynomial decay, extending the result of Linares et al [39] to fractional weights. We then investigate solutions numerically, confirming the well-posedness and extending it to a wider class of functions that includes exponential decay. W…
We consider in this paper modified fractional Korteweg-de Vries and related equations (modified Burgers-Hilbert and Whitham). They have the advantage with respect to the usual fractional KdV equation to have a defocusing case with a different dynamics. We will distinguish the weakly dispersive case where the phase velocity is unbounded for low frequencies and tends to zero at infinity and the strongly dispersive case where the phase velocity vanishes at the origin and goes to infinity at infinity. In the former case, the nonlinear hyperbolic effects dominate for large data, leading to the poss…
We consider the initial value problem (IVP) for the fractional Korteweg-de Vries equation (fKdV) \begin{equation}\label{abstracteq1} \left\{ \begin{array}{ll} \partial_{t}u-D_{x}^α\partial_{x}u+u\partial_{x}u=0, & x,t\in\mathbb{R},\,0<α<1, \\ u(x,0)=u_{0}(x).& \\ \end{array} \right. \end{equation} It has been shown that the solutions to certain dispersive equations satisfy the propagation of regularity phenomena. More precisely, it deals in determine whether regularity of the initial data on the right hand side of the real line is propagated to the left hand side by the flow solution. This pro…
Positive-regularity norm inflation for the 3D hypodissipative Navier--Stokes equations
Capital Normal University（首都师范大学）；Institute of Mathematics, Academy of Mathematics and Systems Science Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学研究所）
Nonlinear PDEs with modulated dispersion IV: normal form approach and unconditional uniqueness
University of Oxford（牛津大学）；Beijing Institute of Technology（北京理工大学）；The University of Edinburgh（爱丁堡大学）；The Maxwell Institute for the Mathematical Sciences（麦克斯韦数学科学研究所）
We show existence of small solitary and periodic traveling-wave solutions in Sobolev spaces ${\mathrm{H}^s}$, ${ s > 0 }$, to a class of nonlinear, dispersive evolution equations of the form \begin{equation*} u_t + \left(Lu+ n(u)\right)_x = 0, \end{equation*} where the dispersion ${L}$ is a negative-order Fourier multiplier whose symbol is of KdV type at low frequencies and has integrable Fourier inverse ${ K }$ and the nonlinearity ${n}$ is inhomogeneous, locally Lipschitz and of superlinear growth at the origin. This generalises earlier work by Ehrnström, Groves & Wahlén on a class of equati…
Nonlinear PDEs with modulated dispersion III: multiplicative noises
CNRS, Laboratoire de mathématiques de Versailles, UVSQ, Université Paris-Saclay（法国国家科学研究中心、凡尔赛数学实验室、巴黎西岱大学、巴黎萨克雷大学）；School of Mathematics, Monash University（蒙纳士大学数学学院）；School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）；The Maxwell Institute for the Mathematical Sciences（麦克斯韦数学科学研究所）
