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SUMMARY:Generic regularity of the two-component Novikov equation
DTSTART;VALUE=DATE-TIME:20260924T114000Z
DTEND;VALUE=DATE-TIME:20260924T120000Z
DTSTAMP;VALUE=DATE-TIME:20260914T055134Z
UID:indico-contribution-539@indico.bitp.kiev.ua
DESCRIPTION:Speakers: Yan Rybalko (B.Verkin Institute for Low Temperature 
 Physics and Engineering of the National Academy of Sciences of Ukraine)\nT
 his talk concerns the Cauchy problem for the following two-component Novik
 ov system:\n$\\begin{align}\n&m_t+(uvm)_x+u_xvm=0\,\n&& m=m(t\,x)\,\\\,u=u
 (t\,x)\,\\\,v=v(t\,x)\,\\\\\n&n_t+(uvn)_x+uv_xn=0\,\n&& n=n(t\,x)\,\\\\\n&
 m=u-u_{xx}\,\\\,\\\,n=v-v_{xx}\,\n&& u\,v\\in\\mathbb{R}\,\\quad\nt\,x\\in
 \\mathbb{R}\,\n\\end{align}$\n\nIt is well-known that this system admits t
 he wave-breaking phenomena: the slope of solution corresponding to the smo
 oth initial data can blow-up in finite time.\n\nThe goal is to establish t
 he generic regularity for the two-component Novikov system. More in detail
 \, we show that there exists an open dense subset of initial data\, such t
 hat the corresponding global weak solutions (which exist beyond blow-up) r
 etain certain regularity properties in the $t\,x$ plane. Applying the Bres
 san and Chen's approach [1]\, originally developed for the nonlinear varia
 tional wave equation\, we prove the following result [2].\n\n**Theorem.** 
 Consider the regular initial data $(u_0\,v_0)$ in the metric space $\\Upsi
 lon^k\\subset\\left(C^k\\right)^2\\cap\\left(H^1\\right)^2$\, $k\\geq5$\, 
 see [2] for details. Then for any $T>0$ there exists an open in $\\Upsilon
 ^k$ and dense in $\\Upsilon^{k-1}\\cap C^k_{\\mathrm{loc}}$ subset $\\math
 cal{M}_T\\subset\\Upsilon^k$\, such that for any initial data $(u_0\,v_0)\
 \in\\mathcal{M}_T$ the corresponding global weak solution $(u\,v)(t)$ sati
 sfies the following regularity properties. There exist two sets of piecewi
 se $C^{k-1}$ curves\, say $\\left\\{\\mathbf{C}_i^W\\right\\}_{i=1}^{N_1}\
 , \\left\\{\\mathbf{C}_j^Z\\right\\}_{j=1}^{N_2}\\subset[-T\,T]\\times\\ma
 thbb{R}$\, such that\n\n - both components $u$ and $v$ are $C^k$ regular a
 way from the characteristic sets $\\mathbf{C}_i^W$ and $\\mathbf{C}_j^Z$\n
  - $u$ is of class $C^1$ along each $\\mathbf{C}_j^Z$ and $v$ is of class 
 $C^1$ along each $\\mathbf{C}_i^W$.\n\n\n\n[1] A. Bressan\, G. Chen. Gener
 ic regularity of conservative solutions to a nonlinear wave equation. Ann.
  I. H. Poincaré 34 (2017) 335--354.\n[2] K.H. Karlsen\, Ya. Rybalko. Gene
 ric regularity and Lipschitz metric for a two-component Novikov system. ar
 Xiv :2512.13305v2 (2026).\n\nhttps://indico.bitp.kiev.ua/event/18/contribu
 tions/539/
LOCATION:Bogolyubov Institute for Theoretical Physics Conference Hall
URL:https://indico.bitp.kiev.ua/event/18/contributions/539/
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