Quintic NLW/NLKG on R3: Difference between revisions

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{{equation
| name = Quintic NLW/NLKG on R^3
| equation = <math>\Box u = m^2 u \pm u^5</math>
| fields = <math>u: \R^{1+3} \to \mathbb{C}</math>
| data = <math>u[0] \in H^s \times H^{s-1}(\R^3)</math>
| hamiltonian = [[Hamiltonian]]
| linear = [[free wave equation|wave]]
| nonlinear = [[semilinear]]
| critical = <math>\dot H^1 \times L^2(\R^3)</math>
| criticality = energy-critical
| covariance = [[Lorentzian]]
| lwp = <math>H^s \times H^{s-1}(\R)</math> for <math>s \geq 1</math>
| gwp = <math>H^s \times H^{s-1}(\R)</math> for <math>s \geq 1</math> (+)<br> <math>s \geq 1</math> and sub-ground-state energy (-)
| parent = [[Quintic NLW/NLKG]]
| special = -
| related = -
}}
* Scaling is <math>s=1</math>.  Thus this equation is energy-critical.
* Scaling is <math>s=1</math>.  Thus this equation is energy-critical.
* LWP for <math>s \geq 1</math> by Strichartz estimates (see e.g. [[LbSo1995]]; earlier references exist)
* LWP for <math>s \geq 1</math> by Strichartz estimates (see e.g. [[LbSo1995]]; earlier references exist)

Latest revision as of 21:57, 4 March 2007

Quintic NLW/NLKG on R^3
Description
Equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Box u = m^2 u \pm u^5}
Fields Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u: \R^{1+3} \to \mathbb{C}}
Data class Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u[0] \in H^s \times H^{s-1}(\R^3)}
Basic characteristics
Structure Hamiltonian
Nonlinearity semilinear
Linear component wave
Critical regularity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \dot H^1 \times L^2(\R^3)}
Criticality energy-critical
Covariance Lorentzian
Theoretical results
LWP Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s \times H^{s-1}(\R)} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \geq 1}
GWP Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s \times H^{s-1}(\R)} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \geq 1} (+)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \geq 1} and sub-ground-state energy (-)
Related equations
Parent class Quintic NLW/NLKG
Special cases -
Other related -


  • Scaling is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s=1} . Thus this equation is energy-critical.
  • LWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \geq 1} by Strichartz estimates (see e.g. LbSo1995; earlier references exist)
    • When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s=1} the time of existence depends on the profile of the data and not just on the norm.
    • For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s<s_c} one has instantaneous blowup in the focusing case, and unbounded growth of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s} norms in the defocusing case (CtCoTa-p2).
  • GWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s=1} in the defocussing case (Gl1990, Gl1992). The main new ingredient is energy non-concentration (Sw1988, Sw1992).
    • Further decay estimates and scattering were obtained in BaSa1998, Na1999d, Ta2006; global Lipschitz dependence was obtained in BaGd1997.
    • For smooth data GWP and scattering was shown in Gl1992; see also SaSw1994
    • For radial data GWP and scattering was shown in Sw1988
    • For data with small energy this was shown for general quintic non-linearities (and for either NLW or NLKG) in Ra1981.
    • Global weak solutions can be constructed by general methods (e.g. Sr1989, Sw1992); uniqueness was shown in Kt1992
    • In the focussing case there is blowup from large data by the ODE method.
    • When there is a convex obstacle GWP for smooth data is known SmhSo1995.
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