SMALL GLOBAL SOLUTIONS FOR NONLINEAR COMPLEX GINZBURG–LANDAU EQUATIONS AND NONLINEAR DISSIPATIVE WAVE EQUATIONS IN SOBOLEV SPACES

2011 ◽  
Vol 23 (08) ◽  
pp. 903-931 ◽  
Author(s):  
MAKOTO NAKAMURA
Keyword(s):  
Sobolev Spaces ◽  
Besov Spaces ◽  
Wave Equations ◽  
Global Solutions ◽  
Regularity Of Solutions ◽  
Cauchy Problems ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equations ◽  
Local Solutions ◽  
Scaling Arguments

The Cauchy problems for nonlinear complex Ginzburg–Landau equations and nonlinear dissipative wave equations are considered in Sobolev spaces. The relation between the order of the nonlinear terms and the regularity of solutions is considered in terms of the scaling arguments, and the existence of local solutions and small global solutions is shown in Sobolev and Besov spaces.

scholarly journals Wave Equations of Cooper Pairs as Quasiparticles in Superconducting Fermion Systems

1997 ◽  
Vol 50 (6) ◽  
pp. 1035
Author(s):  
Nguyen Van Hieu
Keyword(s):  
Variational Principle ◽  
Effective Action ◽  
Vector Fields ◽  
Wave Equations ◽  
Cooper Pairs ◽  
Ginzburg Landau ◽  
Fermion Systems ◽  
Scalar And Vector Fields ◽  
Ginzburg Landau Equations ◽  
Superconducting Pairing

The superconducting pairing of fermions is studied in the framework of the functional intergral approach. The bi-local composite scalar and vector fields are introduced to describe the singlet and triplet pairings. The static (time-independent) fields are the superconducting order parameters. From the variational principle for the effective action of the composite fields we derive the generalised Ginzburg–Landau equations. They are also the extensions of the BCS gap equations.

scholarly journals The Dirichlet problem for the Jacobian equation in critical and supercritical Sobolev spaces

2021 ◽  
Vol 60 (1) ◽  
Author(s):  
André Guerra ◽  
Lukas Koch ◽  
Sauli Lindberg
Keyword(s):  
Dirichlet Problem ◽  
Sobolev Spaces ◽  
Regularity Of Solutions ◽  
Jacobian Equation ◽  
L Log L

AbstractWe study existence and regularity of solutions to the Dirichlet problem for the prescribed Jacobian equation, $$\det D u =f$$ det D u = f , where f is integrable and bounded away from zero. In particular, we take $$f\in L^p$$ f ∈ L p , where $$p>1$$ p > 1 , or in $$L\log L$$ L log L . We prove that for a Baire-generic f in either space there are no solutions with the expected regularity.

PATTERNS, DEFECTS, AND EVOLUTION OF BÉNARD–MARANGONI CELLS

1996 ◽  
Vol 06 (09) ◽  
pp. 1665-1671 ◽  
Author(s):  
J. BRAGARD ◽  
J. PONTES ◽  
M.G. VELARDE
Keyword(s):  
Fluid Layer ◽  
Ambient Air ◽  
Finite Geometry ◽  
Amplitude Equations ◽  
Ginzburg Landau ◽  
Weakly Nonlinear ◽  
Numerical Exploration ◽  
Ginzburg Landau Equations ◽  
Rigid Plane ◽  
Selection Of

We consider a thin fluid layer of infinite horizontal extent, confined below by a rigid plane and open above to the ambient air, with surface tension linearly depending on the temperature. The fluid is heated from below. First we obtain the weakly nonlinear amplitude equations in specific spatial directions. The procedure yields a set of generalized Ginzburg–Landau equations. Then we proceed to the numerical exploration of the solutions of these equations in finite geometry, hence to the selection of cells as a result of competition between the possible different modes of convection.

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