Page 669 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 669
PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS
the lubrication model; if the fluid pressure is not dominant (that is, it is of order O(1)), and the
tractions are known on the upper and lower surfaces, we must use the shallow water model. In
the first case we will say that the fluid is “driven by the pressure” and in the second that it is
“driven by the velocity”.
References
[1] P. Azérad, F. Guillén; Mathematical justification of the hydrostatic approximation in the
primitive equations of geophysical fluid dynamics, SIAM J. Math. Anal. 33(4) (2001), pp.
847–859, https://doi.org/10.1137/S0036141000375962.
[2] G. Bayada, M. Chambat; The Transition Between the Stokes Equations and the
Reynolds Equation: A Mathematical Proof, Appl. Math. Optim. 14 (1986), pp. 73–93,
https://doi.org/10.1007/BF01442229.
[3] J. M. Rodríguez, R. Taboada-Vázquez; Bidimensional shallow water model with polyno-
mial dependence on depth through vorticity, Journal of Mathematical Analysis and Appli-
cations 359(2) (2009), pp. 556-569, https://doi.org/10.1016/j.jmaa.2009.06.003.
[4] J. M. Rodríguez, R. Taboada-Vázquez; Derivation of a new asymptotic viscous shallow
water model with dependence on depth, Applied Mathematics and Computation 219(7)
(2012), pp. 3292-3307, https://doi.org/10.1016/j.amc.2011.08.053.
[5] J. M. Rodríguez, R. Taboada-Vázquez; Asymptotic analysis of a thin fluid layer flow be-
tween two moving surfaces, arXiv:2101.07862 (https://arxiv.org/abs/2101.07862), 2021,
sent to Journal of Mathematical Analysis and Applications.
Method of energy estimates for studying of singular boundary regimes in
quasilinear parabolic equations
Yevgeniia Yevgenieva, yevgeniia.yevgenieva@gmail.com
Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Ukraine
In the cylindrical domain Q = (0, T ) × Ω, 0 < T < ∞, where Ω ⊂ Rn is a bounded domain
such that ∂Ω ∈ C2, the following problem is considered:
(|u|q−1u)t − ∆pu = 0, p q > 0, (1)
u(0, x) = u0 in Ω, u0 ∈ Lq+1(Ω),
u(t, x) = f (t, x),
∂Ω
where f generates boundary regime with singular peaking, namely,
f (t, x) → ∞ as t → T, ∀ x ∈ K ⊂ ∂Ω, K = ∅. (2)
Function f is called a localized boundary regime (S-regime) if
Ω \ Ω0 = ∅, where Ω0 := x ∈ Ω : sup u(t, x) = ∞
t→T
for an arbitrary weak solution u of problem (1). Sharp conditions of localization of boundary
regime were obtained by some version of local energy estimates (see [1] and references therein).
667
the lubrication model; if the fluid pressure is not dominant (that is, it is of order O(1)), and the
tractions are known on the upper and lower surfaces, we must use the shallow water model. In
the first case we will say that the fluid is “driven by the pressure” and in the second that it is
“driven by the velocity”.
References
[1] P. Azérad, F. Guillén; Mathematical justification of the hydrostatic approximation in the
primitive equations of geophysical fluid dynamics, SIAM J. Math. Anal. 33(4) (2001), pp.
847–859, https://doi.org/10.1137/S0036141000375962.
[2] G. Bayada, M. Chambat; The Transition Between the Stokes Equations and the
Reynolds Equation: A Mathematical Proof, Appl. Math. Optim. 14 (1986), pp. 73–93,
https://doi.org/10.1007/BF01442229.
[3] J. M. Rodríguez, R. Taboada-Vázquez; Bidimensional shallow water model with polyno-
mial dependence on depth through vorticity, Journal of Mathematical Analysis and Appli-
cations 359(2) (2009), pp. 556-569, https://doi.org/10.1016/j.jmaa.2009.06.003.
[4] J. M. Rodríguez, R. Taboada-Vázquez; Derivation of a new asymptotic viscous shallow
water model with dependence on depth, Applied Mathematics and Computation 219(7)
(2012), pp. 3292-3307, https://doi.org/10.1016/j.amc.2011.08.053.
[5] J. M. Rodríguez, R. Taboada-Vázquez; Asymptotic analysis of a thin fluid layer flow be-
tween two moving surfaces, arXiv:2101.07862 (https://arxiv.org/abs/2101.07862), 2021,
sent to Journal of Mathematical Analysis and Applications.
Method of energy estimates for studying of singular boundary regimes in
quasilinear parabolic equations
Yevgeniia Yevgenieva, yevgeniia.yevgenieva@gmail.com
Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Ukraine
In the cylindrical domain Q = (0, T ) × Ω, 0 < T < ∞, where Ω ⊂ Rn is a bounded domain
such that ∂Ω ∈ C2, the following problem is considered:
(|u|q−1u)t − ∆pu = 0, p q > 0, (1)
u(0, x) = u0 in Ω, u0 ∈ Lq+1(Ω),
u(t, x) = f (t, x),
∂Ω
where f generates boundary regime with singular peaking, namely,
f (t, x) → ∞ as t → T, ∀ x ∈ K ⊂ ∂Ω, K = ∅. (2)
Function f is called a localized boundary regime (S-regime) if
Ω \ Ω0 = ∅, where Ω0 := x ∈ Ω : sup u(t, x) = ∞
t→T
for an arbitrary weak solution u of problem (1). Sharp conditions of localization of boundary
regime were obtained by some version of local energy estimates (see [1] and references therein).
667
