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Phương pháp Front -Tracking cho định luật bảo toàn doc
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•
u
t
+ H(u)
x
= 0 (x, t) ∈ R × (0, ∞).
•
v
t
+ H(v
x
) = 0 (x, t) ∈ R × (0, ∞),
v(x, 0) = v
0
(x) x ∈ R.
u ∈ L
∞
(R × (0, ∞))
u
t
+ H(u)
x
= 0 (x, t) ∈ R × (0, ∞),
u(x, 0) = u
0
(x) x ∈ R,
∞
−∞
∞
0
(uϕ
t
+ H(u)ϕ
x
)dxdt +
∞
−∞
u
0
(x)ϕ(x, 0)dx = 0.
ϕ(x, t) ∈ C
1
0
(R × [0, ∞))
u C
1
u
∞
−∞
∞
0
(uϕ
t
+ H(u)ϕ
x
)dxdt = 0,
ϕ(x, t) ∈ C
1
0
(R × (0, ∞))
C (x, t)
u C
1
u
H(u
l
) − H(u
r
) =
˙s(u
l
− u
r
) C,
x = s(t) C
u
l,r
= lim
x→s(t)
−,+
u(x, t)
C
σ u
l
, u
r
, H(u
l
), H(u
r
)
C
[H(u)]
= H(u
l
) − H(u
r
) σ
[u]
=
˙s(u
l
− u
r
)
u ∈ L
∞
(R × [0, ∞))
u
∞
−∞
∞
0
(|u − k|ϕ
t
+ sign(u − k)(H(u) − H(k))ϕ
x
)dxdt+
+
∞
−∞
|u
0
(x) − k|ϕ(x, 0)dx ≥ 0,
ϕ(x, t) ∈ C
∞
0
(R × [0, ∞)), ϕ ≥ 0 k ∈ R
R×(0, T ]
∞
−∞
T
0
(|u − k|ϕ
t
+ sign(u − k)(H(u) − H(k))ϕ
x
)dxdt+
+
∞
−∞
|u
0
(x) − k|ϕ(x, 0)dx −
∞
−∞
|u(x, T ) − k|ϕ(x, T )dx ≥ 0
ϕ(x, t) ∈ C
∞
0
(R × [0, T ]), ϕ ≥ 0 k ∈ R
u
t
+ divH(u) = 0 (x, t) ∈ Ω × (0, T )
u(x, 0) = u
0
(x) x ∈ Ω,
u(x, t) = r(x, t)
∂Ω × (0, T )
H = (H
1
, H
2
, , H
n
) T > 0
sign(γu(x, t)−k)(H(γu(x, t))−H(k)).n(x) ≥ 0, ∀k ∈ I(r(x, t), γu(x, t)),
(x, t) ∈ ∂Ω×(0, T ) I(α, β)
α β γu L
1
u
u(x, t) ∈ L
∞
(Ω × (0, ∞))
u
L
ϕ
(u) =
Ω
T
0
[|u − k|ϕ
t
+ sign(u − k)(H(u) − H(k)).∇
x
ϕ]dxdt−
−
∂Ω
T
0
[sign(r − k)(H(γu) − H(k))ϕ].n(x)dxdt+
+
Ω
|u
0
(x) − k|ϕ(x, 0)dx ≥ 0,
ϕ(x, t) ∈ C
∞
0
(Ω × [0, T )), ϕ ≥ 0 k ∈ R
u
u
t
+ H(u
x
) = 0 x ∈ R, t > 0;
u(x, 0) = u
0
(x) x ∈ R.
u = g R × {t = 0},
u ϕ ∈ C
∞
R × (0, ∞)
u − ϕ (x
0
, t
0
) ∈ R × (0, ∞)
ϕ
t
(x
0
, t
0
) + H(ϕ
x
(x
0
, t
0
)) ≤ 0,
u ϕ ∈ C
∞
R × (0, ∞)
u −ϕ (x
0
, t
0
) ∈ R
n
× (0, ∞)
ϕ
t
(x
0
, t
0
) + H(ϕ
x
(x
0
, t
0
)) ≥ 0.
{u
n
(x)}
n
(a, b) C > 0
n
||u
n
||
L
∞
≤ C T V (u
n
) ≤ C
{u
n
k
(x)} {u
n
(x)}
n
u
n
k
(x) → u(x) hi k → ∞, ∀x ∈ (a, b).
u (a, b)
T V (u) ≤ lim inf
n→∞
T V (u
n
).
{u
n
(x, t)}
n
(a, b) × [0, ∞)
C > 0 n
||u
n
(., t)||
L
∞
≤ C; T V (u
n
(., t)) ≤ C, ∀t ∈ [0, ∞)
||u
n
(., t) − u
n
(., s)||
L
1
≤ C|t − s|, ∀t, s ∈ [0, ∞)
{u
n
k
(x, t)} {u
n
(x, t)}
n
u
n
k
(x, t) → u(x, t) hi k → ∞, ∀(x, t) ∈ (a, b) × [0, ∞);
u
n
k
(., t) → u(., t) rong L
1
loc
(a, b) hi k → ∞, ∀t ∈ [0, ∞).
u x
(a, b)
||u(., t)||
L
∞
≤ C; T V (u(., t)) ≤ C, ∀t ∈ [0, ∞)
||u(., t) − u(., s)||
L
1
≤ C|t − s|, ∀t, s ∈ [0, ∞).
u
t
+ H(u)
x
= 0 (x, t) ∈ R × (0, T ]
u(x, 0) = u
0
(x) x ∈ R,
H u
0
H
H
u
0
u
0
(x) =
u
l
x < 0,
u
r
x ≥ 0.
H
∗
H u
l
u
r
H
∗
(u; u
l
, u
r
) := sup
g
g(u)| g(u) g(u) ≤ H(u), ∀u u
l
u
r
.
H
∗
(u; u
l
, u
r
) := inf
g
g(u)| g(u) g(u) ≥ H(u), ∀u u
l
u
r
.
H u
l
u
r
H(u; u
l
, u
r
) =
H
∗
(u; u
l
, u
r
) u
l
< u
r
,
H
∗
(u; u
l
, u
r
) u
l
> u
r
.
H
H
u
1
= u
l
, u
N
= u
r
H N − 2 u
l
u
r
u
2
, , u
N−1
u
i
< u
i+1
u
l
< u
r
u
i
> u
i+1
u
l
> u
r
σ
0
= −∞, σ
N
= +∞
σ
i
=
∼
H(u
i+1
) −
∼
H(u
i
)
u
i+1
− u
i
=
H(u
i+1
) − H(u
i
)
u
i+1
− u
i
i = 1, , N − 1.
i = 1, , N Ω
i
Ω
i
=
(x, t) | 0 ≤ t ≤ T, tσ
i−1
< x ≤ tσ
i
.
u(x, t) = u
i
(x, t) ∈ Ω
i
,
u
u
u
t
+ H(u)
x
= 0, (x, t) ∈ R × (0, T ]; u(x, 0) = u
0
(x), x ∈ R,
u
0
u(x, t)
(x, t)
u
0
H
t > 0
u(x, t)
||H||
Lip
H
||u(., t)||
∞
≤ ||u
0
||
∞
t > 0
T V (u(., t)) ≤ T V (u
0
) t > 0
||u(., t) − u(., s)||
1
≤ ||H||
Lip
T V (u
0
)|t − s| s, t > 0
H
u
0
u
t
+ H(u)
x
= 0 (x, t) ∈ R × (0, ∞)
u(x, 0) = u
0
(x) x ∈ R,
H [m, M]
L u
0
L
1
(R) ∩ BV (R)
[m, M] u
0
H
u
u(x, t)
||u(., t)||
∞
≤ ||u
0
||
∞
t > 0
T V (u(., t)) ≤ T V (u
0
) t > 0
||u(., t) − u(., s)||
1
≤ ||H||
Lip
T V (u
0
)|t − s| s, t > 0
u
t
+ H(u)
x
= 0, (x, t) ∈ (a, b) × (0, T ],
u(x, 0) = u
0
(x) x ∈ (a, b),
u(a, t) = u
a
(t) t ∈ (0, T ),
u(b, t) = u
b
(t) t ∈ (0, T ),
H u
0
, u
a
u
b
(a, b)
sign(γu(a, t) − k)(H(γu(a, t)) − H(k)) ≤ 0, ∀k ∈ I(u
a
(t), γu(a, t)),
sign(γu(b, t) − k)(H(γu(b, t)) − H(k)) ≥ 0, ∀k ∈ I(u
b
(t), γu(b, t)).
u
a
(t), u
b
(t)
u
a
, u
b
u
0
(x)
H
u(x, 0) = u
0
(a
+
) x > a
u(a, t) = u
a
t ∈ (0, T ).
a
u(x, 0) =
u
a
x ≤ a,
u
0
(a
+
) x > a
(a, ∞).
x = a
u(x, 0) = u
0
(b
−
) x < b
u(b, t) = u
b
t ∈ (0, T ).
x = b
u(x, 0) =
u
0
(b
−
) x < b
u
b
x ≥ b
(−∞, b).
x = b
H(u)
[m, M] m, M
u
0
(x), u
a
(t), u
b
(t)
[m, M]
u
t
+ H(u)
x
= 0 (x, t) ∈ (a, b) × (0, T ),
u(x, 0) = u
0
(x) x ∈ (a, b),
u(a, t) = u
a
(t) t ∈ (0, T )
u(b, t) = u
b
(t) t ∈ (0, T )
u(t, x)
t
x u(x, t)
{u
0
(x), u
a
(t), u
b
(t)} ∪ { H}.
u
u x = a, b,
||u(x, t)||
∞
≤ max{||u
0
||
∞
, ||u
a
||
∞
, ||u
b
||
∞
}
T V (u(., t)) ≤ T V (u
0
) + |u
a
(t) − u
0
(a
+
)| + |u
b
(t) − u
0
(b
−
)|
