@article{63246,
abstract = {{Abstract
The hyperbolic-parabolic model
$$\begin{aligned} \left\{ \begin{array}{ll} u_{tt} = u_{xx} - \big (f(\Theta )\big )_x, \qquad & x\in \Omega , \ t>0, \\ \Theta _t = \Theta _{xx} - f(\Theta ) u_{xt}, \qquad & x\in \Omega , \ t>0, \end{array} \right. \end{aligned}$$
u
tt
=
u
xx
-
(
f
(
Θ
)
)
x
,
x
∈
Ω
,
t
>
0
,
Θ
t
=
Θ
xx
-
f
(
Θ
)
u
xt
,
x
∈
Ω
,
t
>
0
,
for the evolution of the displacement variable
u
and the temperature
$$\Theta \ge 0$$
Θ
≥
0
during thermoelastic interaction in a one-dimensional bounded interval
$$\Omega $$
Ω
is considered. Whereas the literature has provided comprehensive results on global solutions for sufficiently regular initial data
$$(u_0,u_{0t},\Theta _0)=(u,u_t,\Theta )|_{t=0}$$
(
u
0
,
u
0
t
,
Θ
0
)
=
(
u
,
u
t
,
Θ
)
|
t
=
0
when
$$f\equiv id$$
f
≡
i
d
, it seems to have remained open so far how far a solution theory can be built solely on the two fundamental physical principles of energy conservation and entropy nondecrease. The present manuscript addresses this by asserting global existence of weak solutions under assumptions which are energy- and entropy-minimal in the sense of allowing for any initial data
$$u_0\in W_0^{1,2}(\Omega )$$
u
0
∈
W
0
1
,
2
(
Ω
)
,
$$u_{0t} \in L^2(\Omega )$$
u
0
t
∈
L
2
(
Ω
)
and
$$0\le \Theta _0\in L^1(\Omega )$$
0
≤
Θ
0
∈
L
1
(
Ω
)
, and which apply to arbitrary
$$f\in C^1([0,\infty ))$$
f
∈
C
1
(
[
0
,
∞
)
)
with
$$f(0)=0$$
f
(
0
)
=
0
and
$$f'>0$$
f
′
>
0
on
$$[0,\infty )$$
[
0
,
∞
)
.
}},
author = {{Winkler, Michael}},
issn = {{0944-2669}},
journal = {{Calculus of Variations and Partial Differential Equations}},
number = {{1}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Rough solutions in one-dimensional nonlinear thermoelasticity}}},
doi = {{10.1007/s00526-025-03170-8}},
volume = {{65}},
year = {{2025}},
}
@article{53324,
author = {{Ahn, Jaewook and Winkler, Michael}},
issn = {{0944-2669}},
journal = {{Calculus of Variations and Partial Differential Equations}},
keywords = {{Applied Mathematics, Analysis}},
number = {{6}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{A critical exponent for blow-up in a two-dimensional chemotaxis-consumption system}}},
doi = {{10.1007/s00526-023-02523-5}},
volume = {{62}},
year = {{2023}},
}
@article{63267,
author = {{Ahn, Jaewook and Winkler, Michael}},
issn = {{0944-2669}},
journal = {{Calculus of Variations and Partial Differential Equations}},
number = {{6}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{A critical exponent for blow-up in a two-dimensional chemotaxis-consumption system}}},
doi = {{10.1007/s00526-023-02523-5}},
volume = {{62}},
year = {{2023}},
}
@article{34677,
author = {{Black, Tobias and Wu, Chunyan}},
issn = {{0944-2669}},
journal = {{Calculus of Variations and Partial Differential Equations}},
keywords = {{Applied Mathematics, Analysis}},
number = {{3}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Prescribed signal concentration on the boundary: eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation}}},
doi = {{10.1007/s00526-022-02201-y}},
volume = {{61}},
year = {{2022}},
}
@article{63305,
abstract = {{AbstractA no-flux initial-boundary value problem for the doubly degenrate parabolic system $$\begin{aligned} \left\{ \begin{array}{l} u_t = \nabla \cdot \big ( uv\nabla u\big ) + \ell uv, \\ v_t = \Delta v - uv, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$
u
t
=
∇
·
(
u
v
∇
u
)
+
ℓ
u
v
,
v
t
=
Δ
v
-
u
v
,
(
⋆
)
is considered in a smoothly bounded convex domain $$\Omega \subset \mathbb {R}^n$$
Ω
⊂
R
n
, with $$n\ge 1$$
n
≥
1
and $$\ell \ge 0$$
ℓ
≥
0
. The first of the main results asserts that for nonnegative initial data $$(u_0,v_0)\in (L^\infty (\Omega ))^2$$
(
u
0
,
v
0
)
∈
(
L
∞
(
Ω
)
)
2
with $$u_0\not \equiv 0$$
u
0
≢
0
, $$v_0\not \equiv 0$$
v
0
≢
0
and $$\sqrt{v_0}\in W^{1,2}(\Omega )$$
v
0
∈
W
1
,
2
(
Ω
)
, there exists a global weak solution (u, v) which, inter alia, belongs to $$C^0(\overline{\Omega }\times (0,\infty )) \times C^{2,1}(\overline{\Omega }\times (0,\infty ))$$
C
0
(
Ω
¯
×
(
0
,
∞
)
)
×
C
2
,
1
(
Ω
¯
×
(
0
,
∞
)
)
and satisfies $$\sup _{t>0} \Vert u(\cdot ,t)\Vert _{L^p(\Omega )}<\infty $$
sup
t
>
0
‖
u
(
·
,
t
)
‖
L
p
(
Ω
)
<
∞
for all $$p\in [1,p_0)$$
p
∈
[
1
,
p
0
)
with $$p_0:=\frac{n}{(n-2)_+}$$
p
0
:
=
n
(
n
-
2
)
+
. It is next seen that for each of these solutions one can find $$u_\infty \in \bigcap _{p\in [1,p_0)} L^p(\Omega )$$
u
∞
∈
⋂
p
∈
[
1
,
p
0
)
L
p
(
Ω
)
such that, within an appropriate topological setting, $$(u(\cdot ,t),v(\cdot ,t))$$
(
u
(
·
,
t
)
,
v
(
·
,
t
)
)
approaches the equilibrium $$(u_\infty ,0)$$
(
u
∞
,
0
)
in the large time limit. Finally, in the case $$n\le 5$$
n
≤
5
a result ensuring a certain stability property of any member in the uncountably large family of steady states $$(u_0,0)$$
(
u
0
,
0
)
, with arbitrary and suitably regular $$u_0:\Omega \rightarrow [0,\infty )$$
u
0
:
Ω
→
[
0
,
∞
)
, is derived. This provides some rigorous evidence for the appropriateness of ($$\star $$
⋆
) to model the emergence of a strikingly large variety of stable structures observed in experiments on bacterial motion in nutrient-poor environments. Essential parts of the analysis rely on the use of an apparently novel class of functional inequalities to suitably cope with the doubly degenerate diffusion mechanism in ($$\star $$
⋆
).}},
author = {{Winkler, Michael}},
issn = {{0944-2669}},
journal = {{Calculus of Variations and Partial Differential Equations}},
number = {{3}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar}}},
doi = {{10.1007/s00526-021-02168-2}},
volume = {{61}},
year = {{2022}},
}
@article{63359,
author = {{Wang, Yulan and Winkler, Michael and Xiang, Zhaoyin}},
issn = {{0944-2669}},
journal = {{Calculus of Variations and Partial Differential Equations}},
number = {{6}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{The fast signal diffusion limit in Keller–Segel(-fluid) systems}}},
doi = {{10.1007/s00526-019-1656-3}},
volume = {{58}},
year = {{2019}},
}