The North Alpine foreland basin (NAFB) in Germany is characterized by various types of sedimentologic features that make it an excellent terrestrial analog of regions affected by high-energy asteroid impact-quakes on Mars. Impact events have shaped all planetary bodies in the inner Solar System over the past >4 Gyr. The well-preserved Ries impact crater (Baden-Württemberg and Bavaria), formed around 14.8 Ma, has recently been linked to an earthquake-produced seismite horizon in Mid-Miocene NAFB sediments that exhibits typical dewatering structures and is associated with sand spikes, seismically produced pin-shaped pseudo-concretions. In this terrestrial setting, the sand spike tails systematically point away from the Ries crater. On its path across Gale Crater, the Mars rover Curiosity seems to have observed a similar seismite horizon in early Hesperian lacustrine deposits including clastic dikes, convolute bedding, and, likely, sand spikes. Their orientation suggests that the nearby Slagnos impact crater might be the seismic source for the formation of those seismites. The Ries impact–seismite deposits can be traced over a distance of more than 200 km from the source crater (northern Switzerland), which makes the NAFB an excellent terrestrial analog for similar deposits and their sedimentologic inventory within Gale Crater's lake deposits on Mars.
Citation: Elmar Buchner, Volker J Sach, Martin Schmieder. Ries impact deposits in the North Alpine Foreland Basin of Germany as a terrestrial analog site for impact-produced seismites and sand spikes on planet Mars[J]. AIMS Geosciences, 2025, 11(1): 68-90. doi: 10.3934/geosci.2025005
[1] | Khazan Sher, Muhammad Ameeq, Muhammad Muneeb Hassan, Basem A. Alkhaleel, Sidra Naz, Olyan Albalawi . Novel efficient estimators of finite population mean in stratified random sampling with application. AIMS Mathematics, 2025, 10(3): 5495-5531. doi: 10.3934/math.2025254 |
[2] | Tolga Zaman, Cem Kadilar . Exponential ratio and product type estimators of the mean in stratified two-phase sampling. AIMS Mathematics, 2021, 6(5): 4265-4279. doi: 10.3934/math.2021252 |
[3] | Olayan Albalawi . Estimation techniques utilizing dual auxiliary variables in stratified two-phase sampling. AIMS Mathematics, 2024, 9(11): 33139-33160. doi: 10.3934/math.20241582 |
[4] | Xiaoda Xu . Bounds of random star discrepancy for HSFC-based sampling. AIMS Mathematics, 2025, 10(3): 5532-5551. doi: 10.3934/math.2025255 |
[5] | Sohaib Ahmad, Sardar Hussain, Muhammad Aamir, Faridoon Khan, Mohammed N Alshahrani, Mohammed Alqawba . Estimation of finite population mean using dual auxiliary variable for non-response using simple random sampling. AIMS Mathematics, 2022, 7(3): 4592-4613. doi: 10.3934/math.2022256 |
[6] | Said G. Nassr, T. S. Taher, Tmader Alballa, Neema M. Elharoun . Reliability analysis of the Lindley distribution via unified hybrid censoring with applications in medical survival and biological lifetime data. AIMS Mathematics, 2025, 10(6): 14943-14974. doi: 10.3934/math.2025670 |
[7] | Sohail Ahmad, Moiz Qureshi, Hasnain Iftikhar, Paulo Canas Rodrigues, Mohd Ziaur Rehman . An improved family of unbiased ratio estimators for a population distribution function. AIMS Mathematics, 2025, 10(1): 1061-1084. doi: 10.3934/math.2025051 |
[8] | Abdullah Ali H. Ahmadini, Amal S. Hassan, Ahmed N. Zaky, Shokrya S. Alshqaq . Bayesian inference of dynamic cumulative residual entropy from Pareto Ⅱ distribution with application to COVID-19. AIMS Mathematics, 2021, 6(3): 2196-2216. doi: 10.3934/math.2021133 |
[9] | Jessica Lipoth, Yoseph Tereda, Simon Michael Papalexiou, Raymond J. Spiteri . A new very simply explicitly invertible approximation for the standard normal cumulative distribution function. AIMS Mathematics, 2022, 7(7): 11635-11646. doi: 10.3934/math.2022648 |
[10] | Yasir Hassan, Muhammad Ismai, Will Murray, Muhammad Qaiser Shahbaz . Efficient estimation combining exponential and ln functions under two phase sampling. AIMS Mathematics, 2020, 5(6): 7605-7623. doi: 10.3934/math.2020486 |
The North Alpine foreland basin (NAFB) in Germany is characterized by various types of sedimentologic features that make it an excellent terrestrial analog of regions affected by high-energy asteroid impact-quakes on Mars. Impact events have shaped all planetary bodies in the inner Solar System over the past >4 Gyr. The well-preserved Ries impact crater (Baden-Württemberg and Bavaria), formed around 14.8 Ma, has recently been linked to an earthquake-produced seismite horizon in Mid-Miocene NAFB sediments that exhibits typical dewatering structures and is associated with sand spikes, seismically produced pin-shaped pseudo-concretions. In this terrestrial setting, the sand spike tails systematically point away from the Ries crater. On its path across Gale Crater, the Mars rover Curiosity seems to have observed a similar seismite horizon in early Hesperian lacustrine deposits including clastic dikes, convolute bedding, and, likely, sand spikes. Their orientation suggests that the nearby Slagnos impact crater might be the seismic source for the formation of those seismites. The Ries impact–seismite deposits can be traced over a distance of more than 200 km from the source crater (northern Switzerland), which makes the NAFB an excellent terrestrial analog for similar deposits and their sedimentologic inventory within Gale Crater's lake deposits on Mars.
Let
Oε={x=(x∗,xn+1)|x∗=(x1,…,xn)∈Qand0<xn+1<εg(x∗)}, |
where
γ1≤g(x∗)≤γ2,∀x∗∈¯Q. | (1) |
Denote
{dˆuε−Δˆuεdt=(H(t,x,ˆuε(t))+G(t,x))dt+m∑j=1cjˆuε∘dwj,x∈Oε,t>τ,∂ˆuε∂νε=0,x∈∂Oε, | (2) |
with the initial condition
ˆuε(τ,x)=ˆϕε(x),x∈Oε, | (3) |
where
As
{du0−1gn∑i=1(gu0yi)yidt=(H(t,(y∗,0),u0(t))+G(t,(y∗,0)))dt+m∑j=1cju0∘dwj,y∗=(y1,…,yn)∈Q,t>τ,∂u0∂ν0=0,y∗∈∂Q, | (4) |
with the initial condition
u0(τ,y∗)=ϕ0(y∗),y∗∈Q, | (5) |
where
Random attractors have been investigated in [2,5,10,19,9] in the autonomous stochastic case, and in [3,21,22,23] in the non-autonomous stochastic case. Recently, the limiting dynamical behavior of stochastic partial differential equations on thin domain was studied in [16,20,13,14,11,12,17,4]. However, in [17,13], we only investigated the limiting behavior of random attractors in
Let
We organize the paper as follows. In the next section, we establish the existence of a continuous cocycle in
Here we show that there is a continuous cocycle generated by the reaction-diffusion equation defined on
{dˆuε−Δˆuεdt=(H(t,x,ˆuε(t))+G(t,x))dt+m∑j=1cjˆuε∘dwj,x=(x∗,xn+1)∈Oε,t>τ,∂ˆuε∂νε=0,x∈∂Oε, | (6) |
with the initial condition
ˆuετ(x)=ˆϕε(x),x∈Oε, | (7) |
where
H(t,x,s)s≤−λ1|s|p+φ1(t,x), | (8) |
|H(t,x,s)|≤λ2|s|p−1+φ2(t,x), | (9) |
∂H(t,x,s)∂s≤λ3, | (10) |
|∂H(t,x,s)∂x|≤ψ3(t,x), | (11) |
where
Throughout this paper, we fix a positive number
h(t,x,s)=H(t,x,s)+λs | (12) |
for all
h(t,x,s)s≤−α1|s|p+ψ1(t,x), | (13) |
|h(t,x,s)|≤α2|s|p−1+ψ2(t,x), | (14) |
∂h(t,x,s)∂s≤β, | (15) |
|∂h(t,x,s)∂x|≤ψ3(t,x), | (16) |
where
Substituting (12) into (6) we get for
{dˆuε−(Δˆuε−λˆuε)dt=(h(t,x,ˆuε(t))+G(t,x))dt+m∑j=1cjˆuε∘dwj,x=(x∗,xn+1)∈Oε,∂ˆuε∂νε=0,x∈∂Oε, | (17) |
with the initial condition
ˆuετ(x)=ˆϕε(x),x∈Oε. | (18) |
We now transfer problem (17)-(18) into an initial boundary value problem on the fixed domain
x∗=y∗,xn+1=εg(y∗)yn+1. |
It follows from [18] that the Laplace operator in the original variable
Δxˆu(x)=|J|divy(|J|−1JJ∗∇yu(y))=1gdivy(Pεu(y)), |
where we denote by
Pεu(y)=(guy1−gy1yn+1uyn+1⋮guyn−gynyn+1uyn+1−n∑i=1yn+1gyiuyi+1ε2g(1+n∑i=1(εyn+1gyi)2)uyn+1). |
In the sequel, we abuse the notation a little bit by writing
Fε(t,y∗,yn+1,s)=F(t,y∗,εg(y∗)yn+1,s),F0(t,y∗,s)=F(t,y∗,0,s), |
where
{duε−(1gdivy(Pεuε)−λuε)dt=(hε(t,y,uε(t))+Gε(t,y))dt+m∑j=1cjuε∘dwj,y=(y∗,yn+1)∈O,Pεuε⋅ν=0,y∈∂O, | (19) |
with the initial condition
uετ(y)=ϕε(y)=ˆϕε∘T−1ε(y),y∈O, | (20) |
where
Given
θ1,t(τ)=τ+t,for allτ∈R. | (21) |
Then
Ω={ω∈C(R,R):ω(0)=0}. |
Let
θtω(⋅)=ω(⋅+t)−ω(t),ω∈Ω,t∈R. | (22) |
Then
dz+αzdt=dw(t), | (23) |
for
Lemma 2.1. There exists a
limt→±∞|ω(t)|t=0for allω∈Ω′, |
and, for such
z∗(ω)=−α∫0−∞eαsω(s)ds |
is well defined. Moreover, for
(t,ω)→z∗(θtω)=−α∫0−∞eαsθtω(s)ds=−α∫0−∞eαsω(t+s)ds+ω(t) |
is a stationary solution of (23) with continuous trajectories. In addition, for
limt→±∞|z∗(θtω)|t=0,limt→±∞1t∫t0z∗(θsω)ds=0, | (24) |
limt→±∞1t∫t0|z∗(θsω)|ds=E|z∗|<∞. | (25) |
Denote by
˜Ω=Ω′1×⋯×Ω′mand F=m⊗j=1Fj, |
Then
Denote by
SCj(t)u=ecjtu,foru∈L2(O), |
and
T(ω):=SC1(z∗1(ω))∘⋯∘SCm(z∗m(ω))=em∑j=1cjz∗j(ω)IdL2(O),ω∈Ω′. |
Then for every
T−1(ω):=SCm(−z∗m(ω))∘⋯∘SC1(−z∗1(ω))=e−m∑j=1cjz∗j(ω)IdL2(O). |
It follows that
On the other hand, since
limt→±∞1t∫t0‖T(θτω)‖2dτ=E‖T‖2=m∏j=1E(e2cjz∗j)<∞, |
and
limt→±∞1t∫t0‖T−1(θτω)‖2dτ=E‖T−1‖2=m∏j=1E(e−2cjz∗j)<∞. |
Remark 1. We now consider
Next, we define a continuous cocycle for system (19)-(20) in
{dvεdt−1gdivy(Pεvε)=(−λ+δ(θtω))vε+T−1(θtω)hε(t,y,T(θtω)vε(t))+T−1(θtω)Gε(t,y),y∈O,t>τ,Pεvε⋅ν=0,y∈∂O, | (26) |
with the initial conditions
vετ(y)=ψε(y),y∈O, | (27) |
where
Since (26) is a deterministic equation, by the Galerkin method, one can show that if
Φε(t,τ,ω,ϕε)=uε(t+τ,τ,θ−τω,ϕε)=T(θt+τω)vε(t+τ,τ,θ−τω,ψε),for all(t,τ,ω,ϕε)∈R+×R×Ω×N. | (28) |
By the properties of
Let
(Rεˆϕε)(y)=ˆϕε(T−1εy),∀ˆϕε∈L2(Oε). |
Given
ˆΦε(t,τ,ω,ˆϕε)=R−1εΦε(t,τ,ω,Rεˆϕε). |
The same change of unknown variable
{dv0dt−n∑i=11g(gv0yi)yi=(−λ+δ(θtω))v0+T−1(θtω)h0(t,y∗,T(θtω)v0(t))+T−1(θtω)G0(t,y∗),y∗∈Q,t>τ,∂v0∂ν0=0,y∗∈∂Q, | (29) |
with the initial conditions
v0τ(y∗)=ψ0(y∗),y∗∈Q, | (30) |
where
The same argument as above allows us to prove that problem (4) and (5) generates a continuous cocycle
Now we want to write equation (26)-(27) as an abstract evolutionary equation. We introduce the inner product
(u,v)Hg(O)=∫Oguvdy,for allu,v∈N |
and denote by
For
aε(u,v)=(J∗∇yu,J∗∇yv)Hg(O), | (31) |
where
J∗∇yu=(uy1−gy1gyn+1uyn+1,…,uyn−gyngyn+1uyn+1,1εguyn+1). |
By introducing on
‖u‖H1ε(O)=(∫O(|∇y∗u|2+|u|2+1ε2u2yn+1)dy)12, | (32) |
we see that there exist positive constants
η1∫O(|∇y∗u|2+1ε2u2yn+1)dy≤aε(u,u)≤η2∫O(∇y∗u|2+1ε2u2yn+1)dy | (33) |
and
η1‖u‖2H1ε(O)≤aε(u,u)+‖u‖2L2(O)≤η2‖u‖2H1ε(O). | (34) |
Denote by
D(Aε)={v∈H2(O),Pεv⋅ν=0on∂O} |
as defined by
Aεv=−1gdivPεv,v∈D(Aε). |
Then we have
aε(u,v)=(Aεu,v)Hg(O),∀u∈D(Aε),∀v∈H1(O). | (35) |
Using
{dvεdt+Aεvε=(−λ+δ(θtω))vε+T−1(θtω)hε(t,y,T(θtω)vε(t))+T−1(θtω)Gε(t,y),y∈O,t>τ,vετ=ψε. | (36) |
To reformulate system (29)-(30), we introduce the inner product
(u,v)Hg(Q)=∫Qguvdy∗,for allu,v∈M, |
and denote by
a0(u,v)=∫Qg▽y∗u⋅▽y∗vdy∗. |
Denote by
D(A0)={v∈H2(Q),∂v∂ν0=0on∂Q} |
as defined by
A0v=−1gn∑i=1(gvyi)yiv∈D(A0). |
Then we have
a0(u,v)=(A0u,v)Hg(Q),∀u∈D(A0),∀v∈H1(Q). |
Using
{dv0dt+A0v0=(−λ+δ(θtω))v0+T−1(θtω)h0(t,y∗,T(θtω)v0(t))+T−1(θtω)G0(t,y∗),y∗∈Q,t>τ,v0τ(s)=ψ0(s),s∈[−ρ,0]. | (37) |
Hereafter, we set
limt→−∞ect‖Bi(τ+t,θtω)‖Xi=0, |
where
Di={Bi={Bi(τ,ω):τ∈R,ω∈Ω}:Bi is tempered in Xi}. |
Our main purpose of the paper is to prove that the cocycle
limε→0supuε∈ˆAεinfu0∈A0ε−1‖uε−u0‖2H1(Oε)=0. | (38) |
To prove (38), we only need to show that the cocycle
limε→0distH(Aε(τ,ω),A0(τ,ω))=0, |
which will be established in the last section of the paper.
Furthermore, we suppose that there exists
¯γΔ=λ0−2E(|δ(ω)|)>0. | (39) |
Let us consider the mapping
γ(ω)=λ0−2|δ(ω)|. | (40) |
By the ergodic theory and (39) we have
limt→±∞1t∫t0γ(θlω)dl=Eγ=¯γ>0. | (41) |
The following condition will be needed when deriving uniform estimates of solutions:
∫τ−∞e12¯γs(‖G(s,⋅)‖2L∞(˜O)+‖φ1(s,⋅)‖2L∞(˜O)+‖ψ3(s,⋅)‖2L∞(˜O))ds<∞,∀τ∈R. | (42) |
When constructing tempered pullback attractors, we will assume
limr→−∞eσr∫0−∞e12¯γs(‖G(s+r,⋅)‖2L∞(˜O)+‖φ1(s+r,⋅)‖2L∞(˜O)+‖ψ3(s+r,⋅)‖2L∞(˜O))ds=0,∀σ>0. | (43) |
Since
∫τ−∞e12¯γs(‖G(s,⋅)‖2L∞(˜O)+‖ψ1(s,⋅)‖L∞(˜O)+‖ψ3(s,⋅)‖2L∞(˜O))ds<∞,∀τ∈R | (44) |
and
limr→−∞eσr∫0−∞e12¯γs(‖G(s+r,⋅)‖2L∞(˜O)+‖ψ1(s+r,⋅)‖2L∞(˜O)+‖ψ3(s+r,⋅)‖2L∞(˜O))ds=0, | (45) |
for any
In this section, we recall and generalize some results in [17] and derive some new uniform estimates of solutions of problem (36) or (19)-(20) which are needed for proving the existence of
Lemma 3.1. Assume that (8)-(11), (39) and (42) hold. Then for every
sup−1≤s≤0‖vε(τ+s,τ−t,θ−τω,ψε)‖2H1ε(O)≤R2(τ,ω), | (46) |
where
R2(τ,ω)=r1(ω)R1(τ,ω)+c∫0−∞e¯γr‖T−1(θrω)‖2(‖G(r+τ,⋅)‖2L∞(˜O)+‖ψ3(r+τ,⋅)‖2L∞(˜O))dr, | (47) |
where
R1(τ,ω)=c∫0−∞e∫r0γ(θlω)dl‖T−1(θrω)‖2‖G(r+τ,⋅)‖2L∞(˜O)dr+c∫0−∞e∫r0γ(θlω)dl‖T−1(θrω)‖2‖ψ1(r+τ,⋅)‖2L∞(˜O)dr, | (48) |
and
Proof. The proof is similar as that of Lemma 3.4 in [17], so we only sketch the proof here. Taking the inner product of (36) with
12ddt‖vε‖2Hg(O)≤−aε(vε,vε)+(−λ0+δ(θtω))‖vε‖2Hg(O)+(T−1(θtω)hε(t,y,T(θtω)vε(t)),vε)Hg(O)+(T−1(θtω)Gε(t,y),vε)Hg(O). | (49) |
By (13), we have
ddt‖vε‖2Hg(O)+2aε(vε,vε)+λ02‖vε‖2Hg(O)+2α1γ1‖T−1(θtω)‖2‖uε‖pLp(O)≤(−λ0+2δ(θtω))‖vε‖2Hg(O)+2λ0γ2|˜O|‖T−1(θtω)‖2‖G(t,⋅)‖2L∞(˜O)+2γ2|˜O|‖T−1(θtω)‖2‖ψ1(t,⋅)‖L∞(˜O). | (50) |
Then, we have for any
e∫στγ(θlω)dl‖vε(σ)‖2Hg(O)+2∫στe∫rτγ(θlω)dlaε(vε(r),vε(r))dr+λ02∫στe∫rτγ(θlω)dl‖vε(r)‖2Hg(O)dr+2α1γ1∫στ‖T−1(θrω)‖2e∫rτγ(θlω)dl‖uε(r)‖pLp(O)dr≤‖vε(τ)‖2Hg(O)+2λ0γ2|˜O|∫στe∫rτγ(θlω)dl‖T−1(θrω)‖2‖G(r,⋅)‖2L∞(˜O)dr+2γ2|˜O|∫στe∫rτγ(θlω)dl‖T−1(θrω)‖2‖ψ1(r,⋅)‖2L∞(˜O)dr, | (51) |
where
Thus by the similar arguments as Lemma 3.1 in [17] we get for every
‖vετ(⋅,τ−t,θ−τω,ψ)‖2L2(O)≤c∫0−∞e∫r0γ(θlω)dl‖ψ1(r+τ,⋅)‖2L∞(˜O)dr+c∫0−∞e∫r0γ(θlω)dl‖T−1(θrω)‖2‖G(r+τ,⋅)‖2L∞(˜O)dr+c∫0−∞e∫r0γ(θlω)dl‖T−1(θrω)‖2‖ψ1(r+τ,⋅)‖2L∞(˜O)dr. | (52) |
Moreover, taking the inner product of (36) with
12ddtaε(vε,vε)+‖Aεvε‖2Hg(O)≤(−λ0+δ(θtω))aε(vε,vε)+(T−1(θtω)hε(t,y,T(θtω)vε(t)),Aεvε)Hg(O)+(T−1(θtω)Gε(t,y),Aεvε)Hg(O). | (53) |
By (15)-(16) we have
ddtaε(vε,vε)+‖Aεvε‖2Hg(O)≤(c+2δ(θtω))aε(vε,vε)+c‖T−1(θtω)‖2(‖G(t,⋅)‖2L∞(˜O)+‖ψ3(t,⋅)‖2L∞(˜O)), | (54) |
The left proof is similar of that Lemma 3.4 in [17], so we omit it here.
We are now in a position to establish the uniform estimates for the solution
Lemma 3.2. Assume that (8)-(11), (39) and (42) hold. Then for every
sup−1≤s≤0‖uε(τ+s,τ−t,θ−τω,ϕε)‖2H1ε(O)≤r2(ω)R2(τ,ω), | (55) |
where
Lemma 3.3. Assume that (8)-(11), (39) and (42) hold. Then for every
sup−1≤s≤0‖vε(τ+s,τ−t,θ−τω,ψε)‖pLp(O)+∫ττ−ρ‖vε(r,τ−t,θ−τω,ψε)‖2p−2L2p−2(O)dr≤R3(τ,ω), | (56) |
where
Proof. The proof is similar as that of Lemma 3.6 in [14], so we omit it here.
Lemma 3.4. Assume that (8)-(11), (39) and (42) hold. Then for every
∫0−1eγMp−2s∫{y∈O: vε(s+τ,τ−t,θ−τω,ψε)≥2M}|vε(s+τ,τ−t,θ−τω,ψε)|2p−2dyds≤η, | (57) |
∫0−1eγMp−2s∫{y∈O: vε(s+τ,τ−t,θ−τω,ψε)≤−2M}|vε(s+τ,τ−t,θ−τω,ψε)|2p−2dyds≤η. | (58) |
Proof. Let
1pddt‖(vε−M)+‖pLp(O)+(p−1)∫vε≥M(vε−M)p−2aε(vε,vε)dx≤(δ(θtω)vε,(vε−M)p−1+)+(T−1(θtω)hε(t,y,T(θtω)vε),(vε−M)p−1+)+(T−1(θtω)Gε(t,y),(vε−M)p−1+). | (59) |
For the first term on the right side of (59) we have
|(δ(θtω)vε,(vε−M)p−1+)|≤1p|δ(θrω)|p∫O|vε|pdx+p−1p∫O(vε−M)p+dx. | (60) |
For the second term on the right-hand side of (59), by (8), we obtain, for
hε(t,y,T(θtω)vε) (vε−M)p−1+≤−α1‖T(θtω)‖p−1(vε)p−1(v−M)p−1+ |
+‖T(θtω)‖−1ψ1(t,y∗,εg(y∗)yn+1)(vε)−1(vε−M)p−1+ |
≤−12α1Mp−2‖T(θtω)‖p−1(vε−M)p+−12α1‖T(θtω)‖p−1(vε−M)2p−2+ |
+‖T−1(θtω)‖−1|ψ1(t,y∗,εg(y∗)yn+1)|(vε−M)p−2+ |
which implies
(T−1(θtω)hε(t,y,T(θtω)vε), (vε−M)p−1+) |
≤−12α1Mp−2‖T(θtω)‖p−2∫O(vε−M)p+dx−12α1‖T−1(θtω)‖p−2∫O(vε−M)2p−2+dx |
+‖T(θtω)‖−2∫O|ψ1(t,y∗,εg(y∗)yn+1)|(vε−M)p−2+dx |
≤−12α1Mp−2‖T(θtω)‖p−2∫O(vε−M)p+dx−12α1‖T(θtω)‖p−2∫O(vε−M)2p−2+dx |
+p−2p∫O(vε−M)p+dx+2p‖T(θtω)‖−p∫O|ψ1(t,y∗,εg(y∗)yn+1)|p2dy. | (61) |
The last term in (59) is bounded by
(T−1(θtω)Gε(t,y),(vε−M)p−1+)≤18α1‖T(θtω)‖p−2∫O(vε−M)2p−2+dx+2α1‖T(θtω)‖−p∫vε≥M|Gε(t,y)|2dy. | (62) |
All above estimates yield
ddt‖(vε−M)+‖pLp(O)−(2p−3−12pα1Mp−2‖T(θtω)‖p−2)∫O(vε−M)p+dx+14pα1‖T(θtω)‖p−2∫O(vε−M)2p−2+dx≤|δ(θrω)|p∫O|vε|pdx+2‖T(θtω)‖−p∫O|ψ1(t,y∗,εg(y∗)yn+1)|p2dy+2pα1‖T(θtω)‖−p∫O|Gε(t,y)|2dy. | (63) |
Multiplying (63) by
‖(vε(τ,τ−t,ω,ψε)−M)+‖pLp(O) |
+14pα1∫ττ−1‖T(θζω)‖p−2e−∫ζτ(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr |
×∫O(vε(ζ,τ−t,ω,ψε)−M)2p−2+dxdζ |
≤e−∫τ−1τ(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖(vε(τ−1,τ−t,ω,ψε)−M)+‖pLp(O) |
+∫ττ−1|δ(θζω)|pe−∫ζτ(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖vε(ζ,τ−t,ω,ψε)‖pLp(O)dζ |
+2|O|∫ττ−1‖T(θζω)‖−pe−∫ζτ+ξ(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖ψ1(ζ,⋅)‖p2L∞(˜O)dζ. |
+2p|O|α1∫ττ−1‖T(θζω)‖−pe−∫ζτ+ξ(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖G(ζ,⋅)‖2L∞(˜O)dζ, | (64) |
where
14pα1∫0−1‖T(θζω)‖p−2e−∫ζ0(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr |
×∫O(vε(ζ+τ,τ−t,θ−τω,ψε)−M)2p−2+dxdζ |
≤e−∫−10(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖(vε(τ−1,τ−t,θ−τω,ψε)−M)+‖pLp(O) |
+∫0−1|δ(θζ+ξω)|pe−∫ζ0(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖vε(ζ+τ,τ−t,θ−τω,ψε)‖pLp(O)dζ |
+2|O|∫0−1‖T(θζω)‖−pe−∫ζ0(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖ψ1(ζ+τ,⋅)‖p2L∞(˜O)dζ. |
+2p|O|α1∫0−1‖T(θζω)‖−pe−∫ζ0(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖G(ζ+τ,⋅)‖2L∞(˜O)dζ. | (65) |
Since
c1≤12pα1‖T(θrω)‖p−2≤c2 for all r∈[−ρ−1,0]. | (66) |
By (66) we obtain
ec2Mp−2ζ≤e∫ζ+ξξ12pα1Mp−2‖T(θrω)‖p−2dr≤ec1Mp−2ζ for all ζ∈[−1,0]andξ∈[−ρ,0]. | (67) |
For the left-hand side of (65), by (67) we find that there exists
14pα1∫0−1‖T(θζω)‖p−2e−∫ζ0(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr |
∫O(vε(ζ+τ,τ−t,θ−τω,ψε)−M)2p−2+dxdζ |
≥c3∫0−1ec2Mp−2ζ∫O(vε(ζ+τ,τ−t,θ−τω,ψε)−M)2p−2+dxdζ. | (68) |
For the first term on the right-hand side of (65), by (67) we obtain
e−∫−10(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖(vε(τ−1,τ−t,θ−τω,ψε)−M)+‖pLp(O) |
≤e2p−3e−c1Mp−2‖(vε(τ−1,τ−t,θ−τω,ψε)−M)+‖pLp(O) |
≤e2p−3e−c1Mp−2‖vε(τ−1,τ−t,θ−τω,ψε)‖pLp(O). | (69) |
Similarly, for the second terms on the right-hand side of (65), we have from (67) there exists
∫0−1|δ(θζω)|pe−∫ζ0(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖vε(ζ+τ,τ−t,θ−τω,ψε)‖pLp(O)dζ |
≤c4∫0−1ec1Mp−2ζ‖vε(ζ+τ,τ−t,θ−τω,ψε)‖pLp(O)dζ | (70) |
Since
2|O|∫0−1‖T(θζω)‖−pe−∫ζ0(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖ψ1(ζ+τ,⋅)‖p2L∞(˜O)dζ. |
+2p|O|α1∫0−1‖T(θζω)‖−pe−∫ζ0(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖G(ζ+τ,⋅)‖2L∞(˜O)dζ |
≤c5∫0−1ec1Mp−2ζdζ≤c−11c5M2−p. | (71) |
By (68)-(71) we get from (65) that
c3∫0−1ec2Mp−2ζ∫O(vε(ζ+τ,τ−t,θ−τω,ψε)−M)2p−2+dydζ |
≤e2p−3e−c1Mp−2‖vε(τ−1,τ−t,θ−τω,ψε)‖pLp(O) |
+c4∫0−1ec1Mp−2ζ‖vε(ζ+τ,τ−t,θ−τω,ψε)‖pLp(O)dζ+c−11c5M2−p, |
which together with Lemma 3.2 and Lemma 3.3 implies that there exist
c3∫0−1ec2Mp−2ζ∫O(vε(ζ+τ,τ−t,θ−τω,ψε)−M)2p−2+dxdζ |
≤c6e−c1Mp−2+c6∫0−1ec1Mp−2ζdζ+c−11c5M2−p≤c6e−c1Mp−2+c−11(c5+c6)M2−p. | (72) |
Since
∫0−1ec2Mp−2ζ∫O(vε(ζ+τ,τ−t,θ−τω,ψε)−M)2p−2+dydζ≤η. | (73) |
Note that
∫0−1ec2Mp−2ζ∫{y∈O: vε(ζ+τ,τ−t,θ−τω,ψε)≥2M}|vε(ζ+τ,τ−t,θ−τω,ψε)|2p−2dydζ≤22p−2∫0−1ec2Mp−2ζ∫O(vε(ζ+τ,τ−t,θ−τω,ψε)−M)2p−2+dxdζ≤22p−2η. | (74) |
Similarly, one can verify that there exist
∫0−1ec2Mp−2ζ∫{y∈O: vε(ζ+τ,τ−t,θ−τω,ψε)≤−2M}|vε(ζ+τ,τ−t,θ−τω,ψε)|2p−2dydζ≤22p−2η. | (75) |
Then Lemma 3.4 follows from (3) and (75) immediately.
Note that
0≤λε1≤λε2≤…≤λεn≤⋯→+∞, |
and their associated eigenfunctions
It follows from Corollary 9.7 in [8] that the eigenvalues and the eigenfunctions of
Next, we introduce the spectral projections. We use
Pεn(u)=m∑i=1(u,ϖεi)Yεϖεiforu∈Yε. |
We use
aε(u,u)=(Aεu,u)Hg(O)≤λεn(u,u)Hg(O),∀u∈PεnD(A1/2ε). | (76) |
and
aε(u,u)=(Aεu,u)Hg(O)≥λεm+1(u,u)Hg(O),u∈QεmD(A1/2ε). | (77) |
Let
Lemma 3.5. Assume that (8)-(11), (39) and (42) hold. Then for every
‖uε2(τ,τ−t,θ−τω,ϕε)‖H1(O)≤η. |
Proof. Taking the inner product (36) with
12ddtaε(vε2,vε2)+‖Aεvε2‖2≤(δ(θtω)vε2,Aεvε2)+(QεnT−1(θtω)hε(t,y,T(θtω)vε),Aεvε2)+(QεnT−1(θtω)Gε(t,y),Aεvε2). | (78) |
For the first term on the right-hand side of (78), we have
(δ(θtω)vε2,Aεvε2)≤18‖Aεvε2‖2+2|δ(θtω)|2‖vε2‖2. | (79) |
For the superlinear term, we have from (9) that
(QεnT−1(θtω)hε(t,y,T(θtω)vε),Aεvε2)≤18‖Aεvε2‖2+2‖T−1(θtω)‖2∫O|hε(t,y,T(θtω)vε)|2dy≤18‖Aεvε2‖2+2α2‖T−1(θtω)‖2∫O(|T(θtω)vε|p−1+ψ2(t,y∗,εg(y∗)yn+1))2dy≤18‖Aεvε2‖2+4α2‖T(θtω)‖2p−4‖v‖2p−22p−2+4α2|O|‖T−1(θtω)‖2‖ψ2(t,⋅)‖2L∞(˜O). | (80) |
For the last term on the right-hand side of (78), we have
(QεnT−1(θtω)Gε(t,y),Aεvε2)≤18‖Aεvε2‖2+2|O|‖T−1(θtω)‖2‖G(t,⋅)‖2L∞(˜O) | (81) |
Noting that
ddtaε(vε2,vε2)+λεn+1aε(vε2,vε2)≤4δ2(θtω)‖vε2‖2+8α2‖T(θtω)‖2p−4‖vε‖2p−22p−2+c‖T−1(θtω)‖2(‖ψ2(t,⋅)‖2L∞(˜O)+‖G(t,⋅)‖2L∞(˜O)). | (82) |
Taking
aε(vε2(τ,τ−t,θ−τω,ψε),vε2(τ,τ−t,θ−τω,ψε))≤∫ττ−1eλεn+1(r−τ)aε(vε2(r,τ−t,θ−τω,ψε),vε2(r,τ−t,θ−τω,ψε))dr+4δ2∫ττ−1eλεn+1(r−τ)δ2(θr−τω)aε(vε2(r,τ−t,θ−τω,ψε),vε2(r,τ−t,θ−τω,ψε))dr+8α2∫ττ−1eλεn+1(r−τ)‖T(θr−τω)‖2p−4‖vε(r,τ−t,θ−τω,ψε)‖2p−22p−2dr+c∫ττ−1eλεn+1(r−τ)‖T−1(θr−τω)‖2(‖ψ2(r,⋅)‖2L∞(˜O))dr+c∫ττ−1eλεn+1(r−τ)‖T−1(θr−τω)‖2‖G(r,⋅)‖2L∞(˜O)dr. | (83) |
Since
aε(vε2(τ,τ−t,θ−τω,ψε),vε2(τ,τ−t,θ−τω,ψε))≤c∫0−1eλεn+1r‖vε(r+τ,τ−t,θ−τω,ψε)‖2p−22p−2dr+c∫0−1eλεn+1raε(vε(r+τ,τ−t,θ−τω,ψε),vε(r+τ,τ−t,θ−τω,ψε))dr |
+c∫0−1eλεn+1rdr≤c∫0−1e(λ0n+1−1)r‖vε(r+τ,τ−t,θ−τω,ψε)‖2p−22p−2dr+c∫0−1e(λ0n+1−1)raε(vε(r+τ,τ−t,θ−τω,ψε),vε(r+τ+s,τ−t,θ−τω,ψε))dr+c∫0−1e(λ0n+1−1)r‖vε(r+τ−ρ0(r+τ+s),τ−t,θ−τω,ψε)‖2dr+c∫0−1e(λ0n+1−1)rdr. | (84) |
Given
c∫0−1e(λ0n+1−1)r‖vε(r+τ,τ−t,θ−τω,ψε)‖2p−22p−2dr≤c∫0−1e(λ0n+1−1)r∫{y∈O:|vε|≥2M}|vε(r+τ,τ−t,θ−τω,ψε)|2p−2dydr+c∫0−1e(λ0n+1−1)r∫{y∈O:|vε|<2M}|vε(r+τ,τ−t,θ−τω,ψε)|2p−2dydr≤c∫0−1eγMp−2r∫{y∈O:|vε|≥2M}|vε(r+τ,τ−t,θ−τω,ψε)|2p−2dydr+c∫0−1e(λ0n+1−1)r∫{y∈O:|vε|<2M}|vε(r+τ,τ−t,θ−τω,ψε)|2p−2dydr≤η+c22p−2M2p−2|O|∫0−1e(λ0n+1−1)rdr≤η+c22p−2M2p−2|O|1λ0n+1−1. | (85) |
For the last three terms on the right-hand side of (84), by Lemma 3.1, we find that there exist
c∫0−1e(λ0n+1−1)raε(vε(r+τ,τ−t,θ−τω,ψε),vε(r+τ,τ−t,θ−τω,ψε))dr+c∫0−1e(λ0n+1−1)rdr≤c1∫0−1e(λ0n+1−1)rdr≤c11λ0n+1−1. | (86) |
Since
aε(vε2(τ+s,τ−t,θ−τω,ψε),vε2(τ+s,τ−t,θ−τω,ψε))≤2η, |
which together
In this subsection, we establish the existence of
Lemma 4.1. Suppose (8)-(11), (39) and (43) hold. Then the cocycle
Proof. We first notice that, by Lemma 3.2,
K(τ,ω)={u∈H1(O):‖u‖2H1(O)≤L(τ,ω)}, | (87) |
where
Φε(t,τ−t,θ−tω,D(τ−t,θ−tω))⊆K(τ,ω). |
Thus we find that
Lemma 4.2. Assume that (8)-(11), (39) and (43) hold. Then, the cocycle
Proof. We will show that for every
‖uε2(τ,τ−tn,θ−τω,ϕε)‖H1(O)=‖Qm0uε(τ,τ−tn,θ−τω,ϕε)‖H1(O)<η4. | (88) |
On the other hand, by Lemma 3.2 we find that the sequence
Theorem 4.3. Assume that (8)-(11), (39) and (43) hold. Then, the cocycle
Proof. First, we know from Lemma 4.1 that
Analogous results also hold for the solution of (4)-(5). In particular, we have:
Theorem 4.4. Assume that (8)-(11), (39) and (43) hold. Then, the cocycle
The following estimates are needed when we derive the convergence of pullback attractors. By the similar proof of that of Theorem 5.1 in [14], we get the following lemma.
Lemma 5.1. Assume that (8)-(11) and (39) hold. Then for every
∫tτ‖vε(r,τ,ω,ψε)‖2H1ε(O)dr≤c‖ψε‖2N+c∫τ+Tτ(‖G(r,⋅)‖2L∞(˜O)+‖ψ1(r,⋅)‖2L∞(˜O))dr, |
where
Similarly, one can prove
Lemma 5.2. Assume that (8)-(11) and (39) hold. Then for every
∫tτ‖v0(r,τ,ω,ψ0)‖2H1(Q)dr≤c‖ψ0‖2M+c∫τ+Tτ(‖G(r,⋅)‖2L∞(˜O)+‖ψ1(r,⋅)‖2L∞(˜O))dr, |
where
In the sequel, we further assume the functions
‖Gε(t,⋅)−G0(t,⋅)‖L2(O)≤κ1(t)ε | (89) |
and
‖Hε(t,⋅,s)−H0(t,⋅,s)‖L2(O)≤κ2(t)ε, | (90) |
where
By (12) and (90) we have, for all
‖hε(t,⋅,s)−h0(t,⋅,s)‖L2(O)≤κ2(t)ε. | (91) |
Since
Theorem 5.3. Suppose (8)-(11), (39), and (89)-(90) hold. Given
limn→∞‖Φεn(t,τ,ω,ϕεn)−Φ0(t,τ,ω,ϕ0)‖N=0. |
Proof. Since
‖vεn(t)−v0(t)‖2N≤c‖ϕεn−ϕ0‖2N+cmaxν∈[τ,t]ξ(θνω)∫tτ‖vεn(s)−v0(s)‖2Nds |
+cεnmaxν∈[τ,t]‖T−1(θνω)‖∫tτ(‖vεn(s)‖2H1εn(O)+‖v0(s)‖2H1(Q))ds+cεnmaxν∈[τ,t]‖T−1(θνω)‖∫tτ(κ21(s)+κ22(s))ds+cεn∫tτ(‖vεn(s)‖2H1εn(O)+‖v0(s)‖2H1(Q))ds, | (92) |
where
‖vεn(t)−v0(t)‖2N≤ec(1+maxν∈[τ,τ+T]ξ(θνω))T‖ϕεn−ϕ0‖2N+ϱεnec(1+maxν∈[τ,τ+T]ξ(θνω))T[‖ψ0‖2M+‖ψεn‖2N+∫τ+Tτ(κ21(s)+κ22(s))ds+∫τ+Tτ(‖G(s,⋅)‖2L∞(˜O)+‖ψ1(s,⋅)‖2L∞(˜O))ds]. | (93) |
Notice that, for all
‖uεn(t,τ,ω,ϕε)−u0(t,τ,ω,ϕ0)‖2N≤maxν∈[τ,τ+T]‖T(θνω)‖2‖vεn(t,τ,ω,T−1(θτω)ϕε)−v0(t,τ,ω,T−1(θτω)ϕ0‖2N, |
which together with (93) implies the desired results.
The next result is concerned with uniform compactness of attractors with respect to
Lemma 5.4. Assume that (8)-(11), (39) and (43) hold. If
limn→∞‖uεn−u‖H1(O)=0. |
Proof. Take a sequence
uεn=Φεn(tn,τ−tn,θ−tnω,ϕεn). | (94) |
By Lemma 4.1, we have
‖QεnN1uεn(τ,τ−tn,θ−τω,ϕεn)‖H1(O)≤η. | (95) |
By Lemma 3.2, we have
‖PεnN1uεn(τ,τ−tn,θ−τω,ϕεn)‖H1(O)<M. | (96) |
It follows from (95) and (96) that
limn→∞‖uεn−u‖H1(O)=0. | (97) |
This completes the proof.
Now we are in a position to prove the main result of this paper.
Theorem 5.5. Assume that (8)-(11), (39), (43), and (89)-(90) hold. The attractors
limε→0distH1(O)(Aε(τ,ω),A0(τ,ω))=0. |
Proof. Given
‖u‖2H1ε(O)≤L(τ,ω)for all 0<ε<ε0 and u∈Aε(τ,ω), | (98) |
where
distH1(O)(zn,A0(τ,ω))≥δfor alln∈N. | (99) |
By Lemma 5.4 there exists
limn→∞‖zn−z∗‖H1(O)=0. | (100) |
By the invariance property of the attractor
zn=Φεn(t,τ−t,θ−tω,ytn). | (101) |
By Lemma 5.4 again there exists
limn→∞‖ytn−yt∗‖H1(O)=0. | (102) |
It follows from Theorem 5.3 that for every
limn→∞Φεn(t,τ−t,θ−tω,ytn)=Φ0(t,τ−t,θ−tω,yt∗)inN. | (103) |
By (100), (101), (103) and uniqueness of limits we obtain
z∗=Φ0(t,τ−t,θ−tω,yt∗)inH1(O). | (104) |
Notice that
limsupn→∞‖ytn‖H1(O)≤‖K(τ−t,θ−tω)‖H1(O)≤L(τ−t,θ−tω). | (105) |
By (102) and (105) we get, for every
‖yt∗‖H1(Q)≤L(τ−t,θ−tω). | (106) |
By
distH1(Q)(z∗,A0(τ,ω))=distH1(Q)(Φ0(t,τ−t,θ−tω,yt∗),A0(τ,ω))≤distH1(Q)(Φ0(t,τ−t,θ−tω,K0(τ−t,θ−tω)),A0(τ,ω))→0,ast→∞. | (107) |
This implies that
distH1(O)(zn,A0(τ,ω))≤distH1(O)(zn,z∗)→0, |
a contradiction with (99). This completes the proof.
The authors would like to thank the anonymous referee for the useful suggestions and comments.
[1] |
Sach VJ, Buchner E, Schmieder M (2020) Enigmatic earthquake-generated large-scale clastic dyke in the Biberach area (SW Germany). Sediment Geol 398: 105571. https://doi.org/10.1016/j.sedgeo.2019.105571 doi: 10.1016/j.sedgeo.2019.105571
![]() |
[2] |
Buchner E, Sach VJ, Schmieder M (2020) New discovery of two seismite horizons challenges the Ries–Steinheim double‑impact theory. Sci Rep 10: 22143. https://doi.org/10.1038/s41598-020-79032-4 doi: 10.1038/s41598-020-79032-4
![]() |
[3] |
Stöffler D, Artemieva NA, Wünnemann K, et al. (2013) Ries crater and suevite revisited—Observations and modeling. Part Ⅰ: Observations. Meteorit Planet Sci 48: 515–589. https://doi.org/10.1111/maps.12086 doi: 10.1111/maps.12086
![]() |
[4] |
Artemieva NA, Wünnemann K, Krien F, et al. (2013) Ries crater and suevite revisited—Observations and modeling. Part Ⅱ: Modeling. Meteorit Planet Sci 48: 590–627. https://doi.org/10.1111/maps.12085 doi: 10.1111/maps.12085
![]() |
[5] |
Schmieder M, Kennedy T, Jourdan F, et al. (2018) A high-precision 40Ar/39Ar age for the Nördlinger Ries impact crater, Germany, and implications for the accurate dating of terrestrial impact events. Geochim Cosmochim Acta 220: 146–157. https://doi.org/10.1016/j.gca.2017.09.036 doi: 10.1016/j.gca.2017.09.036
![]() |
[6] |
Schmieder M, Kennedy T, Jourdan F (2018) Response to comment on "A high-precision 40Ar/39Ar age for the Nördlinger Ries impact crater, Germany, and implications for the accurate dating of terrestrial impact events" by Schmieder et al. (Geochim. Cosmochim. Acta 220 (2018) 146–157). Geochim Cosmochim Acta 238: 602–605. https://doi.org/10.1016/j.gca.2018.07.025 doi: 10.1016/j.gca.2018.07.025
![]() |
[7] | Sach VJ (1999) Litho- und biostratigraphische Untersuchungen in der Oberen Süßwassermolasse des Landkreises Biberach an der Riß (Oberschwaben). Stuttgarter Beitr Naturk B 276: 1–167. |
[8] | Sach VJ (2014) Strahlenkalke (Shatter-Cones) aus dem Brockhorizont der Oberen Süßwassermolasse in Oberschwaben (Südwestdeutschland)—Fernauswürflinge des Nördlinger-Ries-Impaktes, Pfeil Verlag, München, 1–17. |
[9] | Hofmann B, Hofmann F (1992) An impactite horizon in the upper freshwater molasse in Eastern Switzerland: Distal Ries ejecta. Eclogae Geol Helv 85: 788–789. |
[10] |
Letsch D (2017) Diamictites and soft sediment deformation related to the Ries (ca. 14.9 Ma) meteorite impact: the "Blockhorizont" of Bernhardzell (Eastern Switzerland). Int J Earth Sci 107: 1379–1380. https://doi.org/10.1007/s00531-017-1542-1 doi: 10.1007/s00531-017-1542-1
![]() |
[11] |
Holm-Alwmark S, Alwmark C, Ferrière L, et al. (2021) Shocked quartz in distal ejecta from the Ries impact event (Germany) found at ~ 180 km distance, near Bernhardzell, eastern Switzerland. Sci Rep 11: 7438. https://doi.org/10.1038/s41598-021-86685-2 doi: 10.1038/s41598-021-86685-2
![]() |
[12] |
Buchner E, Sach VJ, Schmieder M (2022) Event- and biostratigraphic evidence for two independent Ries and Steinheim asteroid impacts in the Middle Miocene. Sci Rep 12: 18603. https://doi.org/10.1038/s41598-022-21409-8 doi: 10.1038/s41598-022-21409-8
![]() |
[13] |
Collins G, Melosh HJ, Marcus R (2005) Earth impact effects program: a web-based computer program for calculating the regional environmental consequences of a meteoroid impact on Earth. Meteorit Planet Sci 40: 817–840. https://doi.org/10.1111/j.1945-5100.2005.tb00157.x doi: 10.1111/j.1945-5100.2005.tb00157.x
![]() |
[14] |
Schmieder M, Sach VJ, Buchner E (2021) The Chöpfi pinnacles near Winterthur, Switzerland: Long-distance effects of the Ries impact-earthquake? Int J Earth Sci 111: 145–147. https://doi.org/10.1007/s00531-021-02082-0 doi: 10.1007/s00531-021-02082-0
![]() |
[15] |
Buchner E, Sach VJ, Schmieder M (2021) Sand spikes pinpoint powerful palaeoseismicity. Nat Commun 12: 6731. https://doi.org/10.1038/s41467-021-27061-6 doi: 10.1038/s41467-021-27061-6
![]() |
[16] |
Maurer H, Buchner E (2007) Rekonstruktion fluvialer Systeme der Oberen Süßwassermolasse im Nordalpinen Vorlandbecken SW-Deutschlands. German J Geosci (ZdGG) 158: 249–270. https://doi.org/10.1127/1860-1804/2007/0158-0249 doi: 10.1127/1860-1804/2007/0158-0249
![]() |
[17] | Heider J, Wegele A, Amstutz GC (1976) Beobachtungen über Sandrosen und Zapfensande aus der Süßwassermolasse Südwürttembergs. Der Aufschluß 27: 297–307. |
[18] | Sanborn WB (1976) Oddities of the Mineral World. Van Nostrand Reinhold Company, NY, USA. 142. Available from: http://allanmccollum.net/amcimages/sanborn.html, (last accessed February 19, 2025). |
[19] |
Akçiz SO, Grant Ludwig L, Arrowsmith JR, et al. (2010) Century-long average time intervals between earthquake ruptures of the San Andreas fault in the Carrizo Plain, California. Geology 38: 787–790. https://doi.org/10.1130/G30995.1 doi: 10.1130/G30995.1
![]() |
[20] |
McBride EF, Picard MD, Folk RL (1994) Orientated Concretions, Ionian Coast, Italy: Evidence of Groundwater flow direction. J Sediment Res A64: 535–540. https://doi.org/10.1306/D4267DFC-2B26-11D7-8648000102C1865D doi: 10.1306/D4267DFC-2B26-11D7-8648000102C1865D
![]() |
[21] | McCullough LN, Ritter JB, Zaleha MJ, et al. (2003) Habit, formation, and implications of elongeate, calcite concretions, Victoria, Australia. Department of Geology, Wittenberg University, Ohio, USA. Published Senior Honors Thesis. 25. |
[22] | Grant JA, Wilson SA (2018) Possible Geomorphic and Crater Density Evidence for Late Aqueous Activity in Gale Crater. LPI Contrib. 49th Annual Lunar and Planetary Science Conference. |
[23] |
Metz J, Grotzinger J, Okubo C, et al. (2010) Thin‐skinned deformation of sedimentary rocks in Valles Marineris, Mars. J Geophys Res Planets 115: E11004. https://doi.org/10.1029/2010JE003593 doi: 10.1029/2010JE003593
![]() |
[24] | NASA Mars Science Laboratory, Curiosity Rover, 2024. Available from: https://mars.nasa.gov/msl/mission/science/. |
[25] |
Wray JJ (2013) Gale Crater: The Mars Science Laboratory/Curiosity rover landing site. Int J Astrobiol 12: 25–38. https://doi.org/10.1017/S1473550412000328 doi: 10.1017/S1473550412000328
![]() |
[26] |
Grotzinger JP, Sumner DY, Kah LC, et al. (2014) A habitable fluvio-lacustrine environment at Yellowknife Bay, Gale Crater, Mars. Science 343: 6169. https://doi.org/10.1126/science.1242777 doi: 10.1126/science.1242777
![]() |
[27] |
Grotzinger JP, Gupta S, Malin MC, et al. (2015) Deposition, exhumation, and paleoclimate of an ancient lake deposit, Gale Crater, Mars. Science 350: 6257. https://doi.org/10.1126/science.aac7575 doi: 10.1126/science.aac7575
![]() |
[28] |
Buz J, Ehlmann BL, Pan L, et al. (2017) Mineralogy and stratigraphy of the Gale crater rim, wall, and floor units. J Geophys Res Planets 122: 1090–1118. https://doi.org/10.1002/2016JE005163 doi: 10.1002/2016JE005163
![]() |
[29] |
Schwenzer SP, Abramov O, Allen CC, et al. (2012) Gale Crater: Formation and post-impact hydrous environments. Planet Space Sci 70: 84–95. https://doi.org/10.1016/j.pss.2012.05.014 doi: 10.1016/j.pss.2012.05.014
![]() |
[30] |
Montenat C, Barrier P, Ott d'Estevou P, et al. (2007) Seismites: An attempt at critical analysis and classification. Sediment Geol 196: 5–30. https://doi.org/10.1016/j.sedgeo.2006.08.004 doi: 10.1016/j.sedgeo.2006.08.004
![]() |
[31] | Hargitai H, Levi T (2015) Clastic dikes, Encyclopedia of Planetary Landforms, Hargitai H and Kereszturi A, Eds., Encyclopedia of Planetary Landforms, Springer, NY, USA. |
[32] |
Sleep NH, Olds EP (2018) Remote faulting triggered by strong seismic waves from the Cretaceous-Paleogene asteroid impact. Seismol Res Lett 89: 570–576. https://doi.org/10.1785/0220170223 doi: 10.1785/0220170223
![]() |
[33] |
DePalma RA, Smit J, Burnham DA, et al. (2019) A seismically induced onshore surge deposit at the K-Pg. boundary, North Dakota. PNAS 116: 8190–8199. https://doi.org/10.1073/pnas.1817407116 doi: 10.1073/pnas.1817407116
![]() |
[34] |
Vaniman DT, Bish DL, Ming DW, et al. (2014) Mineralogy of a mudstone at Yellowknife Bay, Gale Crater, Mars. Science 343: 6169. https://doi.org/10.1126/science.1243480 doi: 10.1126/science.1243480
![]() |
[35] |
Ehlmann BL, Buz J (2015) Mineralogy and fluvial history of the watersheds of Gale, Knobel, and Sharp craters: A regional context for MSL Curiosity's exploration. Geophys Res Lett 42: 264–273. https://doi.org/10.1002/2014GL062553 doi: 10.1002/2014GL062553
![]() |
[36] |
Carter J, Viviano-Beck C, Loizeau D, et al. (2015) Orbital detection and implications of akaganéite on Mars. Icarus 253: 296–310. https://doi.org/10.1016/j.icarus.2015.01.020 doi: 10.1016/j.icarus.2015.01.020
![]() |
[37] |
Tohver E, Schmieder M, Lana C, et al. (2018) End-Permian impactogenic earthquake and tsunami deposits in the intracratonic Paraná Basin of Brazil. GSA Bull 130: 1099–1120. https://doi.org/10.1130/B31626.1 doi: 10.1130/B31626.1
![]() |
[38] |
Weatherley DK, Henley RW (2013) Flash vaporization during earthquakes evidenced by gold deposits. Nature Geosci 6: 294–298. https://doi.org/10.1038/ngeo1759 doi: 10.1038/ngeo1759
![]() |
[39] |
Simms JM (2003) Uniquely extensive seismite from the latest Triassic of the United Kingdom: evidence for bolide impact? Geology 31: 557–560. https://doi.org/10.1130/0091-7613(2003)031<0557:UESFTL>2.0.CO;2 doi: 10.1130/0091-7613(2003)031<0557:UESFTL>2.0.CO;2
![]() |
[40] |
Banham SG, Gupta S, Rubin DM, et al. (2018) Ancient Martian aeolian processes and palaeomorphology reconstructed from the Stimson formation on the lower slope of Aeolis Mons, Gale crater, Mars. Sedimentology 65: 993–1042. https://doi.org/10.1111/sed.12469 doi: 10.1111/sed.12469
![]() |
[41] |
Bohacs KM, Carrol AR, Neal JE (2003) Lessons from large lake systems—Thresholds, nonlinearity, and strange attractors. Special Papers-Geological Society of America, 75–90. https://doi.org/10.1130/0-8137-2370-1.75 doi: 10.1130/0-8137-2370-1.75
![]() |
[42] |
Brož P, Oehler D, Mazzini A, et al. (2023) An overview of sedimentary volcanism on Mars. Earth Surf Dynam 11: 633–661. https://doi.org/10.5194/egusphere-2022-1458 doi: 10.5194/egusphere-2022-1458
![]() |
[43] |
Sturm S, Wulf G, Jung D, et al. (2013) The Ries impact, a double-layer rampart crater on Earth. Geology 41: 531–534. https://doi.org/10.1130/G33934.1 doi: 10.1130/G33934.1
![]() |
[44] |
Wilson SA, Morgan AM, Howard AD, et al. (2021) The global distribution of craters with alluvial fans and deltas on Mars. Geophys Res Lett 48: e2020GL091653. https://doi.org/10.1029/2020GL091653 doi: 10.1029/2020GL091653
![]() |
[45] |
Grotzinger JP, Crisp J, Vasavada AR, et al. (2012) Mars Science Laboratory mission and science investigation. Space Sci Rev 170: 5–56. https://doi.org/10.1007/s11214-012-9892-2 doi: 10.1007/s11214-012-9892-2
![]() |