Top Global University Project, Waseda University

Waseda Workshop on Partial Diﬀerential Equations 2019
Date: March 22-23, 2019
Venue: Large Conference Room, 1st Floor, 55N Bldg., Waseda University,Nishi-Waseda CampusૣҴాେֶ ੢ૣҴాΩϟϯύεɹ 55 ߸‫ ؗ‬N ౩ 1 ֊ େձٞࣨAccess: https://www.waseda.jp/fsci/en/access/˗ Daoyuan Fang (Zhejiang Univ.) ˗March 22, Friday 10:30-11:10

Almost global existence for the Kirchhoﬀ-type equation
with the periodic boundary conditions
This is the joint work with Zheng Han. In this paper, we prove an almost globalexistence result for the Kirchhoﬀ-type equation on Td with Cauchy data of small amplitude
. We show a lower bound −N for the existence time with any natural number N . This
generalizes a preceding result concerning a lower bound −4 . The proof relies on themethod of normal forms and induction. The diﬃculty is to ﬁnd out the structure of thenonlinearity so that the process of normal forms can be performed up to any order.

˗ Shuichi Kawashima (Waseda Univ.) ˗March 22, Friday 11:20-12:00

Nonlinear waves for a model system of hyperbolic balance laws
We consider a simple model system of hyperbolic balance laws:
ut + f (u)x + qx = 0,
qt + q + ux = 0,(1)where u and q are unknown functions of x ∈ R and t > 0, and the ﬂux f is a smoothconvex function of u. We can regarded u and q as the variables describing macroscopic
and microscopic states in kinetic theory, respectively, and apply the Chapman-Enskog
expansion to the system (1). Then, as the ﬁrst order approximation (Euler level), wehave q = 0 and a hyperbolic conservation law
ut + f (u)x = 0.

(2)

Also, as the second order approximation (Navier-Stokes level), we have q = −ux and a
viscous conservation law(3)
ut + f (u)x = uxx .
We discuss the asymptotic stability of nonlinear waves for the system (1). We ﬁrst
observe that (1) admits a shock proﬁle of the form (u, q) = (ũ, q̃)(x − st) which connects

the constant states (u± , 0) with u− > u+ , where s is a constant. This shock proﬁle is anexact solution of the system (1). Under suitable assumptions we can prove the asymptoticstability of this shock proﬁle (ũ, q̃)(x − st) as t → ∞.
Next we consider a rarefaction wave of the form (u, q) = (ur , 0)(x/t), where ur (x/t) isthe centered rarefaction wave (an exact solution) of the hyperbolic conservation law (2)which connects the constant states u± with u− < u+ . We note that (u, q) = (ur , 0)(x/t) is
not an exact solution of (1). Under suitable assumptions we can also prove the asymptotic
stability of this rarefaction wave (u, q) = (ur , 0)(x/t) as t → ∞.
˗ Sijia Zhong (Southeast Univ.) ˗March 22, Friday 13:30-14:10

Well-posedness of supercritical wave equation on a compact Riemannian
manifold with Dirichlet boundary condition
In this talk, we will study the global well-posedness of wave equation ∂t2 u − u +|u| u = 0 1 < p < 5 on a compact Riemannian manifold with Dirichlet boundary
condition with some special kind of randomized initial data. We will show that this< s < 1 if 3 < p < 5, andequation will be well-posed in H s (M ) × H s−1 (M ) for p−3
p−10 < s < 1 if 1 < p ≤ 3, and get some polynomial growth of the sobolev norm of thesolution.
p−1

˗ Chengbo Wang (Zhejiang Univ.) ˗March 22, Friday 14:20-15:00

The Strauss conjecture on negatively curved backgrounds
We provide a simple and geometric proof of small data global existence for the shifted
wave equation on hyperbolic space involving nonlinearities of the form ±|u|p or ±|u|p−1 u.
It is based on the weighted Strichartz estimates of Georgiev-Lindblad-Sogge (or Tataru)
on Euclidean space. We also prove a small data existence theorem for variably curved
backgrounds which extends earlier ones for the constant curvature case of Anker-Pierfelice,
Anker-Pierfelice-Vallariono and Metcalfe-Taylor. It is based on the joint work with Yannick Sire and Chris Sogge.

˗ Masashi Misawa (Kumamoto Univ.) ˗March 22, Friday 15:30-16:10

Finite singularity for the m-harmonic ﬂow
The m-harmonic ﬂow is the evolution of the m-harmonic maps between two smoothcompact Riemannian manifolds, where the number m is the integer of the dimension ofdomain manifold. The m-harmonic maps are critical points of the m-energy for maps

between two smooth compact Riemannian manifolds.
The m-harmonic map is an extension of the two-dimensional harmonic map to the higherdimension. In such case that the integral exponent is equal to the domain dimension,
we have a scaling transformation, under which the local energy and the equation areinvariant. We know that a weak solution of the m-harmonic map is regular, because of
the scaling invariance and the small energy regularity theorem. The m-harmonic ﬂow is
the negative directed gradient ﬂow of the m-energy and has a fundamental application
to the m-harmonic maps. In this talk, we consider the regularity of the m-harmonic ﬂow
and in particular we study an estimation of size of the singular set. Our main theorem
is the global existence of a weak solution to the m-harmonic ﬂow with large energy data,
which is regular except at most ﬁnitely many points.

˗ Makoto Nakamura (Yamagata Univ.) ˗March 22, Friday 16:20-17:00

Remarks on a semilinear diﬀusion equation
in homogeneous and isotropic spaces
We consider the Cauchy problem (P  ) given by⎧⎨ ∂u(s, x) − Δu(s, x) = A(s)f (u)(s, x) for (s, x) ∈ [0, S0 ) × Rn ,(P ) ∂s⎩u(0, ·) = u (·),
0where we have put
A(s) := (1 − 2Hs)n(p−1)/4−1
andS0 :=12H∞

if H > 0,
if H < 0.

(0.1)

(0.2)

In this talk, we consider the existence of the solutions for small data, and the asymptotic behaviors of them. The existence part is given as follows for some function space
X(R).
Theorem 1. Let H ∈ R, n ≥ 1, 1 < p < ∞.
(1)(Small global solutions for H > 0) Let H > 0. If H −1/(p−1) u0 L∞ is suﬃciently
small, then there exists a unique global solution u for (P  ) in X(R) for some R > 0.
(2)(Small global solutions for H < 0) Let H < 0. If M 1/(p−1) u0 L1 ∩L∞ is suﬃciently
small, then there exists a unique global solution u for (P  ) in X(R) for some R > 0. Here,
we have put⎧(−2H)n(p−1)/4−1 if p ≥ 1 + n4 , H < − 12 ,⎪⎪⎪⎨(−2H)n(p−1)/4−1 if p < 1 + 4 , − 1 ≤ H < 0,n2
M :=4⎪
1if p ≥ 1 + n , − 12 ≤ H < 0,⎪⎪⎩1if p < 1 + n4 , H < − 12 .

˗ Hiroshi Takeda (Fukuoka Inst. Tech.) ˗March 22, Friday 17:10-17:50

Asymptotic expansion of solutions for wave equations
with structural damping and its application
Wave equations with damping term arise from many context including mathematicalphysics. In this talk, we ﬁrstly study the Cauchy problem of wave equations with structural damping term;
 2t > 0, x ∈ Rn ,∂t u − Δu + ν(−Δ)σ ∂t u = 0,(1)x ∈ Rn ,u(0, x) = u0 (x), ∂t u(0, x) = u1 (x),where u : (0, ∞) × Rn → R, 0 ≤ σ ≤ 1 and ν > 0. Our main interest is the asymptotic
proﬁles of the solutions to (1) as t → ∞ and they are summarized as follows:
Theorem 1. Let n ≥ 2 and k0 ≥ 0. Assume that (u0 , u1 ) ∈ (H k0 +1 ∩ L1 ) × (H k0 ∩ L1 ).
Then, the asymptotic proﬁles of solutions to (1) is classiﬁed into 6 patterns depending on
ν, σ, a and b.
We note that our main contribution to Theorem 1 is in the case 12 < σ < 1.
As a consequence of Theorem 1, we can conclude the lower bounds of the solutions.
Corollary 2. If
u1 (x)dx = 0, there exists C > 0 such thatRn−1 −γσ,k −t≤ ∂t ∇kx u(t)2 ≤ Ct−γσ,k − , 0 < σ ≤ 12 ,C −1 log(t + 1) ≤ u(t)2 ≤ C log(t + 1), 12 < σ ≤ 1 and n = 2,C

C −1 t−γσ,k − 2σ ≤ ∂t ∇kx u(t)2 ≤ Ct−γσ,k − 2σ ,for t1≤σ≤12and otherwise1,  ∈ {0, 1} and k ∈ [0, k0 + 1] with k +  ≤ k0 + 1, wherenσk− 1−σ+ 2(1−σ), 0 ≤ σ ≤ 12 ,4(1−σ)γσ,k :=n1k− 2σ+ 2σ, 12 ≤ σ ≤ 1.
4σ

Next, we apply Theorem 1 to the elastic waves with structural damping: 2∂t u − a2 Δu − (b2 − a2 )∇(div u) + B∂t u = 0, t > 0, x ∈ Rn ,(2)u(0, x) = f (x), ∂t u(0, x) = g(x), x ∈ Rn ,where u = (u1 , u2 , · · · , un ), B = (−Δ)σ or (−a2 Δ−(b2 −a2 )∇div)σ , f (x) = (f1 , f2 , · · · , fn )and g(x) = (g1 , g2 , · · · , gn ), which are recently studied by [1], [2] and [3]. We will deduce
the asymptotic proﬁles of solutions to (2) is given by 8 patterns for the suitable initial
data.
References[1] Ikehata, R., Charão, R.C. and da Luz, C. R., J. Evol. Equ. 14 (2014) 197-210.
[2] Reissig, M., Math. Methods Appl. Sci. 39 (2016), 4618-4628.
[3] Wu, M., Chai, S. and Li, S., J. Math. Anal. Appl. 452 (2017), 361-377.

˗ Ting Zhang (Zhejiang Univ.) ˗March 23, Saturday 10:30-11:10

Local and global existence of pathwise solution for the stochastic Boussinesq
equations with multiplicative noises
In this talk, we consider the stochastic Boussinesq equations in Td with the nonlinear
multiplicative noises. At ﬁrst, we establish the local existence of pathwise solutions. Then,we establish the global existence of pathwise solution when the noises are non-degenerate,
which show that the linear multiplicative noises would provide a regularizing eﬀect: theglobal existence of solution occurs with high probability if the initial data are suﬃciently
small, or if the noise coeﬃcients are suﬃciently large. (Based on the work with LihuaiDu)

˗ Jiang Xu (Nanjing Univ. of Aeronautics and Astronautics) ˗March 23, Saturday 11:20-12:00

The optimal time-decay for the compressible Navier-Stokes equations in thecritical Lp framework
The global existence issue for the isentropic compressible Navier-Stokes equations in
the critical regularity framework has been addressed by R. Danchin more than ﬁfteen
years ago. However, whether (optimal) time-decay rates could be shown in general criticalspaces and any high dimensions has remained an open question. In this talk, we survey
recent results not only in the L2 critical framework but also in the more general Lp criticalframework, which were exactly as ﬁrstly observed by A. Matsumura and T. Nishida in
the case p = 2 and d = 3, for solutions with high Sobolev regularity.

References[1] R. Danchin and J. Xu: Optimal time-decay estimates for the compressible Navier-Stokes equationsin the critical Lp framework, Arch. Rational Mech. Anal., 224, 53-90 (2017).
[2] Z. Xin and J. Xu: Optimal decay for the compressible Navier-Stokes equations without additional
smallness assumptions, arXiv:1812.11714v1 (2019).
[3] J. Xu: A low-frequency assumption for optimal time-decay estimates to the compressible NavierStokes equations, Comm. Math. Phys. (2019).

˗ Guixiang Xu (Beijing Normal Univ.) ˗March 23, Saturday 13:30-14:10

Recent developments about the stability of the solitary waves for the(generalized) derivative Schrödinger equation

We will survey some recent developments about the stability of the solitary waves forthe (generalized) derivative Schrödinger equation. We ﬁrstly show the existence/nonexistenceof traveling waves for NLS with derivative (DNLS) by the structure analysis, ODE argument and the variational argument. Secondly, we consider the orbital stability of thesolitary waves for the derivative Schrödinger equation in the energy space. Last, we willshow the orbital instability of the solitary waves for the generalized derivative Schrödinger
equation in the degenerate case. They are joint works with Prof. Changxing Miao andXingdong Tang.

˗ Ruizhao Zi (Central China Normal Univ.) ˗March 23, Saturday 14:20-15:00

Convergence to equilibrium for the solution of the full compressible
Navier-Stokes equations
We study the convergence to equilibrium for the full compressible Navier-Stokes equations on the torus T3. Under the conditions that both the density ρ and the temperatureθ possess uniform in time positive lower and upper bounds, it is shown that global regularsolutions converge to equilibrium with exponential rate. We improve the previous result
obtained by Villani in [Mem. Amer. Math. Soc., 202(2009), no. 950] on two levels:
weaker conditions on solutions and faster decay rates. This is a joint work with Prof.
Zhifei Zhang.

˗ Masayuki Hayashi (Waseda Univ.) ˗March 23, Saturday 15:30-16:10

Potential well theory for the derivative nonlinear Schrödinger equation
We study the derivative nonlinear Schrödinger equation by variational approach. The
main aim of this talk is to investigate global existence from the viewpoints of the solitons.
By using variational arguments inspired from Payne–Sattinger’s classical work we clarify
the connection between mass condition for global existence and potential well generated
by the solitons. We see that the eﬀect of the momentum plays an essential role in theargument.

˗ Erika Ushikoshi (Yokohama National Univ.) ˗March 23, Saturday 16:20-17:00

Hadamard variational formula for the multiple eigenvalues of the Stokes
equations with friction slip boundary conditions
We consider the perturbation problem for the eigenvalue of the Stokes equations in abounded domain Ω ⊂ R3 with a smooth boundary ∂Ω. More precisely, for any ε ≥ 0 andfor any ρ ∈ C ∞ (∂Ω), we deﬁne a perturbed domain by Ωε , whose boundary is∂Ωε := {x + ερ(y)νy ; y ∈ ∂Ω}.
Under such a smooth perturbation, we consider the eigenvalue problem of the Stokes
equations with the slip friction boundary condition;⎧⎪in Ωε ,⎨ΔΦε + λ(ε)Φε = ∇qε(1)in Ωε ,
div Φε = 0⎪⎩(2e(Φε )νx + κΦε )τ = 0 on ∂Ω,where Φε = (Φ1ε (x), Φ2ε (x), Φ3ε (x)) and q = q(x) is unknown velocity ﬁelds and pressurre
at x ∈ Ωε , respectively. Furthermore, κ is a positive friction constant, and e(u) ={eij (u)}1≤i,j≤3 is the strain tensor deﬁned byeij (u) :=12∂ui ∂uj+ i∂y j∂y,i, j = 1, 2, 3.

It is a known fact that the spectral set of (1) consists of the descrete sequence of positive
real numbers. Then, if we arrange them with counting multiplicity, we see that0 < λ1 (ε) ≤ λ2 (ε) ≤ · · · ≤ λk (ε) ≤ · · · → ∞,where the k-th eigenvalue is denoted by λk (ε). In this talk, we consider the domain
dependency of the multiple eigenvalue λ(ε). Indeed, we shall establish the Hadamard
variational formula for the multiple eigenbalue of (1). This is a joint work with Professor
Shuichi Jimbo in Hokkaido University.