Semiclassical limits of ground state solutions to Schrödinger systems

This paper is concerned with the existence and concentration properties of the ground state solutions to the following coupled Schrödinger systems $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\varDelta u+u+V(x)v=W(x)G_{v}(z)~\hbox { in }\ {\mathbb {R}}^N,\\ -\varepsilon ^2\varDelta v+v+V(x)u=W(x)G_{u}(z)~\hbox {in } \ {\mathbb {R}}^N,\\ u(x)\rightarrow 0\ \hbox {and }v(x)\rightarrow 0\ \hbox {as } \ |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$ and $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\varDelta u+u+V(x)v=W(x)(G_{v}(z)+|z|^{2^*-2}v)~\hbox {in } \ {\mathbb {R}}^N,\\ -\varepsilon ^2\varDelta v+v+V(x)u=W(x)(G_{u}(z)+|z|^{2^*-2}u)~\hbox {in } \ {\mathbb {R}}^N,\\ u(x)\rightarrow 0\ \hbox {and }v(x)\rightarrow 0\ \hbox {as } \ |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$ where \(z=(u,v)\in {\mathbb {R}}^2\) , \(G\) is a power type nonlinearity, having superquadratic growth at both \(0\) and infinity but subcritical, \(V\) can be sign-changing and \(\inf W>0\) . We prove the existence, exponential decay, \(H^2\) -convergence and concentration phenomena of the ground state solutions for small \(\varepsilon >0\) .     

Limit theorems for von Mises statistics of a measure preserving transformation

For a measure preserving transformation \(T\) of a probability space \((X,\mathcal{F },\mu )\) and some \(d \ge 1\) we investigate almost sure and distributional convergence of random variables of the form $$\begin{aligned} x \rightarrow \frac{1}{C_n} \sum _{0\le i_1,\ldots ,\,i_d where \(C_1, C_2,\ldots \) are normalizing constants and the kernel \(f\) belongs to an appropriate subspace in some \(L_p(X^d\!,\, \mathcal{F }^{\otimes d}\!,\,\mu ^d)\) . We establish a form of the individual ergodic theorem for such sequences. Using a filtration compatible with \(T\) and the martingale approximation, we prove a central limit theorem in the non-degenerate case; for a class of canonical (totally degenerate) kernels and \(d=2\) , we also show that the convergence holds in distribution towards a quadratic form \(\sum _{m=1}^{\infty } \lambda _m\eta ^2_m\) in independent standard Gaussian variables \(\eta _1, \eta _2, \ldots \) .     

The Hardy–Rellich Inequality and Uncertainty Principle on the Sphere

Let \(\Delta _0\) be the Laplace–Beltrami operator on the unit sphere \(\mathbb {S}^{d-1}\) of \({\mathbb R}^d\) . We show that the Hardy–Rellich inequality of the form $$\begin{aligned} \mathop \int \limits _{\mathbb {S}^{d-1}} \left| f (x)\right| ^2 \mathrm{d}{\sigma }(x) \le c_d \min _{e\in \mathbb {S}^{d-1}} \mathop \int \limits _{\mathbb {S}^{d-1}} (1- {\langle }x, e {\rangle }) \left| (-\Delta _0)^{\frac{1}{2}}f(x) \right| ^2 \mathrm{d}{\sigma }(x) \end{aligned}$$ holds for \(d =2\) and \(d \ge 4\) but does not hold for \(d=3\) with any finite constant, and the optimal constant for the inequality is \(c_d = 8/(d-3)^2\) for \(d =2, 4, 5,\) and, under additional restrictions on the function space, for \(d\ge 6\) . This inequality yields an uncertainty principle of the form $$\begin{aligned} \min _{e\in \mathbb {S}^{d-1}} \mathop \int \limits _{\mathbb {S}^{d-1}} (1- {\langle }x, e {\rangle }) |f(x)|^2 \mathrm{d}{\sigma }(x) \mathop \int \limits _{\mathbb {S}^{d-1}}\left| \nabla _0 f(x)\right| ^2 \mathrm{d}{\sigma }(x) \ge c'_d \end{aligned}$$ on the sphere for functions with zero mean and unit norm, which can be used to establish another uncertainty principle without zero mean assumption, both of which appear to be new.     

On the non-existence of maximal difference matrices of deficiency 1

A \(k\times u\lambda \) matrix \(M=[d_{ij}]\) with entries from a group \(U\) of order \(u\) is called a \((u,k,\lambda )\) -difference matrix over \(U\) if the list of quotients \(d_{i\ell }{d_{j\ell }}^{-1}, 1 \le \ell \le u\lambda ,\) contains each element of \(U\) exactly \(\lambda \) times for all \(i\ne j.\) Jungnickel has shown that \(k \le u\lambda \) and it is conjectured that the equality holds only if \(U\) is a \(p\) -group for a prime \(p.\) On the other hand, Winterhof has shown that some known results on the non-existence of \((u,u\lambda ,\lambda )\) -difference matrices are extended to \((u,u\lambda -1,\lambda )\) -difference matrices. This fact suggests us that there is a close connection between these two cases. In this article we show that any \((u,u\lambda -1,\lambda )\) -difference matrix over an abelian \(p\) -group can be extended to a \((u,u\lambda ,\lambda )\) -difference matrix.     

A study of saturated tensor cone for symmetrizable Kac–Moody algebras

Let \(\mathfrak {g}\) be a symmetrizable Kac-Moody Lie algebra with the standard Cartan subalgebra \(\mathfrak {h}\) and the Weyl group \(W\) . Let \(P_+\) be the set of dominant integral weights. For \(\lambda \in P_+\) , let \(L(\lambda )\) be the integrable, highest weight (irreducible) representation of \(\mathfrak {g}\) with highest weight \(\lambda \) . For a positive integer \(s\) , define the saturated tensor semigroup as $$\begin{aligned} \Gamma _s:= \{(\lambda _1, \dots , \lambda _s,\mu )\in P_+^{s+1}: \exists \, N\ge 1 \,\text {with}\,L(N\mu )\subset L(N\lambda _1)\otimes \dots \otimes L(N\lambda _s)\}. \end{aligned}$$ The aim of this paper is to begin a systematic study of \(\Gamma _s\) in the infinite dimensional symmetrizable Kac-Moody case. In this paper, we produce a set of necessary inequalities satisfied by \(\Gamma _s\) . These inequalities are indexed by products in \(H^*(G^{\mathrm{min }}/B; \mathbb {Z})\) for \(B\) the standard Borel subgroup, where \(G^{\mathrm{min }}\) is the ‘minimal’ Kac-Moody group with Lie algebra \(\mathfrak {g}\) . The proof relies on the Kac-Moody analogue of the Borel-Weil theorem and Geometric Invariant Theory (specifically the Hilbert-Mumford index). In the case that \(\mathfrak {g}\) is affine of rank 2, we show that these inequalities are necessary and sufficient. We further prove that any integer \(d>0\) is a saturation factor for \(A^{(1)}_1\) and 4 is a saturation factor for \(A^{(2)}_2\) .     

The theme of a vanishing period

Consider a multivalued formal function of the type 1 $$\begin{aligned} \varphi (s) : = \sum _{j=0}^k\,c_j(s).s^{\lambda + m_j}.(\mathrm{Log}\,s)^j, \end{aligned}$$ where \(\lambda \) is a positive rational number, \(c_j\) is in \({{\mathrm{\mathbb {C}}}}[[s]]\) and \(m_j \in \mathbb {N}\) for \(j \in [0,k-1]\) . The theme associated with such a \(\varphi \) is the “minimal filtered integral equation” satisfied by \(\varphi \) , in a sense which is made precise in this article. We study such objects and show that their isomorphism classes may be characterized by a finite set of complex numbers, when we assume the Bernstein polynomial of \(\varphi \) to be fixed. For a given \(\lambda \) , to fix the Bernstein polynomial is equivalent to fix a finite set of integers associated with the logarithm of the monodromy in the geometric situation described below. Our aim is to construct some analytic invariants, for instance in the following situation, let \(f : X \rightarrow D\) be a proper holomorphic function defined on a complex manifold \(X\) with values in a disc \(D\) . We assume that the only critical value is \(0 \in D\) and we consider this situation as a degenerating family of compact complex manifolds to a singular compact complex space \(f^{-1}(0)\) . To a smooth \((p+1)\) -form \(\omega \) on \(X\) such that \(\mathrm{d}\omega = 0 = \mathrm{d}f \wedge \omega \) and to a vanishing \(p\) -cycle \(\gamma \) chosen in the generic fiber \(f^{-1}(s_0), s_0 \in D \setminus \{0\}\) , we associated a “vanishing period” \(F_{\gamma }(s) : = \int _{\gamma _s} \omega \big /\mathrm{d}f \) which has an asymptotic expansion at \(0\) of the form \((1)\) above, when \(\gamma \) is chosen in the spectral subspace of \(H_p(f^{-1}(s_0), {{\mathrm{\mathbb {C}}}})\) for the eigenvalue \(\mathrm{e}^{2i\pi .\lambda }\) of the monodromy of \(f\) . Here \((\gamma _s)_{s \in D^*}\) is the horizontal multivalued family of \(p\) -cycles in the fibers of \(f\) obtained from the choice of \(\gamma \) . The aim of this article was to study the module generated by such a \(\varphi \) over the algebra \(\tilde{\mathcal {A}}\) , which is the \(b\) -completion of the algebra \(\mathcal {A}\) generated by the operators \(\mathrm{a} : = \times s\) and \(\mathrm{b} : = \int _{0}^{s}\) .     

On a subsemigroup of the universal covering of Lie semigroups

Let \(G\) be a connected Lie group and \(S\) a generating Lie semigroup. An important fact is that generating Lie semigroups admit simply connected covering semigroups. Denote by \(\widetilde{S}\) the simply connected universal covering semigroup of \(S\) . In connection with the problem of identifying the semigroup \(\Gamma (S)\) of monotonic homotopy with a certain subsemigroup of the simply connected covering semigroup \(\widetilde{S}\) we consider in this paper the following subsemigroup $$\begin{aligned} \widetilde{S}_{L}=\overline{\left\langle \mathrm {Exp}(\mathbb {L} (S))\right\rangle } \subset \widetilde{S}, \end{aligned}$$ where \(\mathrm {Exp}:\mathbb {L}(S)\rightarrow S\) is the lifting to \( \widetilde{S}\) of the exponential mapping \(\exp :\mathbb {L}(S)\rightarrow S\) . We prove that \(\widetilde{S}_{L}\) is also simply connected under the assumption that the Lie semigroup \(S\) is right reversible. We further comment how this result should be related to the identification problem mentioned above.     

Mean Squared Error Minimization for Inverse Moment Problems

We consider the problem of approximating the unknown density \(u\in L^2(\Omega ,\lambda )\) of a measure \(\mu \) on \(\Omega \subset \mathbb {R}^n\) , absolutely continuous with respect to some given reference measure \(\lambda \) , only from the knowledge of finitely many moments of \(\mu \) . Given \(d\in \mathbb {N}\) and moments of order \(d\) , we provide a polynomial \(p_d\) which minimizes the mean square error \(\int (u-p)^2d\lambda \) over all polynomials \(p\) of degree at most \(d\) . If there is no additional requirement, \(p_d\) is obtained as solution of a linear system. In addition, if \(p_d\) is expressed in the basis of polynomials that are orthonormal with respect to \(\lambda \) , its vector of coefficients is just the vector of given moments and no computation is needed. Moreover \(p_d\rightarrow u\) in \(L^2(\Omega ,\lambda )\) as \(d\rightarrow \infty \) . In general nonnegativity of \(p_d\) is not guaranteed even though \(u\) is nonnegative. However, with this additional nonnegativity requirement one obtains analogous results but computing \(p_d\ge 0\) that minimizes \(\int (u-p)^2d\lambda \) now requires solving an appropriate semidefinite program. We have tested the approach on some applications arising from the reconstruction of geometrical objects and the approximation of solutions of nonlinear differential equations. In all cases our results are significantly better than those obtained with the maximum entropy technique for estimating \(u\) .     

Construction of relative difference sets and Hadamard groups

‘There exist normal \((2m,2,2m,m)\) relative difference sets and thus Hadamard groups of order \(4m\) for all \(m\) of the form $$\begin{aligned} m= x2^{a+t+u+w+\delta -\epsilon +1}6^b 9^c 10^d 22^e 26^f \prod _{i=1}^s p_i^{4a_i} \prod _{i=1}^t q_i^2 \prod _{i=1}^u \left( (r_i+1)/2)r_i^{v_i}\right) \prod _{i=1}^w s_i \end{aligned}$$ under the following conditions: \(a,b,c,d,e,f,s,t,u,w\) are nonnegative integers, \(a_1,\ldots ,a_r\) and \(v_1,\ldots ,v_u\) are positive integers, \(p_1,\ldots ,p_s\) are odd primes, \(q_1,\ldots ,q_t\) and \(r_1,\ldots ,r_u\) are prime powers with \(q_i\equiv 1\ (\mathrm{mod}\ 4)\) and \(r_i\equiv 1\ (\mathrm{mod}\ 4)\) for all \(i, s_1,\ldots ,s_w\) are integers with \(1\le s_i \le 33\) or \(s_i\in \{39,43\}\) for all \(i, x\) is a positive integer such that \(2x-1\) or \(4x-1\) is a prime power. Moreover, \(\delta =1\) if \(x>1\) and \(c+s>0, \delta =0\) otherwise, \(\epsilon =1\) if \(x=1, c+s=0\) , and \(t+u+w>0, \epsilon =0\) otherwise. We also obtain some necessary conditions for the existence of \((2m,2,2m,m)\) relative difference sets in partial semidirect products of \(\mathbb{Z }_4\) with abelian groups, and provide a table cases for which \(m\le 100\) and the existence of such relative difference sets is open.     

On semi-classical limits of ground states of a nonlinear Maxwell–Dirac system

We study the semi-classical ground states of the nonlinear Maxwell–Dirac system: $$\begin{aligned} \left\{ \begin{array}{l} \alpha \cdot \left( i\hbar \nabla + q(x)\mathbf{A }(x)\right) w-a\beta w -\omega w - q(x)\phi (x) w = P(x)g(\left| w\right| ) w\\ -\Delta \phi =q(x)\left| w\right| ^2\\ -\Delta {A_k}=q(x)(\alpha _k w)\cdot \bar{w}\ \ \ \ k=1,2,3 \end{array} \right. \end{aligned}$$ for \(x\in \mathbb{R }^3\) , where \(\mathbf{A }\) is the magnetic field, \(\phi \) is the electron field and \(q\) describes the changing pointwise charge distribution. We develop a variational method to establish the existence of least energy solutions for \(\hbar \) small. We also describe the concentration behavior of the solutions as \(\hbar \rightarrow 0\) .     

Nonlinear Schrödinger equations near an infinite well potential

The paper deals with standing wave solutions of the dimensionless nonlinear Schrödinger equation where the potential \(V_\lambda :\mathbb {R}^N\rightarrow \mathbb {R}\) is close to an infinite well potential \(V_\infty :\mathbb {R}^N\rightarrow \mathbb {R}\) , i. e. \(V_\infty =\infty \) on an exterior domain \(\mathbb {R}^N\setminus \Omega \) , \(V_\infty |_\Omega \in L^\infty (\Omega )\) , and \(V_\lambda \rightarrow V_\infty \) as \(\lambda \rightarrow \infty \) in a sense to be made precise. The nonlinearity may be of Gross–Pitaevskii type. A standing wave solution of \((NLS_\lambda )\) with \(\lambda =\infty \) vanishes on \(\mathbb {R}^N\setminus \Omega \) and satisfies Dirichlet boundary conditions, hence it solves We investigate when a standing wave solution \(\Phi _\infty \) of the infinite well potential \((NLS_\infty )\) gives rise to nearby solutions \(\Phi _\lambda \) of the finite well potential \((NLS_\lambda )\) with \(\lambda \gg 1\) large. Considering \((NLS_\infty )\) as a singular limit of \((NLS_\lambda )\) we prove a kind of singular continuation type results.     

Existence and uniqueness of nontrivial solutions for eigenvalue boundary value problem of nonlinear fractional differential equation

In this paper, we study the existence and uniqueness of a nontrivial solution to eigenvalue problems for the following nonlinear fractional differential equation of the form $$\begin{aligned} \left\{ \begin{array}{l} -D^{\alpha }_{0^{+}}u(t)=\lambda [f(t, u(t), D^{\beta }_{0^{+}}u(t))+g(t)],~~ 0 where \(\lambda \) is a parameter, \(D^{\alpha }_{0^{+}},D^{\beta }_{0^{+}}\) are two standard Riemann–Liouville fractional derivatives, \(0<\beta <1<\alpha \le 2,\alpha -\beta >1,f: [0,1]\times {\mathbb{R }}\times {\mathbb{R }}\rightarrow {\mathbb{R }}\) is continuous, and \(g(t): (0, 1)\rightarrow [0, +\infty )\) is Lebesgue integrable. We obtain several sufficient conditions of the existence and uniqueness of nontrivial solution of the above eigenvalue problems when \(\lambda \) is in some interval. Our approach is based on the Leray–Schauder nonlinear alternative. In addition, some examples are included to demonstrate the main result.     

Uncertainty principles and sum complexes

Let \(p\) be a prime and let \(A\) be a nonempty subset of the cyclic group \(C_p\) . For a field \({\mathbb F}\) and an element \(f\) in the group algebra \({\mathbb F}[C_p]\) let \(T_f\) be the endomorphism of \({\mathbb F}[C_p]\) given by \(T_f(g)=fg\) . The uncertainty number \(u_{{\mathbb F}}(A)\) is the minimal rank of \(T_f\) over all nonzero \(f \in {\mathbb F}[C_p]\) such that \(\mathrm{supp}(f) \subset A\) . The following topological characterization of uncertainty numbers is established. For \(1 \le k \le p\) define the sum complex \(X_{A,k}\) as the \((k-1)\) -dimensional complex on the vertex set \(C_p\) with a full \((k-2)\) -skeleton whose \((k-1)\) -faces are all \(\sigma \subset C_p\) such that \(|\sigma |=k\) and \(\prod _{x \in \sigma }x \in A\) . It is shown that if \({\mathbb F}\) is algebraically closed then $$\begin{aligned} u_{{\mathbb F}}(A)=p-\max \{k :\tilde{H}_{k-1}(X_{A,k};{\mathbb F}) \ne 0\}. \end{aligned}$$ The main ingredient in the proof is the determination of the homology groups of \(X_{A,k}\) with field coefficients. In particular it is shown that if \(|A| \le k\) then \(\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.\)      

How Far Can Chemotactic Cross-diffusion Enforce Exceeding Carrying Capacities?

We consider nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis-growth system $$\begin{aligned} \left\{ \begin{array}{l} u_t=\varepsilon u_{xx} -(uv_x)_x +ru -\mu u^2, \qquad x\in \Omega , \ t>0, \\ 0=v_{xx}-v+u, \qquad x\in \Omega , \ t>0, \end{array} \right. \quad (\star ) \end{aligned}$$ in \(\Omega :=(0,L)\subset \mathbb {R}\) with \(L>0, \varepsilon >0, r\ge 0\) and \(\mu >0\) , along with the corresponding limit problem formally obtained upon taking \(\varepsilon \searrow 0\) . For the latter hyperbolic–elliptic problem, we establish results on local existence and uniqueness within an appropriate generalized solution concept. In this context we shall moreover derive an extensibility criterion involving the norm of \(u(\cdot ,t)\) in \(L^\infty (\Omega )\) . This will enable us to conclude that in this case \(\varepsilon =0\) ,

• if \(\mu \ge 1\) , then all solutions emanating from sufficiently regular initial data are global in time, whereas
• if \(\mu <1\) , then some solutions blow-up in finite time.

The latter will reveal that the original parabolic–elliptic problem ( \(\star \) ), though known to possess no such exploding solutions, exhibits the following property of dynamical structure generation: given any \(\mu \in (0,1)\) , one can find smooth bounded initial data with the property that for each prescribed number \(M>0\) the solution of ( \(\star \) ) will attain values above \(M\) at some time, provided that \(\varepsilon \) is sufficiently small. In particular, this means that the associated carrying capacity given by \(\frac{r}{\mu }\) can be exceeded during evolution to an arbitrary extent. We finally present some numerical simulations that illustrate this type of solution behavior and that, moreover, inter alia, indicate that achieving large population densities is a transient dynamical phenomenon occurring on intermediate time scales only.     

Classical limit of the tensor product of holomorphic discrete series representations

For three coadjoint orbits \(\mathcal {O}_1, \mathcal {O}_2\) and \(\mathcal {O}_3\) in \(\mathfrak {g}^*\) , the Corwin–Greenleaf function \(n(\mathcal {O}_1 \times \mathcal {O}_2, \mathcal {O}_3)\) is given by the number of \(G\) -orbits in \(\{(\lambda , \mu ) \in \mathcal {O}_1 \times \mathcal {O}_2 \, : \, \lambda + \mu \in \mathcal {O}_3 \}\) under the diagonal action. In the case where \(G\) is a simple Lie group of Hermitian type, we give an explicit formula of \(n(\mathcal {O}_1 \times \mathcal {O}_2, \mathcal {O}_3)\) for coadjoint orbits \(\mathcal {O}_1\) and \(\mathcal {O}_2\) that meet \(\left( [\mathfrak {k}, \mathfrak {k}] + \mathfrak {p}\right) ^{\perp }\) , and show that the formula is regarded as the ‘classical limit’ of a special case of Kobayashi’s multiplicity-free theorem (Progr. Math. 2007) in the branching law to symmetric pairs.     

Systems of variational inequalities in the context of optimal switching problems and operators of Kolmogorov type

In this paper we study the system $$\begin{aligned}&\min \biggl \{-\mathcal H u_i(x,t)-\psi _i(x,t),u_i(x,t)-\max _{j\ne i}(-c_{i,j}(x,t)+u_j(x,t))\biggr \}=0,\\&u_i(x,T)=g_i(x),\ i\in \{1,\ldots ,d\}, \end{aligned}$$ where \((x,t)\in \mathbb R ^{N}\times [0,T]\) . A special case of this type of system of variational inequalities with terminal data occurs in the context of optimal switching problems. We establish a general comparison principle for viscosity sub- and supersolutions to the system under mild regularity, growth, and structural assumptions on the data, i.e., on the operator \(\mathcal H \) and on continuous functions \(\psi _i\) , \(c_{i,j}\) , and \(g_i\) . A key aspect is that we make no sign assumption on the switching costs \(\{c_{i,j}\}\) and that \(c_{i,j}\) is allowed to depend on \(x\) as well as \(t\) . Using the comparison principle, the existence of a unique viscosity solution \((u_1,\ldots ,u_d)\) to the system is constructed as the limit of an increasing sequence of solutions to associated obstacle problems. Having settled the existence and uniqueness, we subsequently focus on regularity of \((u_1,\ldots ,u_d)\) beyond continuity. In this context, in particular, we assume that \(\mathcal H \) belongs to a class of second-order differential operators of Kolmogorov type of the form: $$\begin{aligned} \mathcal H =\sum _{i,j=1}^m a_{i,j}(x,t)\partial _{x_i x_j}+\sum _{i=1}^m a_i(x,t)\partial _{x_i} +\sum _{i,j=1}^N b_{i,j}x_i\partial _{x_j}+\partial _t, \end{aligned}$$ where \(1\le m\le N\) . The matrix \(\{a_{i,j}(x,t)\}_{i,j=1,\ldots ,m}\) is assumed to be symmetric and uniformly positive definite in \(\mathbb R ^m\) . In particular, uniform ellipticity is only assumed in the first \(m\) coordinate directions, and hence, \(\mathcal H \) may be degenerate.     

Distributions and the Global Operator of Slice Monogenic Functions

In this paper we consider functions \(f\) defined on an open set \(U\) of the Euclidean space \(\mathbb{R }^{n+1}\) and with values in the Clifford Algebra \(\mathbb{R }_n\) . Slice monogenic functions \(f: U \subseteq \mathbb{R }^{n+1} \rightarrow \mathbb{R }_n\) belong to the kernel of the global differential operator with non constant coefficients given by \( \mathcal{G }=|{\underline{x}}|^2\frac{\partial }{\partial x_0} \ + \ {\underline{x}} \ \sum _{j=1}^n x_j\frac{\partial }{\partial x_j}. \) Since the operator \(\mathcal{G }\) is not elliptic and there is a degeneracy in \( {\underline{x}}=0\) , its kernel contains also less smooth functions that have to be interpreted as distributions. We study the distributional solutions of the differential equation \(\mathcal{G }F(x_0,{\underline{x}})=G(x_0,{\underline{x}})\) and some of its variations. In particular, we focus our attention on the solutions of the differential equation \( ({\underline{x}}\frac{\partial }{\partial x_0} \ - E)F(x_0,{\underline{x}})=G(x_0,{\underline{x}}), \) where \(E= \sum _{j=1}^n x_j\frac{\partial }{\partial x_j}\) is the Euler operator, from which we deduce properties of the solutions of the equation \( \mathcal{G }F(x_0,{\underline{x}})=G(x_0,{\underline{x}})\) .     

On regularity of weak solutions to the instationary Navier–Stokes system: a review on recent results

Consider a weak instationary solution \(u\) of the Navier–Stokes equations in a domain \(\varOmega \subsetneq \mathbb {R}^3\) with Dirichlet boundary data \(u=0\) on \(\partial \Omega \) , i.e., \(u\) solves the Navier–Stokes system in the sense of distributions and $$\begin{aligned} u \in L^\infty \left( 0,T;L^2(\varOmega )\right) \cap L^2 \left( 0,T;W^{1,2}_0(\varOmega )\right) . \end{aligned}$$ Since the pioneering work of J. Leray 1933/34 it is an open problem whether weak solutions are unique and smooth. The main step—to nowadays knowledge—is to show that the given weak solution is a strong one in the sense of J. Serrin, i.e., \(u \in L^s \left( 0,T;L^q(\varOmega )\right) \) where \(s>2, q>3\) and \(\frac{2}{s}+ \frac{3}{q}=1\) . This review reports on recent progress in this important problem, considering this issue locally on an initial interval \([0,T')\) , \(T'<T\) , i.e., the problem of optimal initial values \(u(0)\) , globally on \([0,T)\) , and from a one-sided point of view \(u \in L^s \left( T'-\varepsilon ,T';L^q(\varOmega )\right) \) or \(u \in L^s\left( T',T'+\varepsilon ;L^q(\varOmega )\right) \) . Further topics deal with the energy (in-)equality, uniqueness of weak solutions, blow-up phenomena and the analysis in critical spaces for the whole space case.     

On Sharp Aperture-Weighted Estimates for Square Functions

Let \(S_{\alpha ,\psi }(f)\) be the square function defined by means of the cone in \({\mathbb R}^{n+1}_{+}\) of aperture \(\alpha \) , and a standard kernel \(\psi \) . Let \([w]_{A_p}\) denote the \(A_p\) characteristic of the weight \(w\) . We show that for any \(1<p<\infty \) and \(\alpha \ge 1\) , $$\begin{aligned} \Vert S_{\alpha ,\psi }\Vert _{L^p(w)}\lesssim \alpha ^n[w]_{A_p}^{\max \left( \frac{1}{2},\frac{1}{p-1}\right) }. \end{aligned}$$ For each fixed \(\alpha \) the dependence on \([w]_{A_p}\) is sharp. Also, on all class \(A_p\) the result is sharp in \(\alpha \) . Previously this estimate was proved in the case \(\alpha =1\) using the intrinsic square function. However, that approach does not allow to get the above estimate with sharp dependence on \(\alpha \) . Hence we give a different proof suitable for all \(\alpha \ge 1\) and avoiding the notion of the intrinsic square function.     

Conical stochastic maximal L^p-regularity for 1\leqslant p<\infty

Let \(A = -\mathrm{div} \,a(\cdot ) \nabla \) be a second order divergence form elliptic operator on \({\mathbb R}^n\) with bounded measurable real-valued coefficients and let \(W\) be a cylindrical Brownian motion in a Hilbert space \(H\) . Our main result implies that the stochastic convolution process $$\begin{aligned} u(t) = \int _0^t e^{-(t-s)A}g(s)\,dW(s), \quad t\geqslant 0, \end{aligned}$$ satisfies, for all \(1\leqslant p<\infty \) , a conical maximal \(L^p\) -regularity estimate $$\begin{aligned} {\mathbb E}\Vert \nabla u \Vert _{ T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n)}^p \leqslant C_p^p {\mathbb E}\Vert g \Vert _{ T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n;H)}^p. \end{aligned}$$ Here, \(T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n)\) and \(T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n;H)\) are the parabolic tent spaces of real-valued and \(H\) -valued functions, respectively. This contrasts with Krylov’s maximal \(L^p\) -regularity estimate $$\begin{aligned} {\mathbb E}\Vert \nabla u \Vert _{L^p({\mathbb R}_+;L^2({\mathbb R}^n;{\mathbb R}^n))}^p \leqslant C^p {\mathbb E}\Vert g \Vert _{L^p({\mathbb R}_+;L^2({\mathbb R}^n;H))}^p \end{aligned}$$ which is known to hold only for \(2\leqslant p<\infty \) , even when \(A = -\Delta \) and \(H = {\mathbb R}\) . The proof is based on an \(L^2\) -estimate and extrapolation arguments which use the fact that \(A\) satisfies suitable off-diagonal bounds. Our results are applied to obtain conical stochastic maximal \(L^p\) -regularity for a class of nonlinear SPDEs with rough initial data.